f.f. assaad
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
F.F. Assaad.
MPI-Stuttgart. Universität-Stuttgart.
21.10.2002
Numerical approaches to the correlated electron problem:
Quantum Monte Carlo.
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
One magnetic impurity. Cu with Fe as impurity.
Fe: 3d 64s 2 Hunds rule: S=2
Invers
e su
scepti
bili
ty.
T
1T
T
Free spin.
Screened spin.
Temperature.Temperature
Resi
stiv
ity.
Resistivity minimum. (Normal: a + bT 2)
)0(
Kondo problem: crosover from free to screened impurity spin. Many body non-perturbative problem.
The Kondo problem is a many-body problem
Impurity spin.
Electrons.
p
k P´ k Spin-flip
scattering of p
P´ k
Spin of p isconserved
The scattering of electron kwill depend on how electronp scattered.
Thus, the impurity spin is a source of correlations between conduction electrons.
+H cc ss
s ,,
,)( k
kk
k SIf cc ssssssN
J',,',',,2 pkk,p σ
k can spin-flip scatter
k cannot spin-flip scatter
Ground state at J/t >>1
+ S IcJH cct
,
,,,
, j
Ijiji
i
S I
f
2
1
.H.c,,
, cct
δI
δI
)(||1
)( ||2
,EEmn
ZS mn
f
I
En
mnf Se
)(S f
t/
T <TK
T >TK
Dynamical f-spin structure factor
Ground state:
Spin singlet
J/t = is relevant fixpoint. Wilson (1975)
00
SSf
I
f
IId
Numerical (Hirsch-Fye impurity algorithm):
T/TK
J/t = 1.2J/t = 1.6J/t = 2.0
T
TK/t 0.21
TK/t 0.06TK/t 0.12
is the only low energy scaleeTJt
K
/
T>>J: Essentially free impurity spin. TI1
Lattices of magnetic impurities.
SS iii
jiji,
fcJcctH
,,
),(
))(( 21
,21
,,,,,,
,,),(
nnUcffcVcctH ff
f iii
iiiii
jiji,
Periodic Anderson model (PAM).
Kondo lattice model (KLM).
Charge fluctuations on f-sites.
Charge fluctuations on f-sites frozen.
UVJ f/2
Conduction orbitals:
Impurity orbitals:
c,i
f
,i
Simulations of the Kondo lattice.
Consider:
cffccckH iiiiJ
ik
kk ,,,,4
2
,,
,)(
We can simulate this Hamiltonian for all band fillings. No constraint on Hilbert space.
How does H relate to the Kondo lattice?
SS fi
ci
ik
kk JcckH
,,
,)( )( H.c.,,,,,
ffccJ iiiii
nnnnJ fi
ci
fcii
i
,
Conservation law: 0)1()1( ][,,,,
, nnnnH ffff
iiiii
TTP ||0
T
HT
H ee KLM ||Chose so that
0)1()1( ,,,, nnnn ffff
iiiii
Let P0 be projection on Hilbert space with :
Then: HHP KLM0
Mean-field for Kondo lattice.
Order parameters:
r
SS
,,,,,,,,
,,,
fccffccf
mm fzczc f
iiiiiiii
iiii
2
4
1
2
4
1,,
,,,,
,,,,SS
fccf
fccfzzcc ff
iiii
iiiiiiii SS
Decoupling:Two energy scales:
J/t = 4, <nc>=0.5
Cv
T/tTK
Below TK r>0.Same as forsingle impurity.
Tcoh
Below Tcoh
Fermi liquid
Tcoh TK e Jt /
(S. Burdin, A. Georges, D.R. Grempel et al. PRL 01)
4r2
2r
1
ε
,,
,,,,,
,,,
JN
J
ff
fccfccHMF
i σ σiσi
σiσiσiσiσi
σkσkσkk
Mean field Hamiltonian (paramagnetic)
0r
HH MFMF
Saddle point.Exact for theSU(N) modelat N
(Ce x La 1-x ) Pb 3
X=1
X=0.6
Crossover to HF state.
Finite Temperature:
Coherence. Single impurity like
),( ππ ),( π0 ),( 00 ),( ππ
E(k
)5.0,2/ ntJ c
k
Ground state (Mean-field).
Luttinger volume: nc+1
Zk
),( ππ ),( π0 ),( 00 ),( ππ
Fermi liquid with large mass or small coherence temperature.
0 fcMean-Field Problems: , magnetism, finite T.
J/t=2 J/t = 4
t
mnDZm 1
Periodic table of elements
I. Coherence. (FFA. PRB 02)
Note: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.
Strong coupling.
Fermi line tJ /
Brillouin zone.
Technical constraint: Conduction band has to be half-filled. Otherwise sign problem.
SSf
R
c
RR
jji
i JcctH
,
),,(,
Model.
Conduction band:Half-filled.
T/t = 1/20T/t = 1/30T/t = 1/15
T/t = 1/60
T/t = 1/2T/t = 1/5
T/t = 1/10
t
T/tO
pti
cal co
nd
uct
ivit
y
R
esi
stiv
ity
Optical conductivity and resistivity. J/t = 1.6
Single impurity likeCoherence.
cm -1
Temperature
Resi
stiv
ity
Schlabitz et al. 86.
Degiorgi et al. 97
L=8L=6
(L=8: 320 orbitals.)
Resi
stiv
ity
T/t
Thermodynamics: J/t = 1.6
L=6
L=8
T/t
T/t
T S
T S
ss
~)(Q
T*
T*
Scales as a function of J/t.
J/t
T/t T*
Ts
Tmin/2
T
EV
nch
Ms C
h
0
s
c
CV
T/t
T/t
Speci
fic
heat.
c
s
L=6
L=4
T S ss
~)(Q
J/t = 0.8
1/81/10
1/151/20
1/30
1/50
1/80
T/t
Resistivity
t
Scales as a function of J/t.
J/t
T/t T*
Ts
Tmin/2
Comparison T* with TK of single impurity probem.
Ts
J/t
Depleted Kondo lattice.
TK
T*
T* TK
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)
T/t
Tcoh ?
No magnetic order-disorder transition since strong coupling metallic state is unstable towards magnetic ordering.
CePd2Si2 (J.D Mathur et al. Nature 98.)
[ See also CeCu6-xAux]
RKKY
J
II Magnetism : Order-disorder transitions
RKKY Interaction
Kondo Effect.
TK ~ e-t/JEnergy scale
)0,(2~)(eff qJqJEnergy scale
Spin susceptibility of conduction electrons.
Competition RKKY / Kondoleads to quantum phase transitions.
Half-filled Kondo lattice.
One conduction electron per impurity spin.(FFA PRL. 99)
ModelModel
Strong coupling limit. J/t >> 1
Spin Singlet
1) Spin gap
EnergyJ
s
2) Quasiparticle gap.
Energy3J/4
qp
QMC , T=0, L
SS R0R
QR ffe
Ni
N
34
lim
m > 0, Q=(,): long range antiferromagnetic order.
3) Magnetism.
( m f )2=
1D eJt
s
/~
4/~ Jpq
(Tsunetsugu et. al. RMP 97)
Spin Dynamics: S(q, )
Fit: Perturbation in t/J.
Fit: Spin waves.
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).
(S. Capponi, FFA PRB 01)
Single particle spectral function. A(k,
Fit: Strong coupling.
Weak coupling ?
(S. Capponi, FFA PRB 01)
Jc
TK
ms
f-Spins are frozen.a)
ππ 0 k
E( k
)E
( k)
ππ 0 k
Magnetic BZ.
Magnetic BZ.
Jc
TKms
b) Partial Kondo screening,remnant magnetic momentorders.
(M. Feldbacher, C. Jureka, F.F.A., W. Brenig PRB submitted.)
J < 0No Kondoeffect.
Single particle spectral function. A(k,
Fit: Strong coupling.
In ordered phase impurity spins are partially screened. Remnant moment orders.
Mean-field interpretation:Coexistence of Kondo screening and magnetism.(Zhang and Yu PRB 00)
(S. Capponi, FFA PRB 01)
/t
k,
/t
k,/t=
t
)(Qs
L = 4L = 6L = 10
TS ~ J2
Origin of quasiparticle gap at weak couplings.
J/t = 0.8
Quasiparticle gap oforder J is of magnetic origin at J < Jc ~ 1.5 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.
Metal Half filling: Insulator. Scale U
Charge is localized. Internal degree offreedom (spin) is still active.
t
Hubbard Model.
cctH jiji
,,,,
),(),( 2
12
1 nnU iii
U
UStrong coupling U/t >>1 (Half filling)
tU
tJ2
~t Magnetic scale:
SS jiji
JH ,
Heisenberg Model.
The Mott Insulator. Half filling (2D,T=0)
Charge.
Quasiparticle gap > 0
F.F. Assaad M. Imada JPSJ 95.
N1
Spin.
Long range magnetic order.Goldstone mode: Spin-waves.
N1
Mott Insulator U/W
Bandwidth W.
-(BEDT-TTF)2 CU[N(CN)2]C (2D)
V 2 O 3 (3D)
F.F. Assaad, M. Imada und D.J. ScalapinoPhys. Rev. Lett. 77 , 4592, (1996)
The Metal-Insulator Transition.
Metal
Doping Cuprates. (2D)[ (La Nd) 2-x Sr x Cu O 4 ]Superconductivity-Stripes.
Titanates (3D)(La xSr 1-xTi O 3 )
F.F. Assaad und M. ImadaPhys. Rev. Lett. 76 , 3176, (1996).Phys. Rev. Lett. 74 , 3868, (1995) .
iN
U
ji
jiN iitH cλccc2
,
)( ,1, cc Niii c
λ ,
,
,
if N/2
if N/2
δ
δ
How can we avoid the sign problem?
N = 4 n. No sign problem irrespective of lattice topologyand doping.
N=2: HN=2 = Hubbard
N = Mean-field
N > 2 Symmetry: SU(N/2) SU(N/2)
2:
U
Nn n
, , )( zm n n
2:
z zU
Nm m
Orbital Picture.
ElementaryCell
N Z.
2
4
6
8
0
2
3
1
N N/2-1
U N
z zU m mH
iN
U
ji
jiN iitH cλccc2
,
,)( ,1, cc Niii c λ ,
N/2if
N/2if
,
,
δ
δ
N = : SDW Mean field. nn iii
S
,,thatso0)(
)(/2)(
)()()( tNtt
t
Sttt
etP SN )(),( so that
eDeSNHN )(
Tr
erTdU
S Hdii
0
)(2
0Tln
2
1)(
4)(
i iii
jiji nncc UtH ,,
,,,,
)()(
with
Langevin:
More Formal.
1/N
E(N)/E(N=2)
D()(N)/D()(N=2)
T=0, 4 X 4, U/t = 4, 2 Löcher.
Lanczos.
Mean-field.
F.F. Assaad et al.PRL submitted.
Test.
Note: <n>=1, U/t >> 1 24/aberMF N
c c tT TU U
Single particle: N=4, T=0, U/t=3. 2D.
()/t
N(
)
=0,30X8
t/U=0 1 12 2)) (( ,, iU ii
H U nn
N(
)
=0, = 0:
U/2-U/2
L=6LL
0
N(
)
=1/6, = -U/2:
U
L=6L-1L-1
2
0
()/t ()/tN
()
N(
)
=1/14,30X8 =1/5,30X8, 30x12
Spin, S(q), charge, N(q), Structure Factors.
Real space (caricature).
Disctance between walls: 1/
=1/5
=0
Phase-shift in Spin Structure.
One dimensional
/2
/2
S(q
)N
(q)
N=4, T=0, U/t =3, 60X1, =1/5
4kf
2kf
N=4, T=0, U/t =3, 30X8
(/2,) (,) (,/2)
S(q)= 0= 1/14= 1/7= 1/5
= 1/4(,)()(,)
N=4, T=0, U/t =3, 30X8N(q)= 1/14 = 1/7= 1/5 = 1/4
()
(x,) Spin.
(0,0)
Charge. (2x,0) x =
Spin-and charge-Dynamics at =0.2 (T=0,N=4,U/t=3)
qx qx
60X130X430X830X12
First charge-excitation at q=(qx,0)
First Spin-excitation at q=(qx,)
Optical conductivity:
30 X 8, =0.2
Ohne VertexMit Vertex
/t
]),/2([' xxx Lq
qqNq xxx2' /),(),(
N(q,): Dynamical charge Structure factor.
Transport
Ly >4: Particle-hole continuum.
S(q
)
N(q
)
()(x,)
(y)
(0,0)
(2x,0)
(0,2y)
Charge.
Spin.
Two dimensionsLy=10, Lx=30, = 0.2
Two-dimensional metallic with no quasiparticles.Elementary excitations: spin and charge collective modes.
qy qy
qxqx
Interpretation of collective modes.
1) Analogy to 1D ?
2) Goldstone Modes. a) SU(2) SU(2) Symmetry is not broken.
0 : 2 und 0.M s
Energy is invariant under Translation:
/ 3 : 0.00001 , 0.8U t A t B t
2 2 c cs s
0.2 : 10.M
s
c
0 ( , )HFE c s 2 20 ( , ) ( 2 ) ( 2 )cos cosHF A BE s sc sc c
0 : und 0.M A
( ) cos( ) und ( ) cos( )n r r S r rQ Q c sc ssc
2 ,GQ Q c s M GQ s
b) Phasons.