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Ryotaro ARITA (RIKE N) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments
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Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

Jan 02, 2016

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Page 1: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

Ryotaro ARITA   (RIKEN)

Methods for electronic structure calculations

with dynamical mean field theory:

An overview and recent developments

Page 2: 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)

Page 3: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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)

Page 4: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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

Page 5: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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

Page 6: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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

Page 7: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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

Page 8: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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

Page 9: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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

Page 10: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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

Page 11: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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

Page 12: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

Projective QMC and its application to DMFT calculation

Page 13: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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

Page 14: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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

Page 15: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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

Page 16: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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M I

HF-QMC

insulating metallic

=16 =40

Application of PQMC to DMFT (2)

Single band Hubbard model

Page 17: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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

Page 18: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

Application of PQMC to LDA+DMFT for LiV2O4

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

Page 19: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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

Page 20: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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)

Page 21: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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

Page 22: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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?

Page 23: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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Results: G() & A()

Page 24: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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))

Page 25: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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?

Page 26: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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

Page 27: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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)

Page 28: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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)

Page 29: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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)

Page 30: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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

Page 31: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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)

Page 32: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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)

Page 33: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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)

Page 34: Ryotaro ARITA (RIKEN) Methods for electronic structure calculations with dynamical mean field theory: An overview and recent developments.

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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 …