Transcript Aleiner Lecture 5 - Lancaster University
Slide 1
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 2
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 3
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 4
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 5
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 6
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 7
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 8
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 9
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 10
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 11
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 12
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 13
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 14
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 15
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 16
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 17
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 18
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 19
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 20
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 21
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 22
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 23
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 24
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 25
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 26
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 27
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 28
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 29
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 30
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 31
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 32
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 33
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 34
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 35
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 36
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 37
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 38
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 39
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 40
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 41
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 2
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 3
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 4
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 5
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 6
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 7
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 8
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 9
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 10
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 11
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 12
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 13
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 14
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 15
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 16
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 17
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 18
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 19
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 20
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 21
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 22
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 23
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 24
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 25
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 26
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 27
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 28
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 29
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 30
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 31
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 32
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 33
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 34
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 35
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 36
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 37
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 38
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 39
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 40
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41
Slide 41
Localization and Critical Diffusion of
Quantum Dipoles in Two Dimensions
I.L. Aleiner (Columbia U, NYC, USA)
B.L. Altshuler (Columbia U, NYC, USA)
K.B. Efetov (Ruhr-Universitaet,Bochum, Germany)
Phys. Rev. Lett. 107, 076401 (2011)
Windsor Summer School
August 25, 2012
Outline:
1) Introduction:
a) “dirty” – Localization in two dimensions
b) “clean” – Dipole excitations in clean system
2) Qualitative discussion and results for localization of dipoles:
Fixed points accessible by perturbative renormalization group.
3) Modified non-linear s-model for localization
4) Conclusions
2
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
Exact solution for one channel:
M.E. Gertsenshtein, V.B. Vasil’ev, (1959)
“Conjecture” for one channel:
Sir N.F. Mott and W.D. Twose (1961)
Exact solution for s(w) for one channel:
V.L. Berezinskii, (1973)
Scaling argument for multi-channel :
D.J. Thouless, (1977)
localized
Exact solutions for multi-channel:
K.B.Efetov, A.I. Larkin (1983)
O.N. Dorokhov (1983)
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
Anderson (1958);
Proof of the stability of the insulator
localized
1. Localization of single-electron wave-functions:
extended
d=1; All states are localized
d=3; Anderson transition
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
localized
Instability of metal with respect to quantum
(weak localization) corrections:
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979)
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
d=2; All states are localized
If no spin-orbit interaction
Thouless scaling + ansatz:
Density of
state per unit
area
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless energy
Dimensionless conductance
/
Level spacing
Conductivity
Diffusion
coefficient
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
First numerical evidence:
A Maccinnon, B. Kramer, (1981)
Locator expansion
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections:
No magnetic field (GOE)
L.P. Gorkov, A.I.Larkin, D.E. Khmelnitskii, (1979); Wegner (1979)
d=2; All states are localized
If no spin-orbit interaction
E. Abrahams, P. W. Anderson, D. C. Licciardello, and
T.V. Ramakrishnan, (1979)
Thouless scaling + ansatz:
1
ansatz
Instability of metal with respect to quantum
(weak localization) corrections: Wegner (1979)
In magnetic field (GUE)
2. Quantum dipoles in clean 2-dimensional systems
Simplest example:
Square lattice:
z
Each site can be in four excited states, a
+
+
-
-
-
x
Short-range part
# of dipoles is
not conserved
+
+
Single dipole spectrum
+
+
-
-
+
+
-
+ -
+ +
-
+
+
-
+ -
+ -
-
+
-
-
-
+
-
Degeneracy protected by the lattice symmetry
-
+
-
+
+
Single dipole spectrum
Alone does nothing
Qualitatively change
E-branch
Degeneracy protected by the lattice symmetry
Single dipole long-range hops
Second order coupling:
+
-
Fourier transform:
+
-
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Degeneracy protected by the lattice symmetry lifted by long-range hops
Single dipole spectrum
Similar to the transverse-longitudinal
splitting in exciton or phonon polaritons
Goal:
To build the scaling theory of localization including long-range hops
Dipole two band model and disorder
disorder
… and disorder and magnetic field
disorder
Approach from metallic side
Only important new parameter:
Scaling results
Used to be for A=0
No magnetic field (GOE)
1
ansatz
Scaling results
Critical diffusion
(scale invariant)
A>0
No magnetic field (GOE)
1
Instability of insulator,
L.S.Levitov, PRL, 64, 547 (1990)
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
Used to be for A=0
In magnetic field (GUE)
1
ansatz
Scaling results
A>0
“Metal-Instulator”
transition (scale
invariant)
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Summary of RG flow:
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
23
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
(2)
(1)
Qualitative consideration
1) Long hops (Levy flights)
Consider two wave-packets
R
Rate:
(1)
Does not depend on the
shape of the wave-function
Levy flights
(2)
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
Destructive interference
2) Weak localization (first loop) due to the short-range hops
[old story: Gorkov, Larkin, Khmelnitskii (1979); Wegner (1979)]
Constructive interference
No magnetic field (GOE)
0
in magnetic field (GUE)
3) Weak localization (second loop) short hops;
In magnetic
field; Wegner (1979)
0 no magnetic field (GOE)
4) New interference term: Second loop: short hops and
Levy flight interference:
No magnetic field (GOE)
Scaling results
A>0
No magnetic field (GOE)
1
ansatz
Stable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Scaling results
A>0
In magnetic field (GUE)
1
ansatz
Unstable critical fixed point
Accessible by perturbative RG
for
is not renormalized
Standard non-linear s-model for localization
See textbook by K.B. Efetov,
Supersymmetry in disorder and chaos, 1997
Any correlation function
- supersymmetry
Standard non-linear s-model for localization
Free energy functional (form fixed by symmetries) (GOE):
Only running constant (one parameter scaling)
Beyond standard non-linear s-model for
localization (long range hops)
Any correlation
function
- supersymmetry
Beyond standard non-linear s-model for
localization (long range hops)
Orthogonal ensemble:
universal conductance
(independent of disorder)
Unitary ensemble:
metal-insulator transition
37
Conclusions.
1. Dipoles move easier than particles due to long-range hops.
2. Non-linear sigma-model acquires a new term contributing
to RG.
3. RG analysis demonstrates criticality for
any disorder for the orthogonal ensemble and
existence of a metal-insulator transition for
the unitary one.
38
Renormalization group in two dimensions.
Integration over fast modes:
Q0
fast,
~
V
~
~
Q V Q0 V
slow
Expansion in
V
and integration over
New non-linear s -model with renormalized
Gell-Mann-Low equations:
Q0
~
D
~
and
w
~
w w
A consequence of the supersymmetry
Physical meaning: the density of states is constant.
39
Renormalization group (RG) equations.
(t )
dt
d
a - 1, 0 ,1
at 2
1
2
(
t 1-a
3
2
)
t D
-1
t
t0
1 a t 0 ln( w 0 / w )
For the orthogonal, unitary and symplectic ensembles
Orthogonal: localization
Unitary: localization but with a much
larger localization length
Symplectic: “antilocalization”
Unfortunately, no exact solution for 2D has been obtained.
Reason: non-compactness of the symmetry group of Q.
40
The explicit structure of Q
Q UQ 0 U
u
U
0
0
v
u,v contain all Grassmann
variables
All essential structure is in
^
cos
Q0
^
- i
sin
- ie
sin
^
- cos
^
ie
i
^
Q0
0
^
0
i
0
0
(unitary ensemble)
Mixture of both compact and non-compact symmetries
rotations: rotations on a sphere and hyperboloid glued by the
anticommuting variables.
41