LDA+U and beyondA. Lichtenstein

University of Hamburg

In collaborations with:

V. Anisimov (Ekaterinburg)M. Katsnelson (Nijmegen)

Outline

• Complexity of Transition Metal Systems

• LDA+U: spin-charge-orbital ordering

• LDA+DMFT: dynamical effects

• Conclusions

From Atom to Solids

Theory: interactions vs. hopping

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 f-systems

Multiplets in solids: Hubbard-I1 2 3 4 1 2 3 4| |m m m m eeU m m V m m=< >

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

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

DFT: Computational Material Science

CERAN-plate

LDA-modeling:LiAlSiO4

A.L. & R. Jones

Strong correlations in real system

Local moments above Tc

Multiplets in solids

E

“Real” U?

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

Correlation driven MIT

U/W

photoemission spectra (DOS)A. Fujimori et al.

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

• eg orbitals

• t2g orbitals

Mn (3+) = 3d4

5x3x

2x eg

t2g

3d-ion in cubic crystal field

d

Orbital degrees of freedom

Model Hamiltonians

Hubbard and Anderson models unknown parameters many-body explicit Coulomb correlations

Density Functional Theory

LDA, GGAab-initioone-electron averaged Coulomb interaction

LDA++

Coulomb correlations problem

combined LDA+U and LDA+DMFT approaches(GW, TD-DFT are alternative ways)

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, PRB, 44, 943 (1991)

LDAε

Rotationally invariant LDA+ULDA+U functional

Local screend Coulomb correlations

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

Occupation matrix for correlated electrons:

A. I. Lichtenstein, J. Zaanen, and V. I. Anisimov, PRB 52, R5467 (1995)

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:

Constrained LDA calculation of U and J

Gunnarsson-1989 supercell with cutting hybridisation

Norman-1995 estimation of screening parameter

Full-potential LDA+U: a problem

ELDA+U= LDA + U - DC

= + -

= + - No

OK!

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

Spherical RI-LDA+U

Interchange –possible!

Exchange interaction couplings

Calculation of J from LDA+U results:↓

′↑

′′′′′′′′′′′′′′ −≡χ= ∑ imm

imm

imm

jmm

ijmmmm

}m{

immij VVIIIJ

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

mjlk'n

'ilmk'n

mjlnk

ilmnk

'knn k'nnk

k'nnkijmmmm cccc

ff ′′′

↓↓∗′′

↑∗↑

↓↑

↑↓′′′′′′ ∑ ε−ε

−=χ

ji

2

ijij

jiij

EJSSJEθ∂θ∂

∂== ∑

rr

Heisenberg Hamiltonian parameters:

LDA+U eigenvaluesand eigenfunctions: ∑ >=Ψε σσσ

ilm

ilmnknknk ilm|c;

Exchange interactions from LDA++Heisenberg exchagne:

Magnetic torque:

Exchange interactions:

Spin wave spectrum:

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

Non-collinear magnetism :

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

Orbital order: KCuF3

hole density of the same symmetry

A.Lichtenstein 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

Cooperative Jahn-Teller distortions in KCuF3

Quadrupolar distortion in KCuF3

Superexchange interaction

J(K) Jc JabTheory -240 +6Exp. -202 +3

KCuF3

LSDA gave cubic perovskite crystal structure stable in respect to Jahn-Teller distortion of CuF6 octahedra

Only LDA+U produces total energy minimum for distorted structure

Mechanism: OO – Kugel-Khomskii

Spin and Orbital moments in CoO

LDA+U+SO+non-collinear

L

L

S

I. Solovyev, A. L, and K. Terakura, PRL 80, 5758 (1998)

CoPt: LDA+U calculation of MAE

A. Shik, O. Mryasov, PRB (2003)

LDA+U : Forces and Orbital OrderingAFM FMLaTiOLaTiO33 YTiOYTiO33

S. Okatov, et. al.Europhys. Lett. (2004)

LDA and charge order problem

;nU)nn(;nU)nn(;dndU 00

LSDA200

LSDA1 δ+ε=δ+εδ−ε=δ−ε

ε=

Charge disproportionation in LSDA is unstable due to self-interaction problem

in LDA+U self-interaction is explicitly canceled

)n21(U))nn(

21(U)nn(

)n21(U))nn(

21(U)nn(

0000LSDA22

0000LSDA11

−−ε=δ+−+δ+ε=ε

−−ε=δ−−+δ−ε=ε

Charge order in Fe3O4

half of the octahedral positions is occupied by Fe+3 and other half by Fe+2.

V.Anisimov et al, Phys. Rev.B 54, 4387 (1996)

Fe3O4 has spinelcrystal structure

one Fe+3 ion in tetrahedral position (A)

two Fe+2.5 ions in octahedral positions (B)

Below TV=122K a charge ordering happens Verwey transition

Simultaneous metal-insulator transition:

LDA+U: charge ordering in Fe3O4

Charge and orbital order in experimental low-temperature monoclinic crystal structure Fe3O4

ΔQ=0.1 eΔQm=0.7 e

I.Leonov et al, PRL93,146404 (2004)

CaV2O5 and MgV2O5 CaV3O7 CaV4O9

V.Anisimov et al, Phys.Rev.Lett. 83, 1387 (1999)

Exchange interactions in layered vanadates

n=3: CaV3O7 has unusual long-range spin ordern=4: CaV4O9 is a frustrated (plaquets) system with a spin gap value 107K n=2: CaV2O5 is a set of weakly coupled dimerswith a large spin gap 616 K isostructural MgV2O5 has very small spin gap value < 10K

CaVnO2n+1

QMC solution of Heisenberg model

M. Troyer et. al.: Comparison of the calculated and measured susceptibility

LDA+U in fully-localized limit (LDA+U-FLL)

LDA+U in around mean-field limit (LDA+U-AMF)

( ) ( )( )↓↓↓↑↓↑↓↑↓↑↓↑ −−+−+= nnnnUnnnnnnUnnU mmmmmm '''

“mean-field” = LDA LDA+U

[ ]431

4321

21 24314231 ,,,,21

γγγγγγ

γγ δγγγγγγγγδ nUUnE AFM −= ∑

∑−=+

=−=l

lmmmn

lnnnn σσσ

γγσ

γγ δδγγ ,12

1 , 21

1

2121

Around mean-field limit of LDA+U+SO

m ,γ = σ Spin-orbitals

General LDA+U formulation22

2( )2 2 2

−⎡ ⎤= − − +⎢ ⎥⎣ ⎦∑ ∑U

JnUn U JH S n P nσσ σ σ σ

σ σ

AMF: 1/(2 1), 0 SIC: 0, 1/ 2

S l PS P= + == =

FLL is the right “DFT” mean field for localized systems, nmσ= 1 or 0

AMF is the right “DFT” mean field for for uniform occupancy, nmσ= <nσ>

2 2/(2 1) mm

n l n nσ σ σ+ ≤ ≤∑Generalization: (2l+1)Sσ+Pσ=1

A. Petukhov, et. al., PRB 67, 153106 (2003).

Electronic structure of δ-Pu in AMF LDA+U

Pu f-band configuration in AFM LDA+U is close to f6: f5/2 states are filled and 8 states f7/2 are empty

A. Shick et al. Europhys. Lett. 69, 588 (2005)

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>

Dynamical Mean Field Theory

Σ Σ Σ

Σ Σ

Σ Σ Σ

ΣU

( )ττ ′−0G

W. Metzner and D. Vollhardt PRL(1989)A. Georges and G. Kotliar PRB(1992)Europhysics Prize (2006)

DMFT: Self-Consistent Set of Equations

( ) ∑→

⎟⎠⎞

⎜⎝⎛

Ω=

→BZ

k

nn ikGiG ωω ,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’

Local Dynamics: 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′<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. Lichtenstein, et. al. PRB, 57, 6884 (1998)

A. Poteryaev, A. Lichtenstein, and G. Kotliar, PRL 93, 086401 (2004)S. Biermann, A. Poteryaev, A. I. Lichtenstein, and A. Georges

Phys. Rev. Lett. 94, 026404 (2005)

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

• Fast code packages

• Fails for strong correlations

LDA+DMFT

( ) ( ) ( )1

0ˆˆˆˆ

−→

∑ ⎥⎦

⎤⎢⎣

⎡ Σ−⎟⎠⎞

⎜⎝⎛−+=

BZ

knnn ikHIiiG ωμωω

LDA Models approaches

• Input parameters unknown

• Computationally expensive

• Systematic many-body scheme

dcLDA EkHkH −⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛ →→ ˆˆ

0

Multi band Quantum Monte Carlo

Multiorbital CT-QMC: general U-vertex

-4 -2 0 2 40.0

0.2

0.4

0.6

W=2U=2J=0.2

DO

S (1

/eV)

Energy (eV)

5 orbitals, full U-vertex

τ

G(τ) Uijkl

Udiag

E. Gorelov, et. al. to be published A. Rubtsov and A.L., JETP Lett. (2004)

Spectral function –ARPES and DMFT

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

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

SrSr22RuORuO44

before renormalization

after renormalization

Conclusions

LDA+U is an accurate scheme for realistic description of electronic structure, spin, orbital and charge ordering in complex transition metal systems

LDA+DMFT method is useful for dynamical, short-range non-local Coulomb correlations effects in solids: metal-insulator transition