Ryotaro ARITA 　 (RIKEN)

Methods for electronic structure calculations

with dynamical mean field theory:

An overview and recent developments

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)

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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)

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

Computational scheme for correlated electron materials

Model HamiltoniansModel HamiltoniansDFT/LDA DFT/LDA

systematic many-body approach input parameters unknown

material specific, ab initio fails for strong correlations

Anisimov et al 97, Lichtenstein, Katsnelson 98

Dr Aryasetiawan July 25, Prof. Savrasov July 27

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LDA+DMFT

Transition metal oxides LaTiO3

V2O3, VO2

(Sr,Ca)VO3

LiV2O4

(Sr,Ca)2RuO4

NaxCoO2

Cuprates Manganites …

Transition metals Fe, Ni

Heussler alloys

Organic compounds BEDT-TTF TMTSF

Fullerenes Nanostructure materials

Zeolites f-electron systems

Rare earths: Ce Actinides: Pu …

Application to various correlated materials(reviews) Held et al 03, Kotliar et al 06, etc

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Supplementing LDA with local Coulomb interactions　

LDA+DMFT

Expand Ψ+ w.r.t. a localized basis Φilm :

Downfolding: LDA → effective low-energy Hamiltonian

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Lattice model:

DOS Self Energy

LDA+DMFT

†, ,

,lat i j i i i i

i j i i

H t c c U n n n n

Solve model by DMFT Metzner & Vollhardt 89, Georges & Kotliar 92

( )D ( , )lat nk i

†' ( ) ( ') ( ') ( ( ) )S d d c F c d n n Un n

Effective impurity model:

Hybridization F Self Energy ( )imp ni

Self-consistency:

1 1

( )

( )( )

( )

imp n

latt n latt impn imp n

F i

DG i d F G

i i

F

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Solvers for the DMFT impurity model

Iterated perturbation theory

Perturbation expansion in U

Non-crossing approximation

Perturbation expansion in V

Exact diagonalization for small number of host sites

Max # of orbitals <2

Numerical renormalization group

(logarithmic discretization of host spectrum)

Max # of orbitals <2

Dynamical density matrix renormalization group

Quantum Monte Carlo

…

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1 2

1 2

1 2

......

...

L

L

L

s s ss s s

s s s

ZA A

Z

Suzuki-Trotter decomposition

Hubbard-Stratonovich transformation for Hint

Many-particle system

= (free one-particle system + auxiliary field)

1 2

1 2

......

L

L

s s ss s s

Z Z 1 2 ... 0

1

12

Tr [exp( exp( ]) )L

L

s s s ll

L H s nZ

0 int

L

l 1

= ( 1/ )Tr Tr HHH T LZ e e e

12 ( )

1

( )[ ] 12s

U s n nn n n ne e

(cosh( ) exp[ /2])U

Monte Carlo sampling

Auxiliary-field QMC

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, =L

QMC for the Anderson impurity model ( Hirsch-Fye 86 )

Integrate out the conduction bands

G{s}(), w{s}

Calculate

G0() G{s}(), w{s} …

Updating: numerical effort ~L2

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Numerically expensive for low T: numerical effort ~ 1/T3

Projective QMC (Feldbacher et al 04): A new route to T→0

Sign problem in multi-orbital systems: difficult to treat spin flip terms

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)

norm

Zs can be negative:Norm can be small→ <A>=0/0

Problems & Recent developments

Projective QMC and its application to DMFT calculation

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Projective QMC

0

InteractionIsing fields no interaction

→∞ →∞

Feldbacher et al, PRL 93 136405(2004)

Conventional QMC Projective QMC

• Thermal fluctuations

• effort: ~1/T3

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Projective QMC

Interaction U only in red part

for sufficiently large P:Accurate information onG for light red part

1 1 2 2( )† †1 2( ) ( ) H H HO c c e ce c e

0 1 1 2 2

0

/ 2 /) †

1 2

( 2Tr( , ) lim

Tr

H H H H H H

H H

e e ce c eG

e e

-/2 /2 /2+

0

0

/ 2 / 2

0

Trlim

Tr

H H H

T H H

e e OeO

e e

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Application of PQMC to DMFT (1)

PQMC

1 2( )G 1

01() ( ) )(i G i G i

( )( )

( )n n

DG i d

i i

1 10 ( ) ( ) ( )G i G i i

0 1 2( )G

1ex( p( ))) (A dG

(( )

1 )

nn d

i

AG i

Maximum Entropy Method

(T=0)

(T=0)

DMFT self-consistent loop

Problem: How to obtain (i)?

G()→FT→G(i)? No

only G(),P obtained by PQMC

P

Calculate G only for P

Large Extrapolation byMaximum Entropy Method

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M I

HF-QMC

insulating metallic

=16 =40

Application of PQMC to DMFT (2)

Single band Hubbard model

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Metallic solution obtained for =16 (same numerical effort as HF-QMC with =16)

M I

PQMC

=16 =40

Application of PQMC to DMFT (2)

Single band Hubbard model

Application to

LDA+DMFT

at T→0

Application of PQMC to LDA+DMFT for LiV2O4

RA-Held-Lukoyanov-AnisimovPRL 98 166402 (2007)

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Crossover at T*~20K

・ resistivity: =+AT2 with an enhanced A

・ specific heat coefficient: anomalously large (T→0)~190mJ/V mol ・ K2

(Kadowaki-Woods relation satisfied)

・ : broad maximum (Wilson ratio ～ 1.8)

　　　 cf) CeRu2Si2 ~350mJ/Ce mol ・ K2

　　　 UPt3 ~420mJ/U mol ・K2

heavy mass quasiparticles (m* ～ 25mLDA)

T*

(Urano et al. PRL85, 1052(2000))

LiV2O4: 3d heavy Fermion system

Incoherenet metal

FL(T2 law)

(T→0)~190mJ/Vmol・ K2

CW law at HT S=1/2 per V ion

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(Shimoyamada et al. PRL 96 026403(2006))

PhotoEmission Spectroscopy

A sharp peak appearsfor T<26K

=4meV, 10meV

LiV2O4: 3d heavy Fermion system

LDA+DMFT(HF-QMC) (Nekrasov et al, PRB 67 085111 (2003))

T=750K

LDA+DMFT(PQMC)

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Results

U=3.6, U’=2.4, J=0.6

U U’ U’-J (Hund coupling = Ising)

a1g

eg

T=1200KT=300K

PQMCT=300K

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0 0

FAQA

()

A(

)

T→0

0

G(

)

0

～ exp(-0) G(

)

0

Slow-decay component

0

Large T

Why can we discuss A(→0) without calculating G(→∞) explicitly?

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Results: G() & A()

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))

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-J -J

sign problem

QMC for multi-orbital systems

difficult to treat for multi-orbital systems

12 23 31 12 23 31ˆ ˆ ˆ ˆ ˆ ˆ J J J J J JH H H H H He e e e

HJ : usually neglected

12 23ˆ ˆ[ , ] 0J JH H

⇒ Non-trivial Suzuki-Trotter decomposition?

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Held-Vollhardt, 98

n=1.25, Bethe-lattice,W=4, U=9, U’=5, J=2 (Ising)

J

Ising-type vs Heisenberg-type interaction

DMFT study for ferromagnetism in the 2-band Hubbard model

Ising-type couling:

Ferromagnetic instability overestimated

Sakai, RA, Held, Aoki 06

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PSE with respect to V (V: interaction term) (Rombouts et al, 99)

Same Algorithm as Hirsch-Fye

extention to m>2 straightforward:

For spin flip & pair hopping term:

PSE + Hirsch-Fye QMC Sakai, RA, Held, Aoki PRB 74 155102 (2006)

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0 1 0 1( )H H H He e e

Large U,U’,J <k> becomes large Large L needed

0 40 80

Nk

Nk

0 60 120

PSE only PSE+HF

2-band Hubbard model, n=1.9, =8, U=4.4, U‘=4, J=0.2, W=2

1

20 1 0 2 1 0 0( ) ( ) ( )

0 00

(1 ) (1 ) ( ) k

kkH V H H H

k

d de e V e V e

It is not a good idea to treat all U,U’,J terms as V

H0+HU+HU‘+HIsing≡ H0+H1 → standard HF

HJ 　→　 PSE (<k> is small for H

J)

PSE + Hirsch-Fye QMC Sakai, RA, Held, Aoki PRB 74 155102 (2006)

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Wide region of norm >0.01

Lower T, large J　 can be exploredLower T, large J　 can be explored

2-band, n=2, W=2, U=U’+2J, U’=4

0 0 0

1 2

1

1

L

L

sH H HH

s s ss s

e e Q e Q e Q

PSE+HF:

0 0 0

1 2

1

, 10

L

L

H H HHs s s

s s

e e Q e Q e Q

Conventional HF:

(Sakai et al 04)

We have to consider sn=±1 for every n,

For small HJ, small number of n have sn≠0

Expansion with respect to HJ :HJ→negative sign problem relaxed

Sign problem: less severe

PSE + Hirsch-Fye QMC Sakai, RA, Held, Aoki PRB 74 155102 (2006)

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U=1.2, U’=0.8, J=0.2 [eV]

Ising-type Hund, =70 SU(2) Hund + pair hopping, =40 SU(2) Hund + pair hopping, =40

[Liebsch-Lichtenstein, PRL 84,1591 (2000)]

Application to LDA+DMFT calculation for Sr2RuO4

dxy

dyz/zx

1

0-3 0 1-1-2Energy [eV]

SU(2) symmetric 3-band LDA+DMFT

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Continuous time QMC

Weak coupling expansion:

Perform a random walk in the space of K={k, (arguments of integrals)}

(cf. K={auxiliary spins} for Hirsch-Fye scheme)

1 2

1 2

1 2

1 2

' † † †'

'' 1 1 2

'' 2' ' 'r rr r r

rr rr

r rrS t c c drdr c c c c drdw r dr dr

2 1 2 21 2 1

1 2 2 1 2 1 2

1 1 2 2 1 1 2 20

' '' ' † †0 ' '

Tr exp( ) ' ' ( , ' , , , ' )

( 1)

!k k k

k k k

k k k k kk

kr r rr r r

k r r r r r r

Z T S dr dr dr dr r r r r

Z w w Tc c c ck

, ,r i s

0 i s

dr d

Non-local in time & space

Rubtsov et al, JETP Lett 80 61 (2004), PRB 72 035122 (2005)

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Applications

Poteryaev et al, cond-mat/0701263

LDA+DMFT study for V2O3

(Ising type of Hund coupling)

Correlated Adatom Trimer on a Metal Surface

Savkin et al, PRL 94 026402 (2005)

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Hybridization expansion:

Continuous time QMC (2)

Tr exp( )Z T S 0 1S S S

†1

0

( ')' ( ) ( ')a a aS d d

† † †0

0

ab abcda b a b c dS d U

Numerical effort decreases with increasing U

Allows access to low T, even at large U

Impurity-bath hybridization

(~5U)

(~0.5U)

U

Mat

rix s

ize

=100

Werner et al, PRL 97 076405 (2006), PRB 74 155107 (2006）

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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 …

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 …