Das quantenmechanische Vielteilchen
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Transcript Das quantenmechanische Vielteilchen
Numerical approaches to the correlated electron problem:
Quantum Monte Carlo.
F.F. Assaad.
The Monte Carlo method. Basic.
Spin Systems. World-lines, loops and stochastic series expansions.
The auxiliary field method I
The auxiliary filed method II
Special topics
Magnetic impurities Kondo lattices.
Metal-Insulator transition
21.10.2002
MPI-Stuttgart.
Universität-Stuttgart.
One magnetic impurity.
Cu with Fe as impurity.
Fe: 3d 64s 2 Hunds rule: S=2
Temperature.
Temperature
Resistivity minimum.
(Normal: a + bT 2)
1
T
( 0)
T Free spin.
T
Screened spin.
Kondo problem: crosover from free to screened impurity spin.
Many body non-perturbative problem.
The Kondo problem is a many-body problem
H (k ) ck ,s ck ,s
k ,s
Electrons.
Spin-flip
k
p
J
+ 2N
scattering of p
k,p,s,s' S σs,s ' ck ,s cp,s '
f
I
P´
k
P´
k
k can
spin-flip
scatter
Impurity spin.
Spin of p is
conserved
The scattering of electron k
will depend on how electron
p scattered.
Thus, the impurity spin is a
source of correlations
between conduction electrons.
k cannot
spin-flip
scatter
T>>J: Essentially free impurity spin.
1
T
I
Ground state at J/t >>1
c
H t i
i , c j,
, j ,
+
c
J SI
i , j I
1
2
f
S I
c
t
I , cI δ , H.c.
δ ,
Ground state:
Spin singlet
.
is relevant fixpoint
Wilson (1975)
J/t =
Numerical
(Hirsch-Fye impurity algorithm):
Dynamical f-spin structure factor
0.06
0.12
J/t = 1.2 TK/t
J/t = 1.6 TK/t
J/t = 2.0 TK/t
T <TK
S f ( )
0.21
T >TK
T/TK
T
K
e
t / J
/t
is the only low energy scale
I
d
0
S S 0
f
f
I
I
S f ( )
1
Z n,m
En
e
2
| n | S I | m | ( E n E m)
f
Lattices of magnetic impurities.
Conduction orbitals:
Impurity orbitals:
i ,
c
f i,
Periodic Anderson model (PAM). Charge fluctuations on f-sites.
f
f
1
1)
)(
f
f
(
n
U
c
c
n
H t ci, cj, V
f
i ,
i
,
i
,
2
2
i
,
i ,
i ,
i ,
i
( i, j),
Kondo lattice model (KLM). Charge fluctuations on f-sites frozen.
H t ci, cj, J i Si S
( i, j ),
c
f
i
J V 2 / U f
Simulations of the Kondo lattice.
2
Consider:
H (k ) c ck ,
k ,
k ,
c f f c
i, i, i, i,
J
4 i
We can simulate this Hamiltonian for all band fillings. No constraint on Hilbert space.
How does H relate to the Kondo lattice?
H (k ) c ck , J
k ,
k ,
Sic Sif J
i ,
i
(ci, ci, f i, f i, H.c.)
J
i ,
Conservation law:
nic ni f nic nif
[H , (1n f ) (1n f ) n f n f ] 0
i
i ,
Let P0 be projection on Hilbert space with :
i ,
i ,
i ,
(1ni,f) (1ni,f) ni,f ni,f 0
i
Then:
Chose
P0 H H KLM
P0 | T | T
so that
e
H |
HKLM |
T e
T
Mean-field for Kondo lattice.
Two energy scales:
Decoupling:
c
S S
i
f
i
f ,z
1
Sic, z Si
4
1
4
f
i ,
ci , ci ,
f
ci,
f
f c
i ,
i ,
J/t = 4, <nc>=0.5
2
i ,
2
i ,
f
i ,
S
,z
i
c c f
i ,
f
i ,
i ,
mi f
f
i ,
C
c, z
c
mi ,
i
S
c c f
i ,
i ,
i ,
v
Order parameters:
r
T/t
Mean field Hamiltonian (paramagnetic)
H
MF
Jr
εk c c 2 f c c f
k ,σ
k ,σ
k ,σ
i ,σ
f f
i
σ
i ,σ
i, σ
i ,σ
i ,σ
i ,σ
JN r 2
1 4
i ,σ
Below Tcoh
Fermi liquid
Tcoh
Saddle point.
H MF
H MF
0
r
Tcoh
Exact for the
SU(N) model
at N
TK
Below TK r>0.
Same as for
single impurity.
TK e t / J
(S. Burdin, A. Georges, D.R. Grempel et al. PRL 01)
Ground state (Mean-field).
J / t 2 , n c 0.5
Zk
Finite Temperature:
Crossover to HF state.
(π, π) (0, π) (0,0) (π, π)
(Ce
Luttinger
volume: nc+1
(π, π)
1
m
Z
(0, π)
D n
J/t=2
(0,0)
k
(π, π)
La
1-x
) Pb 3
X=0.6
m
X=1
J/t = 4
/t
Fermi liquid with large mass or
small coherence temperature.
Mean-Field Problems:
x
c
f 0
Coherence.
, magnetism, finite T.
Single impurity like
Periodic table of elements
(FFA. PRB 02)
I. Coherence.
Technical constraint: Conduction band has to be half-filled. Otherwise sign problem.
Strong coupling.
Model.
Conduction band:Half-filled.
Brillouin zone.
Fermi line
J /t
c
H t (i
i , c j , J SR SR
, j ),
R
c
Note:
f
Conduction band is half-filled and particle hole symmetry is present.
Allows sign-free QMC simulations but leads to nesting.
At T=0 magnetic insulator.
Optical conductivity and resistivity. J/t = 1.6
Single impurity like
T/t
Schlabitz et al. 86.
L=8
L=6
T/t = 1/20
T/t = 1/30
T/t = 1/15
Temperature
Degiorgi et al. 97
T/t = 1/60
T/t = 1/2
T/t = 1/5
T/t = 1/10
/t
Optical conductivity
Resistivity
Coherence.
50
100
cm
(L=8: 320 orbitals.)
-1
150
200
Thermodynamics: J/t = 1.6
h0
c n
E
CV T
Resistivity
s M
h
T
S
T*
s (Q) ~ s
s
T
S
T/t
T/t
T*
Scales as a function of J/t.
c
L=6
L=8
T/t
Tmin/2
T*
Ts
J/t
CV
T/t
J/t = 0.8
1/8
1/10
T
Resistivity
1/15
1/20
1/30
T/t
s (Q) ~ s
S
s
T/t
1/50
1/80
/t
Scales as a function of J/t.
c
L=6
T/t
Tmin/2
T*
Ts
J/t
Specific heat.
L=4
T/t
Comparison T* with TK of single impurity probem.
T/t
Depleted Kondo lattice.
T*
TK
TK
T*
Ts
Crossover to the coherent heavy
fermion state is set by the single
impurity Kondo temperature.
Note: CexLa1-xCu6
T* ~ 5-12 K for x: 0.73-1.
TK ~ 3K
(Sumiyawa et al. JPSJ
86)
J/t
Tcoh ?
No magnetic order-disorder transition since strong coupling metallic state
is unstable towards magnetic ordering.
II Magnetism : Order-disorder transitions
RKKY Interaction
Energy scale
CePd2Si2
(J.D Mathur et al. Nature 98.)
J eff ( q ) ~ J 2 ( q , 0 )
Spin susceptibility of
conduction electrons.
Kondo
Effect.
RKKY
Energy scale
TK ~ e-t/J
J
[ See also CeCu6-xAux]
Competition RKKY / Kondo
leads to quantum phase transitions.
QMC , T=0, L
Half-filled Kondo lattice.
Model
One conduction
electron per
impurity spin.
(FFA PRL. 99)
Strong coupling limit. J/t >> 1
3) Magnetism.
Spin
Singlet
1) Spin gap
(m
Ds
Energy
J
2) Quasiparticle gap.
)2=
4
iQ R
f
f
e
S
0 SR
N 3N R
lim
m > 0, Q=(p,p): long range
antiferromagnetic order.
Dqp
1D
Energy
3J/4
f
t / J
Ds ~ e
Dq p ~ J / 4
(Tsunetsugu et. al. RMP 97)
Spin Dynamics: S(q, )
Fit: Spin waves.
(S. Capponi, FFA PRB 01)
Fit: Perturbation in
t/J.
Excitations of disordered phase condense to form the order of the
ordered state. Bond mean-field of Kondo necklace (G.M. Zhang et. al. PRB 00).
Single particle spectral function. A(k,
(S. Capponi, FFA PRB 01)
Fit: Strong coupling.
Weak coupling ?
a)
Magnetic BZ.
f-Spins are frozen.
ms
TK
π
Jc
b)
Partial Kondo screening,
remnant magnetic moment
orders.
0
π k
Magnetic BZ.
TK
ms
Jc
π
(M. Feldbacher, C. Jureka, F.F.A., W. Brenig PRB submitted.)
0
π k
(S. Capponi, FFA PRB 01)
Single particle spectral function. A(k,
Mean-field interpretation:
Coexistence of Kondo
screening and magnetism.
Fit: Strong coupling.
(Zhang and Yu PRB 00)
J<0
No
Kondo
effect.
In ordered phase impurity spins are partially screened. Remnant moment
orders.
Origin of quasiparticle gap at weak couplings.
J/t = 0.8
Ak,, T0
Ak,, T/t=1/12
p,p
0,0
0,p
s (Q)
p,p
L=4
L=6
L = 10
0,0
T/t
0,p
TS ~ J2
p,p
/t
Quasiparticle gap of
order J is of magnetic
origin at J < Jc ~ 1.5 t
p,p
/t
Conclusions.
QMC algorithm for Kondo lattices.
Restriction. Particle-hole symmetric conduction bands.
Depleted lattices.
T*
TK
Half-filled Kondo lattice in 2D.
Pairing. No.
II.Doped Mott insulators.
MPI-Stuttgart.
Universität-Stuttgart.
Hubbard Model.
c i, c j ,
Ht
i , j ,
t
U ( n i , 1 2 ) ( n i , 1 2 )
i
U
Half filling: Insulator. Scale U
Metal
Charge is localized. Internal degree of
freedom (spin) is still active.
Strong coupling U/t >>1 (Half filling)
U
t
t
Magnetic scale:
S
H J
i Sj
i, j
J ~t
2
Heisenberg Model.
U
The Mott Insulator. Half filling (2D,T=0)
Charge.
Spin.
1
1
N
Quasiparticle gap > 0
F.F. Assaad M. Imada JPSJ 95.
N
Long range magnetic order.
Goldstone mode: Spin-waves.
The Metal-Insulator Transition.
Cuprates. (2D)
Doping
[ (La Nd) 2-x Sr x Cu O 4 ]
Superconductivity-Stripes.
Metal
Mott Insulator
Bandwidth W.
k-(BEDT-TTF)2
V2 O
3
CU[N(CN)2]C (2D)
(3D)
F.F. Assaad, M. Imada und D.J. Scalapino
Phys. Rev. Lett. 77 , 4592, (1996)
U/W
Titanates (3D)
(La xSr 1-xTi O 3 )
F.F. Assaad und M. Imada
Phys. Rev. Lett. 76 , 3176, (1996).
Phys. Rev. Lett. 74 , 3868, (1995) .
How can we avoid the sign problem?
2
ci λ ci
H N t ci c j U N
i
i , j
ci ( ci,1 ci, N )
λ ,
Orbital Picture.
δ , if N/2
δ , if N/2
N=2:
HN=2 = Hubbard
N =
Mean-field
N > 2 Symmetry: SU(N/2)
Elementary
Z.
Cell
N
2
0
4
1
6
2
8
3
N
H
SU(N/2)
N = 4 n. No sign problem
irrespective of lattice topology
and doping.
:
2U
N
N/2-1
U
U N
n n
m
z
m
z
z
z
:
2U
N
m m
z
n , n , )
(m
More Formal.
2
U
t
HN
c i c j N ci λ ci
i , j
Tr e
HN
i
N S ( )
D e
Langevin: (t t ) (t )
N = : SDW Mean field.
with
ci
δ , if N/2
, λ ,
δ , if N/2
( ci,1 ci, N )
U
S ( )
4
H ( ) t
0 d
2
( )
i
i
1
dH ( )
ln TrT e 0
2
ci, c j, U i i ( )ni, ni,
i , j ,
S ( )
t 2 t / N (t ) so that
(t )
S ( )
0 so that i
ni, ni,
P( , t ) e NS ( )
Test.
T=0, 4 X 4, U/t = 4, 2 Löcher.
Lanczos.
D(p,p)(N)/D(p,p)(N=2)
E(N)/E(N=2)
Mean-field.
1/N
Note:
<n>=1, U/t >> 1
MF
Tc
F.F. Assaad et al.
PRL submitted.
N 4
U aber T c
2
t /U
Single particle: N=4, T=0, U/t=3. 2D.
=0,30X8
t/U=0
N()
N()
N()
( )/t
( )/t
( )/t
H U U ( n i , 1 2 ) ( n i , 1 2 )
i
=0, = 0:
=1/6, = -U/2:
L
L
L=6
L-1
N()
N()
=1/5,30X8, 30x12
=1/14,30X8
-U/2
0
U/2
L-1
2
0
U
L=6
Spin, S(q), charge, N(q), Structure Factors.
N=4, T=0, U/t =3, 30X8
N=4, T=0, U/t =3, 30X8
= 1/14
= 1/5
(0,p)
=
=
=
=
= 1/7
= 1/4
(p,0)
(0,0)
Charge.
(2x,0)
(0,0)
(p,p)
(p/2,p)
Spin.
(p x,p)
One dimensional
N=4, T=0, U/t =3, 60X1, =1/5
S(q)
p p 2kf
p Phase-shift
0
in Spin Structure.
N(q)
=1/5
= 1/4
(p,p/2)
x =p
(p,p)
Real space (caricature).
=0
0
1/14
1/7
1/5
Disctance between walls: 1/
p/2
p
2 p 4kf
0
p/2
p
Spin-and charge-Dynamics at =0.2 (T=0,N=4,U/t=3)
60X1
30X4
30X8
30X12
First charge-excitation at
q=(qx,0)
First Spin-excitation at
q=(qx,p)
qx
2p
pp
qx
Ly >4: Particle-hole continuum.
Transport
Optical conductivity:
' (q, ) N (q,) / q2
xx
x
' [q (2p / Lx), ]
xx
30 X 8, =0.2
Ohne Vertex
Mit Vertex
N(q,): Dynamical charge
Structure factor.
/t
N(q)
S(q)
Two dimensions
Ly=10, Lx=30, = 0.2
qx
qy
Spin.
Charge.
qx
(p,p)
(p x,p)
(0,2y)
(p,p y)
(2x,0)
(0,0)
Two-dimensional metallic with no quasiparticles.
Elementary excitations: spin and charge collective modes.
qy
Interpretation of collective modes.
1) Analogy to 1D ?
2) Goldstone Modes.
a) SU(2) SU(2) Symmetry is not broken.
b)
Phasons.
n(r ) D c cos( Qc r c) und S (r ) D s cos( Qs r s)
Q c 2 Q s G,
M Q s G
0 : M 2 und s 0.
0.2 : M 10.
HF
E 0 ( c, s)
2
2
HF
E 0 ( c, s) A cos ( 2 c s) B cos ( c 2 s)
U / t 3: A 0.00001t , B 0.8 t
Energy is invariant under Translation:
s
0 : M und A 0.
c
2 c s 2 c s D