Transcript DMFT(QMC) - 東京大学物性研究所

Methods for electronic structure calculations
with dynamical mean field theory:
An overview and recent developments
Ryotaro ARITA (RIKEN)
Thanks to …
S. Sakai (Dept. Applied Phys. Univ. Tokyo)
H. Aoki (Dept. Phys. Univ. Tokyo)
K. Held (Max Planck Inst. Stuttgart)
A. V. Lukoyanov (Ural State Technical Univ.)
V. I. Anisimov (Inst. Metal Phys, Ekaterinburg)
Outline



Introduction
 LDA+DMFT
 Various solvers for DMFT
 IPT, NCA, ED, NRG, DDMRG, QMC, …
Conventional QMC (Hirsch-Fye 86)
 Algorithm
 Problems
 numerically expensive for low T:
numerical effort ~ 1/T3
 sign problem in multi-orbital systems:
difficult to treat spin flip terms
New QMC algorithms
 Projective QMC for T→0 calculations
(Feldbacher et al 04, Application: Arita et al 07)
 Application of various perturbation series expansions for Z
(Sakai et al 06, Rubtsov et al 05, Werner et al 07)
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LDA+DMFT
Dr Aryasetiawan July 25, Prof. Savrasov July 27
Anisimov et al 97,
Lichtenstein, Katsnelson 98
DFT/LDA
Model Hamiltonians
material specific, ab initio
fails for strong correlations
systematic many-body approach
input parameters unknown
Computational scheme for correlated
electron materials
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LDA+DMFT
Application to various correlated materials
(reviews) Held et al 03, Kotliar et al 06, etc
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Transition metal oxides
 LaTiO3
 V2O3, VO2
 (Sr,Ca)VO3
 LiV2O4
 (Sr,Ca)2RuO4
 NaxCoO2
 Cuprates
 Manganites
 …
Transition metals
 Fe, Ni
Heussler alloys
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Organic compounds
 BEDT-TTF
 TMTSF
Fullerenes
Nanostructure materials
 Zeolites
f-electron systems
 Rare earths: Ce
 Actinides: Pu
 …
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LDA+DMFT

Downfolding: LDA → effective low-energy Hamiltonian
Expand Ψ+ w.r.t. a localized basis Φilm :

Supplementing LDA with local Coulomb interactions
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LDA+DMFT
 Solve model by DMFT
Metzner & Vollhardt 89, Georges & Kotliar 92
Lattice model:
DOS D ( ) Self Energy lat (k , in )
Hlat  t  ci†, c j ,  U  nini     ni  ni 
i, j 
i
i
Effective impurity model:
Hybridization F
Self Energy imp (in )
S    d d 'c ( ) F (  ')c† ( ')   d ( (n  n )  Unn )

Self-consistency:
F  imp (in )
D ( )
1
Glatt (in )   d 
 F 1  Glatt
 imp
in      imp (in )
F
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Solvers for the DMFT impurity model

Iterated perturbation theory
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Non-crossing approximation
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Perturbation expansion in V
Exact diagonalization for small number of host sites
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Perturbation expansion in U
Max # of orbitals <2
Numerical renormalization group
(logarithmic discretization of host spectrum)
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Max # of orbitals <2

Dynamical density matrix renormalization group
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Quantum Monte Carlo

…
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Auxiliary-field QMC
 Suzuki-Trotter decomposition
Z  Tr e
 H
L
=Tr  e H0 e Hint
(  1/ T  L )
l1
 Hubbard-Stratonovich transformation for Hint
e
 U [ n n  12 ( n  n )]
 


 12  e
 s ( n  n )
(cosh( )  exp[ U/2])
s 1
 Many-particle system
=

(free one-particle system + auxiliary field)

Z

Z s s ...s 
Zs1s2 ...sL
1 2
s1s2 ...sL
 A 

s1s2 ...sL
Z s1s2 ...sL
Z
L
L
1
2L
 A s1s2 ...sL
 Tr[exp( H


0
) exp( sl n )]
l 1
Monte Carlo sampling
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QMC for the Anderson impurity model
( Hirsch-Fye 86 )
Integrate out the conduction bands
0<1,2<1/T, =L
Calculate
G0(1,2)
G{s}(1,2), w{s}
G{s}(1,2), w{s}
…
Updating: numerical effort ~L2
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Problems & Recent developments

Numerically expensive for low T: numerical effort ~ 1/T3

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Projective QMC (Feldbacher et al 04): A new route to T→0
Sign problem in multi-orbital systems: difficult to treat spin flip terms
norm

Zs can be negative:
Norm can be small
→ <A>=0/0
Application of various perturbation series expansions (Rombouts et al, 99):
less severe sign problem
 Combination with HF algorithm (Sakai et al, 06)
 Continuous time QMC
 weak coupling expansion (Rubtsov et al, 05)
 hybridization expansion (Werner et al, 06)
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Projective QMC and its application to
DMFT calculation
Projective QMC
Conventional QMC
Feldbacher et al, PRL 93 136405(2004)
Projective QMC
• Thermal fluctuations
• effort: ~1/T3
Interaction
Ising fields
0
no interaction

→∞
 

→∞
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Projective QMC
O
T 0
Tre  H0 e H / 2Oe H / 2
  lim
 
Tre  H0 e H
O  c(1 )c† ( 2 )  e1H ce(1  2 ) H c†e 2H
Tre  H0 e H / 21H ce (1  2 ) H c†e 2 H  H / 2
G(1 , 2 )   lim
 
Tre  H0 e H
-/2
/2
/2+
Interaction U only in red part
for sufficiently large P:
Accurate information on
G for light red part
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Application of PQMC to DMFT (1)
DMFT self-consistent loop
PQMC
G0 (1, 2 )
(T=0)
G(i )   d 
G(1, 2 ) (T=0)
(i)  G01 (i)  G1 (i)
Maximum Entropy Method
1
G ( )   A( ) exp( )d 

G(in ) 
A( )
d
  in  
1
D( )
in    (in )  
G01 (i)  G1 (i)  (i)
Problem: How to obtain (i)?
G()→FT→G(i)? No
only G(),<P obtained by PQMC
P
Calculate G only for <P
Large : Extrapolation by
Maximum Entropy Method
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Application of PQMC to DMFT (2)
Single band Hubbard model
I
M
HF-QMC
=16
insulating
=40
metallic
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Application of PQMC to DMFT (2)
Single band Hubbard model
I
M
PQMC
=16
=40
Metallic solution obtained for =16
(same numerical effort as HF-QMC with =16)
Application to
LDA+DMFT
at T→0
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Application of PQMC to LDA+DMFT
for LiV2O4
RA-Held-Lukoyanov-Anisimov
PRL 98 166402 (2007)
LiV2O4: 3d heavy Fermion system
Crossover at T*~20K
FL(T2 law)
g(T→0)~190mJ/Vmol・K2
・ resistivity: r =r0+AT2 with an enhanced A
・ specific heat coefficient: anomalously large
g(T→0)~190mJ/V mol・K2
cf) CeRu2Si2 ~350mJ/Ce mol・K2
UPt3
~420mJ/U mol・K2
(Kadowaki-Woods relation satisfied)
CW law at HT
S=1/2 per V ion
・ c: broad maximum (Wilson ratio～1.8)
T*
(Urano et al. PRL85, 1052(2000))
heavy mass quasiparticles (m*～25mLDA)
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LiV2O4: 3d heavy Fermion system
PhotoEmission Spectroscopy
(Shimoyamada et al. PRL 96 026403(2006))
LDA+DMFT(HF-QMC)
(Nekrasov et al, PRB 67 085111 (2003))
T=750K
A sharp peak appears
for T<26K
=4meV, ~10meV
LDA+DMFT(PQMC)
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Results
U
U’
U’-J
(Hund coupling = Ising)
U=3.6, U’=2.4, J=0.6
a1g
PQMC
T=300K
T=1200K
T=300K
eg
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FAQ
Why can we discuss A(→0) without calculating G(→∞) explicitly?
T→0
A()
A()
Large T
0
～exp(-00)
0
0
G()
G()
0
Slow-decay component
0

0

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Results: G() & A()
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Application of
perturbation series expansions to QMC
・Combination with Hirsch-Fye’s algorithm
(Sakai, RA, Held, Aoki PRB 74 155102 (2006))
・Continuous time QMC
weak coupling expansion
(Rubtsov et al, JETP Lett 80 61 (2004), PRB 72 035122 (2005))
hybridization expansion
(Werner et al, PRL 97 076405 (2006), PRB 74 155107 (2006))
QMC for multi-orbital systems
-J
-J
HJ : usually neglected
sign problem
difficult to treat for multi-orbital systems
e
ˆ 23 ˆ 31
Hˆ 12
J HJ HJ
Hˆ 12
Hˆ J23 Hˆ J31
J
e e
e
ˆ 23 ]  0
[ Hˆ 12
,
H
J
J
⇒ Non-trivial Suzuki-Trotter decomposition?
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Ising-type vs Heisenberg-type interaction
DMFT study for ferromagnetism in the 2-band Hubbard model
n=1.25, Bethe-lattice,
W=4, U=9, U’=5, J=2 (Ising)
J
Held-Vollhardt, 98
Ising-type couling:
Ferromagnetic instability overestimated
Sakai, RA, Held, Aoki 06
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PSE + Hirsch-Fye QMC
Sakai, RA, Held, Aoki PRB 74 155102 (2006)
PSE with respect to V (V: interaction term) (Rombouts et al, 99)
Same Algorithm as
Hirsch-Fye
For spin flip & pair hopping term:
extention to m>2 straightforward:
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PSE + Hirsch-Fye QMC
Large U,U’,J
e
  H 0  (   V )
Sakai, RA, Held, Aoki PRB 74 155102 (2006)
<k> becomes large


2
0
0
Large L needed
  (  )k  d k  d 1e1H0 (1   V )e ( 2 1 ) H0  (1   V )e (   k ) H0
k 0
It is not a good idea to treat all U,U’,J terms as V
e ( H0  H1 )  e H0 e H1
H0+HU+HU‘+HIsing≡ H0+H1 → standard HF
HJ → PSE (<k> is small for HJ)
2-band Hubbard model, n=1.9, =8, U=4.4, U‘=4, J=0.2, W=2
PSE+HF
Nk
Nk
PSE only
0
60
120
0
40
80
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PSE + Hirsch-Fye QMC
Sakai, RA, Held, Aoki PRB 74 155102 (2006)
Sign problem: less severe
Wide region of norm >0.01
Conventional HF:
e
 H

s 1
e
s1 sL
 H0
Qs1 e H0 Qs2  e H0 QsL
2-band, n=2, W=2, U=U’+2J, U’=4

We have to consider sn=±1 for every n,
PSE+HF:
(Sakai et al 04)
e
 H
0,1
  e H0 Qs1 e H0 Qs2  e H0 QsL
s1 sL

For small HJ, small number of n have sn≠0
Lower T, large J can be explored
Expansion with respect to HJ :
~ <HJ> negative sign problem relaxed
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Application to LDA+DMFT calculation for Sr2RuO4
Ising-type Hund, =70
SU(2) Hund + pair hopping, =40
U=1.2, U’=0.8, J=0.2 [eV]
1
dxy
dyz/zx
0-3
-2
-1
0
1
Energy [eV]
[Liebsch-Lichtenstein, PRL 84,1591 (2000)]
SU(2) symmetric 3-band LDA+DMFT
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Continuous time QMC
Weak coupling expansion:
Rubtsov et al, JETP Lett 80 61 (2004),
PRB 72 035122 (2005)
S   t c c drdr '   w
r' † r
r r'
r '1 r '2 † r1 †
r1r2
r '1
r '2
r2
c c c c dr1dr '1 dr2 dr '2
Non-local in time & space

Z  Tr T exp( S )    dr1  dr '1
k 0
(1)k r '1 r '2
k  Z 0
wr1r2
k!
wrr2'k2k1r12rk '2 k Tcr†'1 c r1
 dr  dr '
2k
2k
r   , i, s

dr   d 
0
i
s
k (r1 , r '1 , , r2 k , r '2 k )
cr†'2 k c r2 k
Perform a random walk in the space of K={k, (arguments of integrals)}
(cf. K={auxiliary spins} for Hirsch-Fye scheme)
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Applications
LDA+DMFT study for V2O3
(Ising type of Hund coupling)
Poteryaev et al, cond-mat/0701263
Correlated Adatom Trimer
on a Metal Surface
Savkin et al, PRL 94 026402 (2005) R.Arita
Continuous time QMC (2)
Hybridization expansion: Werner et al, PRL 97 076405 (2006),
Z  Tr T exp(S )
PRB 74 155107 (2006）
S  S0  S1

S0    d     b  U
ab
†
a

   c d  S1   d d ' a ( ) a (   ') a† ( ')
abcd
0
†
a
†
b
0
Impurity-bath hybridization
Matrix size
=100
(~5U)
(~0.5U)
Numerical effort decreases
with increasing U
Allows access to low T, even
at large U
U
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Summary

QMC: A powerful tool for LDA+DMFT, but
 low T not accessible
 sign problem in multi-orbital systems
 …

Recent developments
 Access to low T, strong coupling, multi-orbital systems
 Projective QMC for T→0 calculations
 Application of various perturbation series expansions for Z

Future Problems
 Spatial fluctuations (cluster extensions)
 Coupling to bosonic baths
 …
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