Prospects of LDA+DMFT

Silke BiermannCentre de Physique Theorique

Ecole Polytechnique, Palaiseau, France

(*) LDA = the local density approximation (LDA) of density

functional theory

LDA+DMFT = the combination of dynamical mean field theory

(DMFT) with the LDA. – p.1/69

The Mott transition within DMFTSpectral function ofthe Hubbard model

2

0

2

0

2

0

2

0

2

0

−Im

G

ω−4 −2 0 2 4

U/D=1

U/D=2

U/D=2.5

U/D=3

U/D=4

/D

Fig.30

. – p.2/69

Spectral function – survival kitAdd/remove an electron – at which energy?Two “easy” limiting cases:

1. Non-interacting limit:state of N electrons = Slater determinant(N+1)th electron can jump into any (unoccupied) band

probe unoccupied density of states

0

0.2

0.4

0.6

−2 −1 0 1 2E−E

Fermi

DOS

. – p.3/69

Spectral function – survival kit2. “Atomic limit” (complete localization):probe local Coulomb interaction!

0

0.1

0.2

0.3

0.4

0.5

−10 −5 0 5 10E−E

Fermi

U

In the general, interacting case:Spectral function

describes the possibility ofadding an electron with energy (includingrelaxation effects)

. – p.4/69

The Mott transition within DMFTSpectral function ofthe Hubbard model

2

0

2

0

2

0

2

0

2

0

−Im

G

ω−4 −2 0 2 4

U/D=1

U/D=2

U/D=2.5

U/D=3

U/D=4

/D

Fig.30

. – p.5/69

Mott insulator and correl. metal:YTiO and SrVO

Inte

nsity

(ar

b. u

nits

)

2.0 1.0 0.0Binding Energy (eV)

Sr0.5Ca0.5VO3

SrVO3 (x = 0)

CaVO3 (x = 1)

hν = 900 eV hν = 275 eV hν = 40.8 eV hν = 21.2 eV

2

0

2

0

2

0

2

0

2

0−

ImG

ω−4 −2 0 2 4

U/D=1

U/D=2

U/D=2.5

U/D=3

U/D=4

/D

Fig.30

. – p.6/69

Outline

Reminder: Dynamical Mean Field Theory(DMFT)

The “LDA+DMFT” method

:Reminder: an “effective atom” point of view

A functional point of view

Examples

What can we calculate ?The current status

Beyond LDA+DMFT: the GW+DMFT

scheme

LDA = The local density approximation to Density Functional Theory

LDA+DMFT = The combination of LDA and dynamical mean field theory (DMFT)

(**) GW+DMFT = The combination of Hedin’s GW approximation with DMFT. – p.7/69

The local Green’s function ...... is the central object of DMFT

Definition of Green’s function:

Relation to local spectral function:

. – p.8/69

Green’s function – survival kitQuasi-particles are poles of

All correlations are hidden in the self-energy:

. – p.9/69

Dynamical mean field theory ...... maps a lattice problem

onto a single-site (Anderson impurity) problem

with a self-consistency condition

. – p.10/69

Effective dynamics ...... for single-site problem

with the dynamical mean field

. – p.11/69

DMFT (contd.)Green’s function:

Self-energy (k-independent):

DMFT assumption :

Self-consistency condition for

. – p.12/69

The DMFT self-consistency cycleAnderson impurity model solver

Self-consistency condition:

. – p.13/69

Realistic Approach to CorrelationsCombine DMFT with band structure calculations

(Anisimov et al. 1997, Lichtenstein et al. 1998)

effective one-particle Hamiltonian within LDArepresent in localized basis

add Hubbard interaction term for correlated orbitals

solve within Dynamical Mean Field Theory

. – p.14/69

Hamiltonian formulation

(correl. orb.)

(correl. orb.)

in localized basis set( lecture by F. Lechermann)

all valence electrons

Hubbard terms for “correlated orbitals”. – p.15/69

Dynamical mean field theory ...... maps a solid

onto an “effective atom” problem

with a self-consistency condition

. – p.16/69

The DMFT self-consistency cycleAnderson impurity model solver

Self-consistency condition:

. – p.17/69

What do we mean by this?Represent the Green’s function in localized basis,e.g. LMTO’s:

,

where

double counting correctionsis a matrix in orbital space

is a matrix in the space of the correlated orbitals

lecture by F. Lechermann

. – p.18/69

Hubbard interaction termsScreened Coulomb interaction, onsite terms only:

with

Simplification: keep “diagonal density” terms only:

. – p.19/69

Parametrization of interaction:

with the Slater integrals

, e.g. for d-electrons:

. – p.20/69

Double counting corrections

of Hartree type (cf. “LDA+U”, Anisimov et al.)

Lichtenstein et al., 2001:

Tr

( no zero frequency contribution,cf. LDA Fermi surface Luttinger’s theorem)

. – p.21/69

A functional point of view ?Reminder about Density Functional Theory:Energy is a functional of the density:E= E[n(r)]

Kohn-Sham potential can be viewed as a Lagrangemultiplier to fix the density of the non-interactingreference system

. – p.22/69

A functional point of view ...... of LDA+DMFT

“Spectral Density Functional Theory”:Free energy a functional of(1) the density n(r)(2) the local Green’s function of the correlated orbitals

Construction by introduction of source termslecture by M. Katsnelson

Kotliar, Savrasov, PRB 2004

. – p.23/69

LDA+DMFT as a ...... spectral density functional theoryFree energy functional:

tr

tr

Kotliar, Savrasov, 2004 . – p.24/69

LDA+DMFT – the full scheme

DMFT loop

DMFT preludeDFT part

update

PSfrag replacements

VKS = Vext + VH + Vxc

[

−∇2

2 + VKS

]

|ψkν〉 = εkν |ψkν〉

from charge density ρ(r) constructupdate

|χRm

〉

GKS =[

iωn + µ+ ∇2

2 − VKS

]−1

G0

build GKS =[

iωn + µ+ ∇2

2 − VKS

]−1

construct initial G0

impurity solver

Gimpmm′(τ − τ ′) = −〈T dmσ(τ)d†

m′σ′(τ ′)〉Simp

self-consistency condition: construct Gloc

G−10 = G−1

loc + Σimp

Gloc = P(C)R

[

G−1KS −

(

Σimp − Σdc

)]−1

P(C)R

Σimp = G−10 − G−1

imp

ρ

compute new chemical potential µ

ρ(r) = ρKS(r) + ∆ρ(r)

(Appendix A)

(from F. Lechermann, A. Georges, A. Poteryaev, S. B., M. Posternak, A. Yamasaki, O. K. Ander-

sen, Phys. Rev. B 74 125120 (2006))

. – p.25/69

Some more examples

. – p.26/69

Cerium sesquioxide Ce O Mott insulator,

paramagnetic above 10 K

. – p.27/69

Cerium sesquioxide Ce O

Metallic in LDA:

-6

-4

-2

0

2

4

6

8

10

G K M G A L H A

Ene

rgy

(eV

)Ce2O3 bands (Ce 6p in basis)

Note the narrow f-bands!. – p.28/69

Cerium sesquioxide Ce O

LDA+DMFT splits f-states into upper and lowerHubbard bands:

(L. Pourovskii et al., arXiv:0705.2161). – p.29/69

Spectral functionCe O

25

50

0

25

50

Den

sity

of

stat

es (

1/eV

)

LDA

DMFT-nonSC

DMFT-SC

-6 -4 -2 0 2 4Energy (eV)

0

25

50

(L. Pourovskii et al., arXiv:0705.2161)

. – p.30/69

Total energyCe O

3.6 3.7 3.8 3.9 4

a (Ao

)

0

25

50

75

100

E-E

min

(m

Ry)

LDA DMFT (without charge SC)DMFT (with charge SC)

aexp

=3.89 Ao

(L. Pourovskii et al., arXiv:0705.2161)

. – p.31/69

Vanadium dioxide: VO

High temperature rutile phase: metallic

Low temperature monoclinic phase: insulating

. – p.32/69

Metal-insulator Transition

. – p.33/69

VO in LDA and LDA+DMFT

0

0.5

1.0

−2 0 2 4

Spe

ctra

lFun

ctio

n(a

rb.u

nits

)

ω (eV)

VO2

rutile phase

VO2

monoclinic phase

LDA: dashed lineLDA+DMFT: solid line

S.B., A.Poteryaev, A. Georges, A.Lichtenstein, PRL 2005 . – p.34/69

LDA+DMFT simulations ...for the monoclinic phase have to be done with care:

Inter-site (intra-pair)fluctuations ?

"Cluster-DMFT" :embed V pair in bath non-local self-energy !

. – p.35/69

VO recent photoemission

Koethe et al.,PRL 2006

lecture byL.-H. Tjeng

. – p.36/69

What can we calculate ?

. – p.37/69

What can we calculate ?

total energy

Examples: Ce O Pu

3.6 3.7 3.8 3.9 4

a (Ao

)

0

25

50

75

100

E-E

min

(m

Ry)

LDA DMFT (without charge SC)DMFT (with charge SC)

aexp

=3.89 Ao

(Pourovskii et al., arXiv:0705.2161) (Savrasov et al., Nature 2001)

(see also Held et al., PRL 2001, and Amadon et al., PRL 2006, for total energy calculations for

Cerium)

. – p.38/69

Total energy functional

with:

= crystal energy

= Hartree energy

= exchange correlation energysecond line = energy terms stemming from manybody Hamiltonian

(from B. Amadon et al., PRL 2006)

. – p.39/69

What can we calculate ?

local spectral functions

Examples:VO2-M1 and Ce O

25

50

0

25

50

Den

sity

of

stat

es (

1/eV

)

LDA

DMFT-nonSC

DMFT-SC

-6 -4 -2 0 2 4Energy (eV)

0

25

50

0

0.5

1.0

−2 0 2 4

Spe

ctra

lFun

ctio

n(a

rb.u

nits

)

ω (eV)

VO2

rutile phase

VO2

monoclinic phase

. – p.40/69

What can we calculate ?

total energy

local spectral functions

self-energies

-2

-1

0

1

-3 -2 -1 0 1 2 3

Σ(ω

)

ω [eV]

lecture by D. Vollhardt

. – p.41/69

Self-energy and spectral function

0

0.1

0.2

0.3

0.4

0.5

0.6

-3 -2 -1 0 1 2 3

A(ω

)

ω [eV]

-2

-1

0

1

-3 -2 -1 0 1 2 3

Σ(ω

)

ω [eV]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-4 -2 0 2 4

A(ω

)

ω [eV]

-5

-4

-3

-2

-1

0

1

2

3

-4 -2 0 2 4

Σ(ω

)ω [eV]

(from J.Tomczak, PhD thesis, 2007) . – p.42/69

What can we calculate ?

self-energies

Example: VO insulating phase

-3

-2

-1

0

1

2

3

4

-3 -2 -1 0 1 2 3 4

ω [eV]

Re

Σ (ω

) [e

V]

a1g

a1g-a1g

egπ1

egπ2

-3

-2

-1

0

1

2

3

4

-3 -2 -1 0 1 2 3 4

ω [eV]Im

Σ (

ω)

[eV

]

. – p.43/69

What can we calculate ?

... and k-resolved spectral functions

Example: VO insulating and metallic phases:

0.1

1

ω [e

V]

ΓZCYΓ-2

-1

0

1

2

3

0.1

1

ω [e

V]

ΓZCYΓ-2

-1

0

1

2

3

effective bandstructures (and absence thereof!)

J.Tomczak, F. Aryasetiawan, SB, arXiv:0704.0902 . – p.44/69

What can we calculate ?

phonons

Example:

-Pu

Dai et al., Science 2003

. – p.45/69

What can we calculate ?

local susceptibilities

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

0.0

0.5

1.0

Ni

Fe

χ-1M

eff2 /3

Tc

M(T

)/M

(0)

T/Tc

A. Lichtenstein, M. Katsnelson, G. Kotliar,

PRL 2001

. – p.46/69

What can we calculate ?

total energy

local spectral functions

k-resolved spectral functions

phonons

local susceptibilities

optical properties

. – p.47/69

Optical conductivityIn the Hubbard model within DMFT

from Rozenberg et al., 1995

d

f f

tr

k

k

k

k

see also: Pruschke et al., Blümer PhD thesis, Oudovenko et al., Haule et al.

. – p.48/69

Optical conductivityThe example of insulating VO :

from J.M. Tomczak, PhD thesis, 2007(for full formalism Poster by J. Tomczak)

0

1

2

3

4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Re

σ(ω

) [1

03 (Ω

cm)-1

]

ω [eV]

Experiment LDA+CDMFT ’upfolded’ E || [100]E || [010]E || [x0z]

. – p.49/69

LDA+DMFT – current statusMany successful applications:

Correlation effects in transition metals

Mott transitions in oxides

Volume collapse transitions in rare earth/actinidesystems

etc ...

Still (at least partly) open questions/challenges:

influence of full charge self-consistency

basis sets lecture by F. Lechermann

what’s U ? (Is there a U? On what energy scale?)

... and nuisances: double counting. – p.50/69

What’s U in a solid?... an answer from RPA:

Divide where = polarization of thecorrelated orbitals (e.g. 3d orbitals)Then:

where that does not include 3d-3d screening:

Identify = !F. Aryasetiawan, M. Imada, A. Georges, G. Kotliar, S.B., A. I. Lichtenstein PRB 70 195104 (2004)

. – p.51/69

U and W in a solidScreened Coulomb interaction from RPA

and

for Nickel:

0 10 20 30 40 50 60

ω (eV)

0

10

20

30

40

Re

<dd|

W|d

d> (e

V)

W Eg

Wr E

g

Paramagnetic Ni

(Aryasetiawanet al., 2004)

. – p.52/69

What’s U in a solid?

... what about an answer beyond RPA ??

. – p.53/69

Can we calculate ...... from a (dynamical) impurity model?

Question of representability !

DMFT: calculated from impurity model

What about ?

Self-consistency requirement:

= of the solid

= of the solid

“GW+DMFT”

S.B., Aryasetiawan, Georges, PRL 2003

condmat

. – p.54/69

The GW approximation(Hedin, 1965)

dynamically screened Coulomb interaction

GW successful for sp-metals, semiconductors ...

(Reviews:Onida et al., Rev. Mod. Phys. 2002;

Aryasetiawan et al., Rep. Prog. Phys. 1998)

. – p.55/69

A functional point of view[Almbladh et al. 1999]

Tr

Tr

Tr

Tr

Free energy

is a functional of

one-electron Green’s function

the screened Coulomb interaction

= bare (Hartree) Green’s function

. – p.56/69

Approximations to

?GW:

Extended DMFT (“E-DMFT”):

E-DMFT for local part + GW for nonlocal part:

NB: “local” = “onsite” is a basis-set dependent notion!

. – p.57/69

Extended DMFT ...... maps a lattice problem

onto a single-site (Anderson impurity) problem

with a dynamical interaction

. – p.58/69

Self-consistency loop

. – p.59/69

Challenges and questions

Global self-consistency?

Choice of orbitals? Hamiltonian?

Treat all orbitals – localized and delocalized – onequal footing? Downfolding?

How to solve the dynamical impurity model?here: static approximation

. – p.60/69

A simplified implementationNon-selfconsistent GW + local from static impuritymodel

Tr

Nonlocal part: correct Hartree by GW

Local part: correct LDA by DMFT

. – p.61/69

Simplified GW+DMFTNi band structure

−8

−6

−4

−2

0

Γ

Ene

rgy

[eV

]

XX Minority Majority

Circles: GW+DMFTDashed: LDATriangles: photoemission data

(Bünemann et al. 2002, Mårtensson et al., 1984)

. – p.62/69

Simplified GW+DMFT:Spectral function of Ni

0

2

4

6

−10 −5 0 50

2

4

6

−10 −5 0 5E−EF

[eV]

ρ(E)

6eV

Majority and minority spins

Satellite at 6eV correct!

. – p.63/69

Conclusion and perspectivesCombination of LDA and DMFT ...

a useful tool for electronic structure calculationsof strongly correlated materials, such as

transition metals, their oxides, f-electron materials

VO (Correlation-induced Peierls transition)

Ce O , a Mott insulator

Prospects ?

technical and physical questions concerningimplementations

calculate more quantities, more materials ... !

Beyond LDA+DMFT ? GW+DMFT

Plenty of work left! . – p.64/69

References of recent work

VO : Correlation-induced Peierls transitionS.B., A. Poteryaev, A. Georges, A. Lichtenstein, Phys. Rev. Lett. 94, 026404 (2005)

J. M. Tomczak, SB, Journal of Phys. Cond. Mat., in press

J.M. Tomczak, F. Aryasetiawan, SB, condmat07040902

Ce O : A Mott insulator – questions ofcharge-selfconsistencyL. Pourovskii, B. Amadon, S. Bierman, A. Georges arXiv:0705.2161

U and GW+DMFT:F. Aryasetiawan, M. Imada, A. Georges, G. Kotliar, S.B., A. I. Lichtenstein PRB 70195104 (2004),

S. Biermann, F. Aryasetiawan, A. Georges, PRL 2003, condmat 2004

LDA+DMFT - recent Reviews:D. Vollhardt, G. Kotliar, Physics Today 2004

S. B., in Encyclop. of Mat. Science. and Technol., Elsevier 2005

. – p.65/69

GW+DMFT: local part

calculated from local impuritymodel:

. – p.66/69

GW+DMFT (contd)Combine local self-energy and polarization

with non-local self-energy and polarization:

. – p.67/69

Self-consistency condition

Update Weiss field and impurity interaction:

Iterate until self-consistency ...

. – p.68/69

NickelEvidence for many-body effects:

Exp. LDA GW

LDA+DMFT

bandwidth 3.3 eV 4.5 eV 3.5 eV OK!x-splitting 0.3 eV 0.6 eV 0.6 eV OK!satellite at 6eV NO! NO! YES!

large quasi-particle widthsOpen question: Fermi surface pocket ?

(1) Aryasetiawan, 1992(2) Lichtenstein et al., 2001

. – p.69/69