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 ,  (1n f ) (1n f )  n f n f ]  0
i
i ,
Let P0 be projection on Hilbert space with :
i ,
i ,
i ,
 (1ni,f) (1ni,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


 εk 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
h0
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
Ak,, T0
Ak,, 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 ,
Ht
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
  dH ( )
 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.
(2x,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
pp
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)
(0,2y)
(p,p y)
(2x,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