Efficient Perturbation Theory for Correlated Higher Tc Materials

A. Lichtenstein University of Hamburg

In collaborations with:H. Hafermann, M. Kecker, F. Lechermann, S. Brener (Hamburg)A. Rubtsov (Moscow)M. Katsnelson, I. Di Marco (Nijmegen)

U+t V J V Δ ?

Outline

• Correlated systems: non-perturbative DMFT

• Beyond DMFT: Dual Fermion approach

• Antiferromagnetic pseudogap

• d-wave: BSE

• Conclusions

From Atom to Solid: DMFT

E

N(E)

EF

QPLHB UHB

E

N(E)

EF

Atomic physics Bands effects (LDA)

LDA+DMFT

E

N(E)

EF dndn+ 1| SL>

DMFT: Self-Consistent Set of Equations

( )ττ ′−0G

( ) ( )∑Ω=BZ

knn ikGiG r

rωω ,ˆ1ˆ

( ) ( ) ( )nnn iiGi ωωω Σ+= −− ˆˆˆ 110GΣ Σ Σ

Σ Σ

Σ Σ Σ

ΣUQMC ED

DMRG IPTFLEX

( )ττ ′−0G

( ) ( ) ( )nnnnew iGii ωωω 110 ˆˆˆ −− −=Σ G

Single Impurity Solver

A. Georges and G. Kotliar (1992)W. Metzner and D. Vollhardt (1987)

Real Materials: LDA+DMFT

LDA+UStatic mean-field approximationEnergy-independent potential

|minlVinlm|V̂mm

mm σ′σ= ∑σ′

σ′

LDA+DMFTDynamic mean-field approximation

Energy-dependent self-energy operator

|minl)(inlm|)(ˆmm

mm σ′σ=εΣ ∑σ′

σ′

Applications:Insulators with long-range

spin-,orbital- and charge order

Applications:Paramagnetic, paraorbitalstrongly correlated metals

short range spin and orbital order

Cluster LDA+DMFT approximation

V. Anisimov, et al. J. Phys. CM 9, 7359 (1997)A. L. and M. Katsnelson PRB, 57, 6884 (1998)

A. Poteryaev, A. L., and G. Kotliar, PRL 93, 086401 (2004)

Spectral Function Fe: ARPES vs. DMFT

SP-ARPES:J. Sánchez-Barriga, et al,BESSY

General Projection formalism for LDA+DMFTDELOCALIZED S,P-STATES

CORRELATED D,F-STATES

G. Trimarchi et al. JPCM 20,135227 (2008)B. Amadon et al. PRB 77, 205112 (2008)

|L>|G>

HTSC: from LDA to 1-band modelO.K. Andersen, at alJ. Phys. Chem. Solids56, 1573 (1995)

From LDA “Chemistry”to

Low-energy TB-modelt’/t=r (-0.3 for YBCO)

t⊥∼ (coskx-cosky)2

EF

EF

NMTO-orbitalsO.K. Andersen, et alPhys. Rev. B62, R16219 (2000)

SCF-LDA+DMFT

F. Lechermann, et al, PRB (2007)

Continuous Time QMC formalism

Partition function and action for fermionic system with pair interactions

Tr( )SZ Te−=

1 2 1 2

1 2 1 2

' ''' ' ' 1 1 2 2' ' '

r r r rr rr r r r r rS t c c drdr w c c c c drdr dr dr

+ + += +∫ ∫ ∫ ∫ ∫ ∫{ , , }r i sτ=

0 i s

dr dβ

τ= ∑∑∫ ∫Splitting of the action into

Gaussian part and interaction 0S S W= +

( )( )2 1 2 2 12 1 2 2 1' ' ' ''0 ' 2 2 '' 'r r r r rr rr r r r r r rS t w w dr dr c c drdrα += + +∫ ∫ ∫ ∫( )( )1 2 1 1 2 21 2 1 1 2 2' ' ' ' ' ' 1 1 2 2' 'r r r r r rr r r r r rW w c c c c drdr dr drα α+ += − −∫ ∫ ∫ ∫

'rrα -- additional parameters, which are necessary to minimize the sign problem

arXiv: 0302228

A. Rubtsov “Quantum Monte Carlo determinantal algorithm without Hubbard-Stratonovich transformation: a general consideration ”

Continuous Time QMC formalismFormal perturbation-series:

1 1 2 2 1 1 2 20

' ... ' ( , ' ,..., , ' )k k k k kk

Z dr dr dr dr r r r r∞

=

= Ω∑∫ ∫ ∫ ∫2 1 2 1 21 2

1 2 2 1 2 1 2

' ' ...' '1 1 2 2 0 ' ... '

( 1)( , ' ,..., , ' ) ...!

k k k

k k k

kr r r rr r

k k k r r r r r rr r r r Z w w Dk−

−

−Ω =

( ) ( )1 2 2 21 11 2 1 1 2 2...' ... ' ' ' ' '...k k kk k kr r r rr rr r r r r rD T c c c cα α+ += − −Since S0 is Gaussian one can

apply the Wick theorem D can be presented as a determinant g0

( ) ( )( ) ( )

2 21 1

1 1 2 2

2 21 1

1 1 2 2

' ' ' ' ''

' ' ' '

...( )

...

k k

k k

k k

k k

r rr rrr r r r rr

r r rr rr r r r

Tc c c c c cg k

T c c c c

α α

α α

+ + +

+ +

− −=

− −The Green function can be

calculated as follows

ratio of determinantsIn practice efficient calculation

of a ratio is possible due to fast-update formulas

A. Rubtsov and A.L., JETP Lett. 80, 61 (2004)

Random walks in the k space

Step k+1Step k-11

1

k

k

w Dk D

+

+

1k

k

k Dw D

−

Acceptance ratio

0 20 40 60

0

Dis

tribu

tion

k

decrease increase

Maximum at 2UNβ

k-1 k+1

Z=… Zk-1 + Zk + Zk+1+ ….

Wannier - GW and effective U(ω)

T. Miyake and F. AryasetiawanPhys. Rev. B 77, 085122 (2008)

C-GW

GW

Co on Cu: 5d-orbitals QMC calculation

DOS for Co atom in Cu

E. Gorelov et al, arXiv:0905.3581

G(τ

) LDA

τ

CT-QMC

U=4, b = 10 (T ~ 1/40 W)

Beyond single-site DMFT

1/d expansion: A. Schiller and K. Ingersent PRL’95; A. Georges and G. Kotliar, et. al. RMP’962-site Bethe lattice: G. Moeller, PhD’94, et. al. , PRB’99DCA k-space: M. Jarrell: H. Hettler et. al. , PRB’98Cluster DMFT: A. L. and M. Katsnelson, PRB’00CDMFT: G. Kotliar et al. PRL’01Chain-DMFT: A. Georges, PRB’00, S. Biermann et al, PRL’01

Σ Σ Σ

Σ Σ

Σ Σ

Σ

Σ

Σ

Σ Σ Σ Σ

Cluster DMFT

ΣU

( )ττ ′−0G

ΣU

V

M. Hettler et al, PRB 58, 7475 (1998)A. L. and M. Katsnelson, PRB 62, R9283 (2000)G. Kotliar, et al, PRL 87, 186401 (2001)

AFM and d-wave in HTSC

A.L. and M.Katsnelson, PRB 62, R9283 (2000)

AFM and d-wave in CDMFT (2x2)

In superconducting state:

Coexistence of AFM and d-wave

0 1

23

A.L. and M.Katsnelson, PRB (2000)

S. Kancharla et al, PRB (2008) M. Jarrell et al, EPL (2001)

CDMFT and DCA: phase diagram

CDMFT DCA

Cluster DMFT and beyond

Δ

How to include exact k-dependence for correlated systems?

- Dynamical Vertex Approximation (K. Held, M. Jarrell)- Dual Fermion Approximation (A. Rubtsov)

short-range fluctuations

Beyond DMFT: Dual Fermion scheme

A. Rubtsov, et al, PRB 77, 033101 (2008)

General Lattice Action

Optimal Local Action with hybridization Δω

Lattice-Impurity connection:

Dual FermionsGaussian path-integral

With new Action:

Diagrammatic:

gω and χν,ν‘,ω from DMFT

Basic diagrams for dual self-energy

Lines denote the renormalized Green’s function.

Condition for Δ and relation with DMFTTo determine Δ, we require that Hartree correction in dual variables vanishes.If no higher diagrams are taken into account, one obtains DMFT:

Higher-order diagrams give corrections to the DMFT self-energy, and already the leading-order correction is nonlocal.

Σ(k,ω)

Gd=GDMFT-g

Convergence of Dual Fermions: 2d

=1

H. Hafermann, et al. PRL102, 206401 (2009)

DMFTDF

LDFA

2d: Im Σ(k, ω=0)

10 20 30 40 50 60

10

20

30

40

50

60

Im Σ(0,0)=-0.06

Im Σ(0,π)=-8

kx

kyIm Σ(0,0)=-0.04

Im Σ(0,π)=-0.08

kx

ky Im Σ(0,0)=-0.04

U=1 U=2

A. Rubtsov, et al, PRB 79, 045133 (2009)

Dynamical AF correlations: shadow bands

U=8t=W, T=0.2t, n=1 H. Hafermann, PhD (2009)

Pseudogap in HTSC: Ladder-DF

H. Hafermann, et al. PRL102, 206401 (2009)

n=1

Arcs in HTSC: Dual Fermions

2d: U=W=2

FS, n=0.85 QP-dispersion

A. Rubtsov, et al, PRB 79, 045133 (2009)

e-vH

Bethe-Salpeter EquationElectron-hole channel

Exact transformation to original fermions

Magnetic susceptibility

Susceptibility: 2d – Hubbard model

Bethe-Salpeter equation: pp-channel

AFMdx2-y2p

U=W/2 t’/t=-0.3 x=15%

Weak-coupling perturbation: FLEX

U/W=0.5 t’=0N. Bickers, D. Scalapino and S. White,PRL (1988)

LDFA – strong coupling analog of FLEX

d-wave symmetry of the eigenfunction: DF

0

max

min

U/W=1 t’=0H. Hafermann, et al, J. Supercond. Nov. Magn. 22, 45 (2008)

LDFA?

LDFA can reduce Tc due to formation of pseudogap (in progress)

Conclusions

Dual Fermion expansion around DMFT can efficiently interpolate between weak and strong couplingAntiferromagnetic pseudo-gap and Fermi-arcs describe well in ladder DF-schemed-wave pairing for overdoped regime can be analysed in simple DF, while for underdopedlimit the cluster-DF or ladder-DF is neededRealistic multiorbital LDA+DF for correlated higher-Tc materials (Fe-As) is a next challenge: work in progress.