Transcript Anderson Localization (1957)

Anderson localization: from theoretical
aspects to applications
Antonio M. García-García
[email protected]
http://phy-ag3.princeton.edu
Princeton and ICTP
Analytical approach to the
3d Anderson transition
Theoretical aspects
Existence of a band of
metallic states in 1d
Applications
Collaborators:
Localization in Quantum
Chromodynamics
Emilio Cuevas, Wang Jiao, James Osborn
Problem: Get analytical expressions for
different quantities characterizing the
metal-insulator transition in d  3 such as
, level statistics.
Locator expansions
One parameter scaling theory
Selfconsistent condition
Quasiclassical approach to
the Anderson transition
Anderson localization
50’
70’
Perturbative locator expansion
Self consistent
conditions
Anderson
Abou Chakra, Anderson, Thouless,
Vollhardt, Woelfle
1d
70’
80’
Kotani, Pastur, Sinai, Jitomirskaya, Mott.
Thouless, Wegner, Gang of four, Frolich,
Scaling
Spencer, Molchanov, Aizenman
Dynamical localization
Fishman, Grempel, Prange, Casati
Weak Localization
Field theory
90’
00’
Cayley tree and rbm
Computers
Experiments
Lee
Efetov, Wegner
Efetov, Fyodorov,Mirlin, Klein,
Zirnbauer,Kravtsov
Aoki, Schreiber, Kramer, Shapiro
Aspect, Fallani, Segev
4202 citations!
What if I place a particle in a random potential and wait?
Tight binding model
Vij nearest neighbors, I random potential
Technique: Looking for inestabilities
in a locator expansion
Not rigorous! Small
denominators
Correctly predicts a metal-insulator transition in 3d
and localization in 1d
Interactions?
Disbelief?,
against the spirit
of band theory
But my recollection is that, on the whole,
the attitude was one of humoring me.
Perturbation theory around the
insulator limit (locator expansion).
No control on the approximation.
It should be a good approx for d>>2.
It predicts correctly localization in 1d and a
transition in 3d
The distribution of the self energy Si (E) is
sensitive to localization.
metal
insulator
  Im Si ( E  ih )
>0
=0
metal
insulator
~h
Scaling theory of localization
Energy Scales
  1
Phys. Rev. Lett. 42, 673
(1979), Gang of four.
Based on
Thouless,Wegner,
scaling ideas
1. Mean level spacing:
2. Thouless energy:
ET  h / tT
tT(L) is the travel time to cross a box of size L
Dimensionless
Thouless conductance
Diffusive motion
without localization corrections
ET
g

ET  DL2   Ld g  Ld 2
ET  
g  1
Metal
ET  
g  1
Insulator
Scaling theory of
localization
 (g )
The change in the
conductance with the
system size only depends
on the conductance itself
g
Weak localization
d log g
  (g)
d ln L
g  1 g  Ld 2  ( g )  (d  2)   / g
g  1 g  eL / 
 ( g )  log g  0
Predictions of the scaling
theory at the transition
1. Diffusion becomes anomalous
r (t )  t
2
2/ d
Imry, Slevin
2. Diffusion coefficient become size and
momentum dependent
D(q)  q
d 2
2 d
D(L)  L
Chalker
3. g=gc is scale invariant therefore level statistics
are scale invariant as well
Weak localization
Positive correction to
the resistivity of a metal
at low T
1.Cooperons (Langer-Neal, maximally crossed, responsible for weak localization)
and Diffusons (no localization, semiclassical) can be combined.
3. Accurate in d ~2.
Self consistent condition (Wolfle-Volhardt)
No control on the approximation!
Predictions of the self consistent
theory at the transition
1. Critical exponents:
Vollhardt, Wolfle,1982
| (r) | e
r / 

 | E  Ec |
  d 12
  1/ 2
d 4
d 4
2. Transition for d>2
Disagreement with
numerical simulations!!
3. Correct for d ~ 2
Why?
Why do self consistent
methods fail for d = 3?
1. Always perturbative around the metallic
(Vollhardt & Wolfle) or the insulator state
(Anderson, Abou Chacra, Thouless) .
A new basis for localization is needed
2. Anomalous diffusion at the transition
(predicted by the scaling theory) is not
taken into account.
D( L)  L2  d
D( q )  q d  2
Proposal:
Analytical results combining the scaling theory
and the self consistent condition.
 and level statistics.
Idea Solve the self consistent equation assuming that the diffusion
coefficient is renormalized as predicted by the scaling theory
Assumptions:
1. All the quantum corrections missing in the self
consistent treatment are included by just
renormalizing the coefficient of diffusion
following the scaling theory.
2. Right at the transition the quantum dynamics is well
described by a process of anomalous diffusion with
no further localization corrections.
r 2 (t )  t 2 / d
Technical details:
Critical exponents
2
The critical exponent ν, can be obtained by solving the

above equation for  | E  Ec | with D (ω) = 0.

1
1


2
d 2
Level Statistics:
Starting point:
Anomalous diffusion predicted
by the scaling theory
Semiclassically, only “diffusons”
Two levels
correlation function
s
E  E'

gc  cd / d  2
Shapiro, Abrahams
Aizenman, Warzel
Cayley tree
A linear number
variance in the
3d case was
obtained by
Altshuler et
al.’88
d 
d  D2

2d
 1
Chalker
Kravtsov,Lerner
Comparison with
numerical results
 
 3T  1.5
 4T  1
 5T  0.83
 6T  0.75
| (r) | e
r / 

 | E  Ec |
1
1

2
d 2
 3N
 4N
 5N
 6N
 1.52  0.06
 1.03  0.07
 0.84  0.06
 0.78  0.06
 3T  0.33
 4T  0.5
 5T  0.6
 6T  0.66
 3 N  0.27
 4 N  0.48
 5 N  0.7
 6 N  0.77
1. Critical exponents: Excellent
2, Level statistics: OK? (problem with gc)
3. Critical disorder: Not better than before
Problem: Conditions for the
absence of localization in 1d
Motivation
Quasiperiodic potentials
Nonquasiperiodic potentials
Work in progress in collaboration with E Cuevas
Your intuition about localization
Ea
Random
Eb
V(x)
0
Ec
X
For any of the energies above:
Will the classical motion be strongly affected by quantum
effects?
tt
t
t
Speckle
potentials
The effective 1d random potential
is correlated
Localization/Delocalization in 1d:
Exponential
localization for every
energy and disorder
Random uncorrelated
potential
Periodic potential
Bloch theorem. Absence
of localization. Band
theory
In between?
Quasiperiodic potentials
V ( x)  cos(x)
Jitomirskaya,
Sinai,Harper,Aubry
 1
 1
 1
Metal
Insulator
Critical
Similar results
V ( x)   ak cos(2kx)
k
ak  B exp( Ak)
Jitomirskaya, Bourgain
Conjecture
V (n)  C 3/ 2
What it is the least smooth
potential that can lead to a
band of metallic states?
No metallic band if V(x) is
discontinuous Jitomirskaya, Aubry,
Damanik
V (n)   ak cos(2k (n   ))  C 
k
 0
| a k |
A
k 1 
Fourier space:
Long range hopping
Localization
for >0
Delocalization
in real space
Levitov
Fyodorov
Mirlin
A metallic band can exist for
V (n)  C

 0
Non quasiperiodic potentials
Physics literature
1. Izrailev & Krokhin
 (k )  S (k )
Born approximation
2. Lyra & Moura
S (k )  1 / k 
 2
S (k )   dx exp( ikx ) V ( x)V (0)
Metallic
band if:
S (k )  0
k   kc , kc 
Neither of them is accurate
1a. A vanishing Lyapunov exponent does
not mean metallic behavior.
1b. Higher order corrections make the
Lyapunov exponent > 0
2. Not generic
Decaying and sparse potentials (Kunz,Simon, Soudrillard): transition but non ergodic
Localization in correlated potentials: Luck, Shomerus, Efetov, Mirlin,Titov
Mathematical literature:
Kotani, Simon, Kirsch, Minami,
Damanik.
Kotani’s theory of ergodic operators
Non deterministic
potentials
No a.c. spectrum
Deterministic potentials
More difficult to tell
Discontinuous potentials
Damanik,Stolz,
Sims
No a.c. spectrum
B( x)  V ( x)V (0)  1/ x
No a.c. spectrum
A band of metallic states might
exist provided
B( x)  lim V ( x)V (0)  0
x 
V ( x)  C

 > 0 and V(x) and its  derivative are bounded.
Neighboring values of the potential must be
correlated enough in order to avoid destructive
interference.
According to the scaling theory in the metallic
region motion must be ballistic.
How to
proceed?
Smoothing uncorrelated
random potentials
Finite size scaling
analysis
P(s)   s  i 1  i /  
Spectral correlations are scale
invariant at the transition
Thouless, Shklovski, Shapiro 93’
i
var  s 2  s
2
s n   s n P( s)ds
AGG, Cuevas
Diffusive Metal
Clean metal
Insulator
var(s)  varWD  0.286
var(s)  0
var(s)  varP  1
Savitzsky-Golay
1. Take np values of V(n)
around a given V(n0)
2. Replace V(n0) by the
best fit of the np values to
a polynomial of M degree
3. Repeat for all n0
B( x)  lim V ( x)V (0)  0
x 
Resulting potential
is not continuous
A band of metallic
states does not exist
Fourier filtering
1. Fourier transform of
the uncorrelated noise.
2. Remove k > kcut
3. Fourier transform
back to real space
Resulting potential
is analytic
B( x)  lim V ( x)V (0)  0
x 
A band of metallic
states do exist
Gruntwald Letnikov operator
i  1/ N1/ 2 ,1/ N1/ 2 
B( x)  lim V ( x)V (0)  0
x 
Resulting potential
is C-+1/2
A band of metallic
states exists provided

V (n)  C ,  1/ 2
Is this
generic?
Localization in systems with chiral
symmetry and applications to QCD
1. Chiral phase transition in lattice QCD as a metal-insulator
transition, Phys.Rev. D75 (2007) 034503, AMG, J. Osborn
2. Chiral phase transition and Anderson localization in the
Instanton Liquid Model for QCD , Nucl.Phys. A770 (2006) 141-161, AMG. J.
Osborn
3. Anderson transition in 3d systems with chiral symmetry,
Phys. Rev. B 74, 113101 (2006), AMG, E. Cuevas
4. Long range disorder and Anderson transition in systems
with chiral symmetry , AMG, K. Takahashi, Nucl.Phys. B700 (2004) 361
5. Chiral Random Matrix Model for Critical Statistics, Nucl.Phys.
B586 (2000) 668-685, AMG and J. Verbaarschot
QCD : The Theory of the strong interactions
High Energy g << 1 Perturbative
1. Asymptotic freedom
Quark+gluons, Well understood
Low Energy
g ~ 1 Lattice simulations
The world around us
2. Chiral symmetry breaking
 ~ (240MeV )
3
Massive constituent quark
3. Confinement
Colorless hadrons
V (r )  a / r  r
How to extract analytical information? Instantons , Monopoles, Vortices
Deconfinement and chiral restoration
Deconfinement: Confining potential vanishes:
Chiral Restoration: Matter becomes light:
How to explain these transitions?
1. Effective, simple, model of QCD close to the phase
transition (Wilczek,Pisarski,Yaffe): Universality.
2. Classical QCD solutions (t'Hooft): Instantons (chiral),
Monopoles and vortices (confinement).
We propose that quantum
interference/tunneling plays an important role.
QCD at T=0, instantons and chiral symmetry breaking
tHooft, Polyakov, Callan, Gross, Shuryak, Diakonov, Petrov,VanBaal
Instantons: Non perturbative solutions of the classical Yang Mills equation.
Tunneling between classical vacua.

ins 
μ
D =  μ + gA 
Dψ0 r   0 ψ0 r  1/ r
1. Dirac operator has a zero mode in the field of an instanton
2. Spectral properties of the smallest eigenvalues of the Dirac operator are
controled by instantons
3. Spectral properties related to chiSB. Banks-Casher relation:
 (m)
1
 ( )
1
   Tr ( D  m)   d
  lim 
mm0
V
m  i
V
3
Instanton liquid
models T = 0
Non linear equations
Solution
Multiinstanton
vacuum?
No superposition
Variational principles(Dyakonov),
Instanton liquid model (Shuryak).
3
0


(


)
TIA   d 4 x I ( x  zI )iD  A ( x  z A ) ~ i(u  Rˆ ) I 3A iD  
R
T
AI
ILM T > 0
TIA ~ exp( R / l (T ))
T 

0
IA
QCD vacuum as a conductor (T =0)
Metal: An electron initially bounded to a single atom gets
delocalized due to the overlapping with nearest neighbors
QCD Vacuum: Zero modes initially bounded to an instanton
get delocalized due to the overlapping with the rest of zero
modes. (Diakonov and Petrov)
Differences
Dis.Sys: Exponential decay
QCD vacuum: Power law decay
QCD vacuum as a disordered conductor
Diakonov, Petrov, Verbaarschot, Osborn, Shuryak, Zahed,Janik
Instanton positions and color orientations vary
Ion
T=0
Instantons
TIA~ 1/R,  = 3<4
QCD vacuum is a conductor
Shuryak,Verbaarschot, AGG and Osborn
Electron
Quarks
T>0 TIA~ e-R/l(T)
A transition is possible
QCD Dirac
operator
D =  + gA
QCD
μ
μ
Phys.Rev. D75 (2007) 034503
Nucl.Phys. A770 (2006) 141
with J. Osborn

 D
QCD
 n  in  n
At the same Tc that the
Chiral Phase transition



n
  0
undergo a metal - insulator transition
n
A metal-insulator transition in the Dirac operator
induces the QCD chiral phase transition
Signatures of a metal-insulator transition
1. Scale invariance of the spectral correlations.
A finite size scaling analysis is then carried
out to determine the transition point.
Skolovski, Shapiro, Altshuler
2.

P( s ) ~ s
P(s) ~ e  As
s  1
s  1
var
3. Eigenstates are multifractals.
  (r ) d r ~ L
2q
n
d
 Dq ( q 1)
var  s  s
2
2
s n   s n P( s)ds
Mobility edge
Anderson transition
Metal
insulator
transition
ILM, close to the origin, 2+1
flavors, N = 200
Spectrum is
scale
invariant
ILM with 2+1 massless flavors,
We have observed a metal-insulator transition at T ~ 125 Mev
Localization versus
chiral transition
Instanton liquid model Nf=2, masless
Chiral and localizzation transition occurs at the same temperature
Problem: To determine the
importance of Anderson localization
effects in deterministic (quantum
chaos) systems
Scaling theory in quantum chaos
Metal insulator transition in quantum chaos
What is quantum
chaos?
Quantum chaos studies the quantum
properties of systems whose
classical motion is chaotic (or not)
Bohigas-Giannoni-Schmit conjecture
Classical chaos
Wigner-Dyson
Energy is the only integral of motion
Momentum is not a good quantum number
Delocalization
Gutzwiller-Berry-Tabor conjecture
Integrable
classical
motion
Poisson
statistics
(Insulator)
Integrability
P(s)
Canonical momenta
are conserved
s
System is localized in momentum
space
Dynamical
localization
Fishman, Prange, Casati
Exceptions to the BGS conjecture
1. Kicked systems
H  p  V ( ) (t  nT)
2
Classical
<p2>
Quantum
n
V ( )  K cos
Dynamical localization
in momentum space
t
2. Harper model
3. Arithmetic billiards
Random
Characterization
d>2
Weak disorder
Wigner-Dyson
g
Delocalization
Normal diffusion
d = 1,2
d>2
Strong disorder
g 0
g  gc
Chaotic
motion
Always?
Poisson
Localization
Integrable
motion
Diffusion stops
Bogomolny
Critical statistics
d>2
Critical disorder
Deterministic
Altshuler, Levitov
Casati, Shepelansky
Multifractality
Anomalous
diffusion
??????????
Adapt the one parameter scaling theory in
quantum chaos in order to:
Determine the class of systems in which
Wigner-Dyson statistics applies.
Does this analysis coincide with the BGS
conjecture?
Scaling theory and anomalous diffusion
Compute g
q2  t
g ( L) 
ET

  L d / d
 clas
 clas
L
de fractal dimension
of the spectrum.
e
d 2
 
de 
Universality
L
Wigner-Dyson
(g) > 0
weak localization?
Poisson
(g) < 0
 ( g )   clas  f ( g )
Two routes to the Anderson transition
1. Semiclassical origin
2. Induced by quantum effects
 clas  0
 clas  0
 (g)  0
 quan  clas
 quan  0
Wigner-Dyson statistics in non-random systems
1. Estimate the typical time needed to reach the
“boundary” (in real or momentum space) of the
system.
In billiards: ballistic travel time.
In kicked rotors: time needed to explore a fixed basis.
2. Use the Heisenberg relation to estimate
thedimensionless conductance g(L) .
Wigner-Dyson statistics applies if
g ( L) 
ET

 clas
L
 clas
d 2
  0
de 
q2  t
and
 quan  0
Anderson transition in quantum chaos
Conditions:
1. Classical phase space must be homogeneous.
2. Quantum power-law localization.
3.
g ( L) 
ET

 clas
L
d 2
 clas    0
de 
k
2
t

Examples:
1D =1, de=1/2, Harper model, interval exchange maps (Bogomolny)
=2, de=1, Kicked rotor with classical singularities (AGG, WangJiao)
2D =1, de=1, Coulomb billiard (Altshuler, Levitov).
3D =2/3, de=1, 3D Kicked rotor at critical coupling.
3D kicked rotator
V (1,2 ,3 )  k cos(1 ) cos(2 ) cos(3 )
Finite size scaling analysis shows
there is a transition at kc ~ 2.3
At k = kc ~ 2.3
diffusion is anomalous
p 2 (t )
quan
~ t 2/3
1D kicked rotor with singularities
H  p  V ( ) (t  nT )
2
n
Classical
Motion
V ( )  K cos
Normal diffusion
V ( )   log |  |
Anomalous Diffusion
P( k , t )  1 / k 
Quantum Evolution
Quantum
anomalous
diffusion
P(k, t )  1/ k
|k | t
1
 1
V ( )   |  |

2
2

T


T

Uˆ  exp(
) exp(iV ( ) / ) exp(
)
2
2
4 
4 
'
| k | t
'
No dynamical
localization for <0
g ( L) 
1.
2.
3.
ET

>0
<0
=0
 clas
L
 clas  
|k| t
1
 1
Localization
Poisson
Delocalization Wigner-Dyson
MIT
Critical statistics
Anderson transition for
log and step singularities
AGG, Wang
Jjiao, PRL
2005
Results are stable under perturbations and
sensitive to the removal of the singularity
Possible to test experimentally
Analytical approach: From the kicked rotor to the 1D
Anderson model with long-range hopping

1 2
i ( , t )  
( , t )  V ( ) (t  n)( , t )
2
t
2 
n
Fishman,Grempel, Prange
Tmum  Wr um r  Eum
Tm pseudo
random
1d Anderson
model
r 0
V ( )  K cos
Wr   r ,1   r , 1

V ( )   |  |
Wr 
Always
localization
1
r
 1
Insulator for  0
Explicit analytical results are possible, Fyodorov and Mirlin
Conclusions:
1. Anderson localization depends on the degree of
differentiability of the potential.
2. Critical exponents and level statistics are
acessible to analytical techniques
3. The adaptation of the scaling theory to quantum
chaos provides a powerful tool to predict
localization effects in non random systems
4. Anderson localization plays a role in the
chiral phase transition of QCD
Thanks!
[email protected]
http://phy-ag3.princeton.edu
NEXT
1. Find a way to compute analytically the critical
disorder and others quantities that characterize
the Anderson transition.
2. Adapt localization theories to the peculiarities
of cold atoms.
3. Mathematicians: Prove delocalization
[email protected]
http://phy-ag3.princeton.edu