LDA+DMFT and beyond - IIA. Lichtenstein

University of Hamburg

In collaborations with:A. Poteryaev(Ekaterinburg), M. Rozenberg, S. Biermann, A. Georges (Paris)L. Chioncel (Graz), I. di Marco, M. Katsnelson (Nijmegen)E. Pavarini (Jülich), O.K. Andersen (Stuttgart)G. Kotliar (Rutgers), S. Savrasov (Devis), A. Rubtsov (Moscow)F. Lechermann, H. Hafermann, T. Wehling, C. Jung, M. Karolak (Hamburg)

http://www.physnet.uni-hamburg.de/hp/alichten/lectures/LDA+DMFT-2.pdf

Outline-III

• Non-local Correlation Effects

• Cluster DMFT Scheme

• LDA+cDMFT for Real Systems

• Dual Fermions

• Future of first-principle LDA+DMFT

Magnetic nanostructures

Quantum corralQuantum corralD. D. EiglerEigler (IBM)(IBM)

S. S. OtteOtte et. al.et. al.(IBM)(IBM)Fe

Co

Dynamical Mean-Field Theory

Σ Σ Σ

Σ

Σ

Σ

ΣΣ

U

U

G( ’)τ−τ

ττ’

Review: A. Georges, et. al. RMP, 1996

-

Cluster DMFT schemeFree cluster

Periodic cluster

1G−Σ −-1= G

1

1

[ ( ) ( )]k

G i H k

G

ω μ ω −

−

= + − −Σ

= +Σ

∑-1G

1

1

[ ( ) ( , )]k

ab ai ij jbij

ij ij ia ab bjab

G i H k k

i t G t

G G G G G

ω μ ω

ω μ

−

−

= + − −Σ

= + −

= −

∑

∑

∑

-1G

AFM and d-wave in HTSC

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

AFM+d-wave in CDMFT

In superconducting state:

Coexistence of AFM and d-wave

0 1

23

Magnetism vs. Kondo resonance

-4 -2 0 2 4

0.0

0.2

0.4

0.6

DO

S

Energy

U=2.4, J=-0.2 and J=0, β=64

Three impurity atoms with Hubbard

repulsion and exchange interaction

M. M. CrommieCrommie, PRL(2001), PRL(2001) CTCT--QMC:QMC: single single vs.vs. trimertrimerV. V. SavkinSavkin et al, PRL (2005)et al, PRL (2005)

#1#1 #2#2

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)

Double Bethe lattice: AFM vs. Kondo

Z Z →→ ∞∞

Z=3Z=3

Exact cluster DMFTExact cluster DMFT MIT: G. MMIT: G. Mööllerller, et. al. 1999, et. al. 1999

QMC:QMC:U=2U=2W=2W=2t=0.5t=0.5ββ=10=10 t

m0.5

0.1p

jiU t U

t t

P

J~tJ~tpp

Double-Bethe Lattice: exact C-DMFT

A. RuckensteinPRB (1999)

Self-consistent condition: C-DMFT

AF-between plane AF-plane

Finite temperature phase diagram

• order-disorder transition at t⊥ / t=√2 for large U• MIT for intermediate U

Density of States: large U

Spin-correlations: large U

CT-QMC measuring χ in imaginary time

Dynamical susceptibility

M. Marezio et al., (1972)

TMTM--Oxide VOOxide VO22: singlet formation: singlet formation

Metal

Tem

pera

ture

(K)

Insulator

Rutile structure Monoclinic distortion inthe insulating phase

j

i

G ω( )ij

U

U

tij

U/t

ε εi jb

a

LH

UH

Correlation vs. Bonding

Local Green function

Veg w Ww Veg WW W Va1g

Matrix of intersite Coulomb interaction

IntersiteIntersite interaction: Tiinteraction: Ti22OO33

Corundum structurefor low and high temperature phases

Efa

1g

a1g*

Ve

g

WV

a1g

eg

eg

a1g

a1g

w

V-V pair in V2O3

j

i

G ω( )ij

ae1

The Goodenough diagram

E

O2pO2p

EFa1g

ega1g

Ti2O3 V2O3

eg

eg

Ti2O3 – d1 state1-st order MITAround 470KWidth 250 K

LDA givesMetallic state

Problem:Bonding vs. U

LDA electronic structureLDA electronic structure

gg aat 11 −

gg ea εε −1

TiTid −

0.21 eV

-0.85 eV

2.57 Å

0.23 eV

-0.63 eV

2.72 Å

-2 -1 0 1 2 3Energy, eV

0

1

2

DO

S, s

tate

s/eV

Totale

g

a1g

VOVO22 TiTi22OO33

-2 -1 0 1 2 3Energy, eV

0

1

2

DO

S, s

tate

s/eV

Totale

g

a1g

gg aat 11 −

gg ea εε −1

VVd −

LT HT

0.29 eV

-0.81 eV

2.53 Å

0.15 eV

-0.31 eV

2.75 Å

M1 R

-4 -2 0 2 4 6Energy, eV

0.0

0.5

1.0

1.5

2.0

DO

S, s

tate

s/eV

Totale

g

a1g

-4 -2 0 2 4 6Energy, eV

0.0

0.5

1.0

1.5

2.0

DO

S, s

tate

s/eV

Totale

g

a1g

U = 2, J = 0.5, W = 0.5β = 20 eV-1, LT structure

U = 2, J = 0.5, W = 0.5β = 10 eV-1, HT structure

Cluster LDA+DMFT: TiCluster LDA+DMFT: Ti22OO33

K. Smith et al., PRB 38, 5965 (1988)

10iω

n, eV

-0.2

-0.1

0.0

Σ, e

V

ReΣa

1g-a

1g

ImΣa

1g

ReΣ’a

1g-a

1g

A. Poteryaev, et al, PRL93, 086401 (2004)

Cluster-DMFT results for VO2

0

0.2

0.4

0.6

0.8

1.0

−2 0 2 4

U=4eV J=0.68eV

ρ(ω)

ω[eV]

LDA VO2

rutileDMFT

(dashed)(solid)

0

0.5

1.0

1.5

−4 −2 0 2 4

DOS VO2−M1

LDA

ω [eV]

cluster DMFT

(dashed)

(solid)

U = 4 eV, J=0.68 eV β = 20 eV-1Rutile

M1

New photoemission from Tjeng’s groupT. C. Koethe, et al. PRL (2006)

Sharp peak below the gap is NOT a Hubbard band !

S. Biermann, et al, PRL 94, 026404 (2005)

Beyond DMFT: Dual Fermion scheme

A. Rubtsov, et al, Phys. Rev. B 77, 033101 (2008)

General Lattice Action

Optimal Local Action with hybridization Δω

Lattice-Impurity connection:

Dual FermionsGaussian path-integral

With new Action:

here:

gω and χν,ν‘,ω from DMFT

Dual Fermion Action: Details

Dual and Lattice Green’s Functions

Basic diagrams for dual self-energy

Lines denote the renormalized Green’s function.

A. Rubtsov, et al., Phys. Rev. B 79, 045133 (2009)

Condition for Δ and relation with DMFT

To 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

Self-consistent DF-scheme

2d Hubbard: 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

kyIm Σ(0,0)=-0.04

U=1 U=2

A. Rubtsov, et al, Phys. Rev. B (2008)

AFM – symmetry breaking

Pseudogap in HTSC: dual fermions

A. Rubtsov et alPRB (2009)

2d: U=4 W=2t/t’=-0.3

FSΣ

Quasiparticle dispersion

Cluster Dual Fermions: 1d-test, n=1

H. Hafermann, et al. JETP Lett (2007), arXiv:0707.4022

Two-particle Green-Functions

Bethe-Salpeter Equations

Susceptibility: DF vs.DMFT

Convergence of Dual Fermions: 2d

=1

H. Hafermann, PRL 102, 206401 (2009)

Susceptibility: 2d – Hubbard model

Future of first-princuple DMFT

• Accurate description of screen interaction (GW)

• Exact double-counting correction (Vxc-Σ)

• Optimal local basis set (Wannier)

• Accurate impurity solver (CT-QMC)

• Non-local correction (CDMFT, GW+DMFT, DF)

Conclusions

Cluster generalization of DMFT is easy to do in the Matrix-DMFT formalism Cluster LDA+DMFT method can be useful for the short-range non-local correlations in solids.Dual Fermion formalism can described k-dependence of self-energy for correlated systems