Applications: LDA+DMFT scheme - I

A. 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-1.pdf

Outline-I

• From Atom to Solids: Multiplets in solids

• Functionals: MFT, DFT, SDF

• Multiorbital DMFT for General Lattice

• Matrix version LDA+DMFT Scheme

• Impurity solvers: CT-QMC

• Examples of LDA+DMFT

Control parameters• Bandwidth (U/W)• Band filling• Dimensionality

Degrees of freedom• Charge / Spin• Orbital • Lattice

3d - 4fopen shells

materials

U<<WCharge fluct.

U>>WSpin fluct.

• Kondo• Mott-Hubbard• Heavy Fermions• High-Tc SC• Spin-charge order• Colossal MR

Nd2-xCexCuO4 La2-xSrxCuO4

0.3 0.2 0.10

100

200

300

SC

AFTem

pera

ture

(K)

Dopant Concentration x0.0 0.1 0.2 0.3

SC

AF

Pseudogap

'Normal'Metal

La1-xCaxMnO3

Dopant Concentration x

CMR

FM

I II IIIb IVb Vb VIb VIIb VIIIb Ib IIb III IV V VI VII 0H HeLi Be B C N O F NeNa Mg Al Si P S Cl Ar

Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I XeCs Ba La* Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At RnFr Ra Ac** Rf Db Sg Bh Hs Mt

Lanthanides *Actinides ** Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr

K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr

Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

Strongly Correlated Electron Systems

- How to incorporate atomic physics in the band structure ?

- How good is a local approximation ?

- What is a best solution for atomic problem in effective medium ?

- What is different from one band Hubbard model?

- How to solve a complicated Quantum multiorbital problem ?

- What is the best Tight-Binding scheme for realistic Many-Body calculation for solids?

From Atom to Solids

Theory of everything: t vs. U

Coulomb inraatomic interaction

Multiband Hubbard model (<im|jm0 >=δ ijδ mm0 )

Matrix elements of electron-electron interactions:

Exact diagonalization of atom: tij=0 gives multiplets!Solution with hoppings tij≠0 in solids is unknown!

Strong correlations in real systems

Local moments above Tc

Multiplets in solids Hub-I

E

“Real” U?

1 2 3 4 1 2 3 4| |m m m m eeU m m V m m=< >

A.L. and M. Katsnelson, PRB (1998)

Hubbard-I approximation

Disolving Multiplets in 3d-alkali system

PES: Fe on alkali metalsC. Carbone (Trieste), O. Rader (BESSY), et al DMFT imp.: 5-band in 1-bath

Functionals: MFT- DFT- DMFT

Weiss Mean-Field Theory (MFT) of classical magnetsKohn Density Functional Theory (DFT) of inhomogeneous electron gas in solids Dynamical Mean-Field Theory (DMFT) of strongly correlated electgron systems

G. Kotliar et. al. (2002), A. Georges (2004)

Many-body System

The Euclidian Action: x=(r ,τ,σ)

Functionals: general consideration

The one-electron Green's function

Introduction of the source (constraining field)

Functional derivative:

Functionals: Legendre Transformation

The Functionals of Green's function

The partition function Z

Self-Energy: Constraining field J=Σthe inverse of the exact Green's function

Baym-Kadanoff Functional

Exact representation of Φ

Different Functionals:

DFT: G=ρ J=V=Vh+VxcSDF: G=G(iω) J=Σloc(iω)BKF: G=G(k,iω) J=Σ(k,iω)

DFT-Density Functional TheoryInhomogeneous electron gas in solids ( U=e2/|r-r0| ):

Energy functional with constrained density <n (r) > =ρ (r)

Stationarity in λ insures that:

Construct a functional of ρ (r) only:

DFT: reference systemNon-interacting electrons in effective potential (t=-∇ 2/2):

Minimization with respect to λ (r):

Coupling constant trick:

DFT - Functional

where density-density correlations defined as:

Exact relations for density functional:

Local density approximation (LDA):

Exchange parameters and Functionals

Exchange interactions from DFTHeisenberg exchagne:

Magnetic torque:

Exchange interactions:

Spin wave spectrum:

M. Katsnelson and A. L., Phys. Rev. 61, 8906 (2000)

Non-collinear magnetism :

U/tChemical potential

U

t

∑∑ ↓↑+ +=

iiiji

ijij nnUcctH σσ

Hubbard model for correlated electrons

Dynamical Mean Field Theory

Σ Σ Σ

Σ Σ

Σ Σ Σ

ΣU

( )ττ ′−0G

A.Georges, G.Kotliar, W.Krauth and M.Rozenberg, Rev. Mod. Phys. 68, 13 (1996)G. Kotliar and D. Vollhardt, Physics Today 57, 53 (2004)

DMFT: Self-Consistent Set of Equations

( ) ( )∑Ω=

BZ

knn ikGiG

r

rωω ,ˆ1ˆ

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

QMC ED

DMRG IPTFLEX

( ) ( ) ( )nnnnew iGii ωωω 110

ˆˆˆ −− −=Σ G

Quantum Impurity Solver

Σ Σ Σ

Σ

Σ

Σ

ΣΣ

U

U

G( ’)τ−τ

ττ’

( ) ∑∑ ↓↑+ +⋅+−=

iiiji

ijijij nnUcctH σσμδ

METAL

INSULATOR

~U

LowerHB

UpperHB

Quasi-particle peak

Hans Bethe

Transition from paramagnetic metal to paramagnetic insulator on the Bethe lattice

DMFT solution for the Hubbard model

Ut

A.Georges, G.Kotliar, W.Krauth and M.Rozenberg, Rev. Mod. Phys. ‘96

From Atom to Solid

E

N(E)

EF

QPLHB UHB

E

N(E)

EF

Atomic physics Bands effects (LDA)

LDA+DMFT

E

N(E)

EF dndn+ 1| SL>

• Materials-specific (structure, Z, etc.)

• Fast code packages

• Fails for strong correlations

LDA+DMFT

( ) ( ) ( ) ( )[ ]1

0ˆˆˆ1ˆ

−

∑ Σ−−+Ω

=BZ

knnn ikHiIiG

r

rωωμω

LDA Models approaches

• Input parameters unknown

• Computationally expensive

• Systematic many-body scheme

( ) ( ) dcLDA UkHkH ˆˆˆ0 −=

rr

Multi band Quantum Monte Carlo

Flow diagram for the LDA+DMFT approach:

1,

1,

−− −=∑ locbath GG σσσ

[ ]∑ −− Σ−=κ

σσσ11

,, LDAloc GG

σσσ ∑+= −− 1,

1, locbath GG

Quantum impurity problemBand problem (LDA)

[ ]locGQMC ,: σ

DMFT self-consistency

Specific features of realistic-DMFT

Matrix form of multiorbital bath Green functions – G0mm’(ω)

General form of the electron-electron interactions – Umm’m’’m’’’

Screened (GW) Coulomb interaction - frequency dependent U(ω)

Accurate Wannier description of one-electron band structure

Complicated solution of LDA+DMFT Quantum-Impurity Model

Matrix form of Bath Green Function

eg

eg

gt

gt

gt

mm

GmGm

GmGm

GmmmmmmG

00000000000000000000

5

4

23

22

21

54321'

Simple case of d-orbital GF in the cubic lattice:

Genaral case of non-cubic lattice- Matrix form of G0mm’(ω)

Small cluster in DMFT – e.g. double Bethe lattice:

1 2

^11 12

21 22

( ) ( )( ) ( )

⎛ ⎞= ⎜ ⎟⎝ ⎠

G GG

G Gω ωω ω

Coulomb vertex

The Coulomb interaction matrix for t2g:

033226303225330224

220333223032223301

654321

3

2

1

3

2

1

321321

JUJUUJUJUmJUJUJUUJUmJUJUJUJUUm

UJUJUJUJUmJUUJUJUJUmJUJUUJUJUm

mmmmmmUij

−−−−−−−−−−−−

−−−−−−−−−−−−

↓

↓

↓

↑

↑

↑

↓↓↓↑↑↑

'2)1(/}2/)2()3)[(1({ UJUNNJUJUNNNUUav =−=−−+−−+=

Multi-band QMC-scheme (Hirsch-Fye)

)}(exp{21)]}(

21[exp{

''1

'''''

mmmmS

mmmmmmmmnnSnnnnU

mm

−=+−Δ− ∑±=

λτ

Discrete HS-transformation (Hirsch, 1983)

Number of Ising fields: N M(2M 1)= −m m ' m m '

1 2 1arccos h exp U _ N ote _ or2 L | U | W

⎡ ⎤ β⎛ ⎞λ = Δτ Δτ = <⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Green Functions:

'

1' '

( )

1 1' ' ' '

' ' ''

'

1( , ') ( , ', ) det

( , ', ) ( , ') ( )( ) ( )

1, '1, '

−

− −

= ×

= −

=

+ <⎧= ⎨− >⎩

∑

∑

m m

m m m mS

m m m m m m m

m m m m m m mm

m m

G G S GZ

G S VV S

m mm m

τ

ττ

τ τ τ τ

τ τ τ τ τ δ δ

τ λ τ σ

σ

G

' ''i

ij i j mm m mij mmH t c c U n nσ σσ

+= − +∑ ∑

U

m´

´mm´

τ

m

τ0

β

L Δ τ

Sign problem and DMFTDeterminant ratio in one-band model:

1

1

ˆdet[ ] 1 [1 ( , )](exp( 2 ) 1) 0ˆdet[ ]new

ii iGR G VGσ τ τ

−

−= = + − − − ≥

Green function in arbitrary Ising fields:' 'ˆ ˆ ˆ ˆ' (1 )( 1) ' [1 (1 )( 1)] [0 1]V V VG G G e G e−= + − − ≈ + − − = ÷G G G

τ

-G

0

1

-1

β− β00 ( ) 1< << <iiGτ β

τ

DMFT-SCFBethe lattice bath-Green function:

)()( 2nnn iGtii ωμωω −+=-1G

Energy

DO

S

EF

DMFT-SCFHilbert transform (perovskite lattice):

)()()(

)(

)()(

1nnn

nn

iiGi

iizz

dNzG

ωωω

ωμωεεε

Σ+=

Σ−+=−

=

−

∫

1-G

Energy

DOS

EF

t2g

DMFT-QMC calculation: Sr2RuO4

QMC for 3-orbitals:

15 auxiliary fields

300000 sweeps

128 imaginary times

T=15 meV

Max-Ent for DOS

LDA+DMFT, A. Liebsch and A.L., PRL(2000)

Spectral function –ARPES and DMFT

Van Hove=10 meVm*/m=2.1-2.6

ARPES (A. Damascelli, et al PRL2000)LDA+DMFT, A. Liebsch,et al PRL(2000)

SrSr22RuORuO44

LDA

DMFT

DMFT-SCFLDA+DMFT (orthogonal LMTO-TB):

)(ˆ),(ˆ)(ˆ)(ˆ

),()(

)()(),(1

''

''1

'

αωαω

ωω

ωμωω

α

+

∈∈

∈

−

−

∑∑

∑

=

=

Σ−−+=

UikGUiG

ikGiG

ikHiikG

IBZkn

On

BZknLLnLL

nDMFTLL

LDALLnnLL

h

Energy

DOS

EF

sp

d

ddddpdd

dpppp

dpss

kHΣ+

=Σ+HHH

HHHHHH

s

s

ss

)(ˆ)(ˆ ωr

Correlated d-states:

FBZ-integration

IBZ-integration+symmetrization

DMFT for a general latticeBath Green function for multiband case:

from the cavity construction:

Gij(0) is the Green function with eliminated 0-site:

Using the Fourier transform:

DMFT for a general lattice: GF GF in DMFT:

Here:

Taking into account that ∑t(k)=0 we have:

andUsing this formulas we obtained:

Finally for a bath GF we have:

A.L. and M. Katsnelson, PRB (1998)

LDA+DMFT: Local Dynamics

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′<εΣ>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)S. Biermann, A. Poteryaev, A. L., and A. Georges PRL 94, 026404 (2005)

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

CORRELATED D,F-STATES

G. Trimarchi et al. arXiv:0802.4435, JPCM (2008)B. Amadon et al. arXiv:0801.4353 , PRB (2008)

|L>

|G>

SCF-LDA+DMFT

F. Lechermann, et al, PRB (2007)

LDA+U: static mean-filed approximation

LDA+U functional:

One-electron energies: )n21(U

nE

iLDAii −+ε=

∂∂=ε

Occupied states: 2U1n LDAii −ε=ε⇒=

LDA i j d dij

U UE E n n - n (n -1)2 2

= + ∑

Empty states:2U0n LDAii +ε=ε⇒=

Mott-Hubbard

gap

d

LDAn

U∂ε∂≡

V. Anisimov et al, PRB, 44, 943 (1991), JPCM 9, 767 (1997)

LDAε

Full-potential LDA+U: a problem

ELDA+U= LDA + U - DC

= + -

= + - No

OK!

S. Dudarev et. al. PRB 57, 1505 (1998)

Spherical RI-LDA+U

Interchange –possible!A.L. et al. Springer Series in Materials Science. Volume. 54 (2003)

Static limit: LDA+URotationally invariant LDA+U functional

Local screend Coulomb correlations

LDA-double counting term (nσ =Tr(nmm0σ ) and n=n↑ +n⇓ ):

Occupation matrix for correlated electrons:

Slater parametrization of UMultipole expansion:

Coulomb matrix elements in Ylm basis:

Slater integrals:

Angular part – 3j symbols

Average interaction: U and JAverage Coulomb parameter:

Average Exchange parameter:

For d-electrons:Coulomb and exchange interactions:

• eg orbitals

• t2g orbitals

Mn (3+) = 3d4

5x3x

2x eg

t2g

Cubic Crystal field splitting

Spins ||Atomic Hunds rule

––

––

–– ––

––––

d

Orbital degrees of freedom

Charge transfer TMO insulators

Zaanen-Sawatzky-Allen(ZSA) phase diagram

Mott-Hubbard

Charge-Transfer

Eg

Eg~U

~ Δ

(WM+WL

)/2 Δ

Insulator

U

MW

NiOFeO

LaMnO3

V2O

3

TiO

V 2O5p-

met

al

d-metal

CuO

EFN

(E)

EW

U

Δ

dn-1

pL

n+1d

Orbital order: KCuF3

hole density of the same symmetry

A.L. et al, Phys. Rev.B 52, R5467 (1995);

In KCuF3 Cu+2 ion has d9 configuration

with a single hole in eg doubly degenerate subshell.

Experimental crystal structure

antiferro-orbital order

LDA+U calculations for undistortedperovskite structure

Electronic structure of TMO: LDA+U

0

4

8

12MnO

Den

sity

of S

tate

s (s

tate

s/eV

form

ula

unit)

LSDA

0

4

8U= 5eV

0

4

8U= 9eV

-12 -8 -4 0 4Energy (eV)

0

4

8U= 13eV

NiO

LSDA

U= 5eV

U= 9eV

-12 -8 -4 0 4 8Energy (eV)

U= 13eV0

100

200

300

400

w(q

), m

eV

G Z F G L

U =13LDA5

791113exp

DOS

Spin-waveSpectrum

NiOI. Solovyev

MnO NiO

O2p3d 3d

Constrained LDA calculation of U and J

Gunnarsson-1989 supercell with cutting hybridisation

Norman-1995 estimation of screening parameter

Constrain GW calculations of U

F. Aryasetiawanan et alPRB(2004)

Wannier - GW and effective U(ω)

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

C-GW

GW

Continuous Time QMC formalism: U(ω)

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 r

r 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 1

2 1 2 2 1

' ' ' ''0 ' 2 2 '' 'r r r r rr r

r r r r r r rS t w w dr dr c c drdrα += + +∫ ∫ ∫ ∫

( )( )1 2 1 1 2 2

1 2 1 1 2 2

' '' ' ' ' 1 1 2 2' 'r r r r r r

r 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

A. Rubtsov et al Phys Rev B 72, 035122 (2005)

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 1

1 2 1 1 2 2

...' ... ' ' ' ' '...k k k

k k k

r 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) www.ct-qmc.ru

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+ ….

CT-QMC: fast update k -> k+1

N2 operations

Metal-Insulator transition: Bethe lattice

0.0 0.5 1.0 1.5 2.0 2.5 3.0-7

-6

-5

-4

-3

-2

-1

0

iωiω

Σ(iω)

0.0 0.5 1.0 1.5 2.0 2.5 3.0-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

U=2

U=3

U=2

G(iω

)

U=3

-4 -2 0 2 40.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

DO

S

Energy

Density of states for β=64:U=2; U=2.2; U=2.4; U=3

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-2

-1

0

1

2

3

4

coexistence of the metallic and insulating solutions: U=2.4, β=64, W=2

iω

G(iω

)

Dynamical screening in Hubbard model

( ) ( ) ( )U U U Vτ δ τ τ→ = +

10 ( ) ( )G i iω ω μ −= +

2

2 2( )U U VωωΩ

= −+Ω

Compare to Exact solution

0 2 4 6 80.0

0.1

0.2

0.3

0.4

0.5

U=3, V=2, Ω=1, β=8

G(τ

)

τ

0 1 2 3 40.0

0.1

0.2

0.3

0.4

0.5

U=2, V=0.7, Ω=4, β=4

G(τ

)

τ

Co on Cu: 5d-orbitals CT-QMC calculation

DOS for Co atom in Cu

E. Gorelov et al, to be published

G(τ

) LDA

τ

CT-QMC

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

Advantages of the CT-QMC method

Number of auxiliary spinsin the Hirsch scheme

Short-range interactions Long-range interactions

Local in time interactions Non-local in time interactions

• non-local in time interactions: dynamical Coulomb screening

• non-local in space interactions: multi-band systems, E-DMFT

Auxiliary field (Hirsch) algorithm is time-consuming since it’s necessary to introduce large number of auxiliary fields, while

CT-QMC scheme needs almost the same time as in local case

Miracle of continuous time QMC

Weak coupling expansion (A. Rubtsov et al)

Strong coupling expansion (P. Werner et al)

Path Integral:

Comparison of different CT-QMC

Σ Σ Σ

Σ

Σ

Σ

ΣΣ

U

U

G( ’)τ−τ

ττ’

Double counting in LDA+DMFT

Analytic modelsAround mean fieldFully localized limit

Constraint on particle number

Constraint on self-energyLDA

!

0!

Tr Tr

Tr Tr

GG

GG

=

=

( )

( ) 0 ReTr

00 ReTr !

!

=∞

=

Σ

Σ

Choice of double counting in LDA+DMFTShift of chemical potential for correlated state

Natural choice :

1( , ) ( ) ( ) [ ] [ ( ) ]LDA dc c cG k i H k Eω ω μ δμ ω δμ→ →

− = + − + − − Σ −

1( , ) ( ) ( ) ( )cLDAG k i H kω ω μ ω→ →

− = + − −ΣTransformations:

11

0 01

10( ) ( )

c c

c cc c

G G

G G

δμ

ω ω δμ

−−

−−

= −

Σ = − = Σ −Condition for (Friedel SR)

0[ ] [ ]Tr G Tr G=0.0 0.5 1.0

-1.0

-0.5

0.0

GG0

G(τ

)

τ

Ni Ferro Eg-up

dc cE δμ=

cδμ

DC-test for LDA+DMFT: NiO

LDA part includes entire Hilbert space

Coulomb interaction acting on correlatedsubspace only

Double counting

∑ +=k

kkk ccH εLDA

'int ij i j '

ij

1 1 1H U n n2 2 2

σσσ σ

σ

⎛ ⎞⎛ ⎞= − −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

∑

∑−=σ

σμi

inH DDC

NiO – a charge transfer system

LDA band structure (paramagnetic)

Ni-3d orbitals as correlated subspaceO-2d orbitals as uncorrelated subspace

NiO – double counting

Total particle number (color encoded) as function of chemical potential μ and double counting μDC

(eV)

(eV)

Peak positions and spectral weights I

Fitting of Green functions to DOS with 3 δ-peaksat εi with spectral weight Zi

( ) ( ) ( )[ ]∑ −Θ−= −

iii neZG i εττ τε

F

Peak positions and spectral weights II

Energy of peaks in spectral function of Ni-3d and O-2p orbitals as function of double counting. Line thicknesses correspond spectral weight of each peak.

( ) 0 ReTr !=∞Σ

FLLAMF LDA!

0!

Tr Tr

Tr Tr

GG

GG

=

=

Spectral functions and double counting

Mott insulator

Charge transfer insulator

Almost metallic

eV21DC =μ

eV26DC ≥μ

Itinerant ferromagnetism

Stoner

T=0

T<Tc

T>Tc

Heisenberg Spin-fluctuation

Magnetism of metals: LDA+DMFT

A. L., M. Katsnelson and G. Kotliar, PRL 87, 067205 (2001)

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,20,0

0,2

0,4

0,6

0,8

1,0

1,2

0,0

0,2

0,4

0,6

0,8

1,0

1,2

χ(T)M(T)

M(T) and χ(T): LDA+DMFT

Ni

Fe

χ-1M

eff2 /3

T c

M(T

)/M(0

)

T/Tc

Global spin flip

Exchange interactions in metalsFinite temperature 3d-metal magnetism

Ferromagnetism of transiton metals

-8 -6 -4 -2 0 20.0

0.5

1.0

1.5

2.0

2.5

LDA

DMFT

PES

Ni: LDA+DMFT (T=0.9 Tc)

EF

Den

sity

of s

tate

s, e

V-1

Energy, eV

0 2 4 60.0

0.5

1.0

1.5

τ, eV-1

<S(τ)

S(0)

>

Ferromagnetic Ni DMFT vs. LSDA: • 30% band narrowing• 50% spin-splitting reduction• -6 eV satteliteLDA+DMFT with ME

J. Braun, et alPRL (2006)

A. L, M. Katsnelson and G. Kotliar, PRL (2001)

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,20,0

0,2

0,4

0,6

0,8

1,0

1,2

0,0

0,2

0,4

0,6

0,8

1,0

1,2

χ(T)M(T)

M(T) and χ(T): LDA+DMFT

Ni

Fe

χ-1M

eff2 /3

T c

M(T

)/M(0

)

T/Tc

Ferromagnetic Iron: Spectral Function

Realistic DMFT for Co(111) surfaceFirst model surface DMFT – M.Potthoff (1999)

FPLMTO+DMFT: Igor di Marco PRB (2007)

Orthorhombic 3dOrthorhombic 3d11 PerovskitesPerovskitesSrSrVVOO33 CaCaVVOO33

LaLaTiTiOO33 YYTiTiOO33

Metal

m*/m=2.7

Metal

m*/m=3.6

Insulator

Gap=0.2eV

Insulator

Gap=1.0eV

Crystal-field splittings w/in t2g multiplet:(140,200) meV for LaTiO3 ; (200,330) meV for YTiO3

LDA-NMTO results: DOS

all metallicin LDA

YTiO3

Violet: OxygenOrange: M

Orbital ordering in MTiO3

Occup. LDA DMFT

LaTiO3 0.45 0.88

YTiO3 0.50 0.96

La

Y

E. Pavarini et al. PRL (2004)

LDALDA

+ + DMFTDMFT

m*/m=2.2 m*/m=3.5

1.00.2

LDALDA++DMFT: DMFT: comparison with comparison with experimentsexperiments

Exp=2.7 Exp=3.6

SrVOSrVO33 CaVOCaVO33

LaTiOLaTiO33 YTiOYTiO33

Exp Exp

Conclusions

LDA+DMFT is a perfect scheme for realistic description of electronic structure of correlated electron materials

Matrix DMFT formalism can be used for general lattice or cluster compounds