from lda+u to lda+dmft

University of California DavisKashiwa, July 27, 2007

From LDA+U to LDA+DMFTFrom LDA+U to LDA+DMFT

S. Y. Savrasov,Department of Physics, University of California, Davis

Collaborators:

Q. Yin, X. Wan, A. Gordienko (UC Davis)G. Kotliar, K Haule (Rutgers)


ContentContent

From LDA+U to LDA+DMFT

Extension I: LDA+Hubbard 1 Approximation

Extension II: LDA+Cluster Exact Diagonalization

Applications to Magnons Spectra


T U U

T

( )N E( )N E

U

k kt U

k kt

1ˆ ˆ ˆ ˆ ˆ2ij i j i i

ij i

H t c c Un n

Idea of LDA+U is borrowed from

the Hubbard Hamiltonian:

LDA+ULDA+U

T

T UU

Spectrum of Antiferromagnet at half fillingSpectrum of Antiferromagnet at half filling

FE


LDA+U Orbital Dependent PotentialLDA+U Orbital Dependent Potential

ˆ ˆ ˆ( ) [ ( 1/ 2)]LDA U LDA dm m dmV V r P U n P

The correction acts on the correlated orbitals only:

ˆ ˆ[ ( 1/ 2)]dm m dmP U n P | |

' ' '' '

ˆ| ( ) ( )l l ml m

r Y r

{ }ˆ ˆ( ) ( ) ( ) ( )dm d dm dmSP r r Y r r dV

The Schroedinger’s equation for the electron is solved

2 ˆ( )LDA U kj kj kjV

with orbital dependent potential

( ) ( )kj r A r

When forming Hamiltonian matrix 2 ˆ| |LDA UH V

EF

( )N E( )N E

Up

d

d


Main problem with LDA+UMain problem with LDA+U

LDA+U is capable to recover insulating behavior in magnetically ordered state.

However, systems like NiO are insulators both above and below the Neel temperature. Magnetically disordered state is not described by LDA+U

Another example is 4f materials which show atomic multiplet structure reflecting atomic character of 4f states. Late actinides (5f’s)show similar behavior.

Atomic limit cannot be recovered by LDA+U because LDA+U correction is the Hartree-Fock approximation for atomic self-energy, not the actual self-energy of the electron!


NiO: Comparison with PhotoemissionNiO: Comparison with Photoemission

LSDALDA+UParamagnetic LDA

Electronic Configuration: Ni 2+O2- d8p6 (T2g6Eg

2p6)


LDA+DMFT as natural extension of LDA+ULDA+DMFT as natural extension of LDA+U

In LDA+U correction to the potential

,( )LDA U atomic DC atomic HF DCV V V

is just the Hartree-Fock value of the exact atomic self energy.

Why don’t use exact atomic self-energy itself instead of its Hartree-Fock value? This is so called Hubbard I approximation to the electronic self-energy.

Next step: use self-energy from atom allowing to hybridize with conduction bath, i.e. finding it from the Anderson impurity problem.

Impose self-consistency for the bath: full dynamical mean field theory is recovered.

LDA+U LDA+ ,atomic HF LDA+ ( )atomic

LDA+ ( )atomic LDA+ ( )inpurity

LDA+ ( )inpurity LDA+DMFT


Localized electrons: LDA+DMFTLocalized electrons: LDA+DMFT

2 ˆ ˆ[ ( ) ( ( ) ) ] ( ) ( )LDA d imp DC d kj kj kjV r P V P r r

Electronic structure is composed from LDA Hamiltonian for sp(d) electrons

and dynamical self-energy for (d)f-electrons extracted from solving Anderson impurity model

Poles of the Green function 1( , )

kj

G k

have information about atomic multiplets, Kondo, Zhang-Rice singlets, etc.

N(N())

ddnn->d->dn+1n+1

Better description compared to LDA or LDA+U is obtained

ddnn->d->dn-1n-1

21ˆˆ ( ) [ ( )] kd

imp d impk k

VG


Exact Diagonalization MethodsExact Diagonalization Methods

[ . .]imp d k k k kd kk k

H d d Ud d dd c c V c d c c

For capturing physics of localized electrons combination of LDA and exact diagonalization methods can be utilized:

| |impH E

kd

kdV

The cluster Hamiltonian is exact diagonalized

' '

0

0 | | | | 0( )

( )m m

imp

d dG

E E

The Green function is calculated:

Self-energy is extracted:2

1ˆˆ ( ) [ ( )] kdimp d imp

k k

VG


Exact Diagonalization MethodsExact Diagonalization Methods

In the limit of small hybridization Vkd=0 this is reduced to calculating atomic d(f)-shell self-energy: Hubbard I approximation (Hubbard, 1961). IfHatree Fock estimate is used here, LDA+U method is recovered.

Corrections due to finite hybridization can be alternatively evaluated using QMC, or NCA, OCA, SUNCA approximations (K. Haule, 2003)

21ˆˆ ( ) [ ( )] kd

imp d impk k

VG


Excitations in AtomsExcitations in Atoms

nd

1( 1)

2n dE n Un n

1nd

1

1( 1) ( 1)( 2)

2n dE n U n n

1nd

1

1( 1) ( 1)

2n dE n U n n

Electron Removal Spectrum

1 / 2n n dE E U Electron Addition Spectrum

1 / 2n n dE E U

/ 2d U / 2d U

Im ( )G


Atomic Self-Energies have singularitiesAtomic Self-Energies have singularities

2

1 1( )

( )4( )

dd

d

GU

Self-energy with a pole is required:

Ground state energies for configurations dn, dn+1, dn-1 give rise to electron removal En-En-1 and electron addition En-En+1 spectra. Atoms are always insulators!

1 / 2n n dE E U 1 / 2n n dE E U

1/ 2 1/ 2( )

/ 2 / 2atomicd d

GU U

or two poles in one-electron Green function

Electron removalElectron addition

Coulomb gap

This is missing in DFT effective potential or LDA+U orbital dependent potential


Mott Insulators as Systems near Atomic LimitMott Insulators as Systems near Atomic Limit

Classical systems: MnO (d5), FeO (d6), CoO (d7), NiO (d8). Neel temperatures 100-500K. Remain insulating both below and above TN

1( )LDA

d LDA

GV

1 2

1 1( )

( ) ( )( )

4( )

1/ 2 1/ 2

( ) / 2 ( ) / 2

Hubbardd

d

G kUk

k

k U k U

Frequency dependence in self-energy is required:

LDA/LDA+U, other static mean field theories, cannot access paramagnetic insulating state.


Electronic Structure calculation with LDA+Hub1Electronic Structure calculation with LDA+Hub1

1( )

( )

ss ps ds

LDA Hub sp pp dp

sd pd dd Atomic DC

H H H

H H H H

H H H V

LDA+Hubbard 1 Hamiltonian is diagonalized

Green Function is calculated2

1

| ( ) |( )

( )

kj

LDA Hubj kj

AG k

Density of states can be visualized

1

1( ) Im ( )LDA Hub

k

N G k


LDA+Hub1 Densities of States for NiO and MnOLDA+Hub1 Densities of States for NiO and MnO

Results of LDA+Hubbard 1 calculation: paramagnetic insulating state is recovered

LHB UHB

LHB UHB

U

U

Dielectric Gap

Dielectric Gap


NiO: Comparison with Photoemission NiO: Comparison with Photoemission

Insulator is recovered, satellite is recovered as lower Hubbard bandLow energy feature due to d electrons is not recovered!

LHB UHBU

Dielectric Gap


Americium PuzzleAmericium Puzzle

Density functional based electronic structure calculations: Non magnetic LDA/GGA predicts volume 50% off. Magnetic GGA corrects most of error in volume but gives m~6B

(Soderlind et.al., PRB 2000). Experimentally, Am has non magnetic f6 ground state with J=0 (7F0)

Experimental Equation of State (after Heathman et.al, PRL 2000)

Mott Transition?“Soft”

“Hard”


Photoemission in Am, Pu, SmPhotoemission in Am, Pu, Sm

after J. R. Naegele, Phys. Rev. Lett. (1984).

Atomic multiplet structure emerges from measured photoemission spectra in Am (5f6), Sm(4f6) - Signature for f electrons localization.


Am Equation of State: LDA+Hub1 PredictionsAm Equation of State: LDA+Hub1 Predictions

LDA+Hub1 predictions: Non magnetic f6 ground

state with J=0 (7F0) Equilibrium Volume: Vtheory/Vexp=0.93

Bulk Modulus: Btheory=47 GPa

Experimentally B=40-45 GPa

Theoretical P(V) using LDA+Hub1

Self-consistent evaluations of total energies with LDA+DMFT using matrix Hubbard I method.

Accounting for full atomic multiplet structure using Slater integrals:F(0)=4.5 eV, F(2)=8 eV, F(4)=5.4 eV, F(6)=4 eVNew algorithms allow studies of complex structures.

Predictions for Am II

Predictions for Am IV

Predictions for Am III

Predictions for Am I


Atomic Multiplets in AmericiumAtomic Multiplets in AmericiumLDA+Hub1 Density of States

Exact Diag. for atomic shell

F(0)=4.5 eV F(2)=8.0 eV F(4)=5.4 eV F(6)=4.0 eV

Matrix Hubbard I Method

F(0)=4.5 eV F(2)=8.0 eVF(4)=5.4 eV F(6)=4.0 eV


Many Body Electronic Structure for Many Body Electronic Structure for 77FF00 Americium Americium

Experimental Photoemission Spectrum after J. Naegele et.al, PRL 1984

Insights from LDA+DMFT: Under pressure energies of f6 and f7 states become degenerate which drives Americium into mixed valence regime.Explains anomalous growth in resistivity, confirms ideas pushed forward recently by Griveau, Rebizant, Lander, Kotliar, (2005)


Effective (DFT-like) single particle spectrumalways consists of delta like peaks

Real excitational spectrumcan be quite different

Bringing k-resolution to atomic multipletsBringing k-resolution to atomic multiplets

0[ ( ) ( )] ( , ) 1H k G k


Many Body Calculations with speed of LDA Many Body Calculations with speed of LDA

( ) ( ) i

i i

W

P

(Savrasov, Haule, Kotliar, PRL2006)

0[ ( )] 0kjH

Non-linear over energy Dyson equation

with pole representation for self energy

(1)0 1 2

(2)1 1

(3)

2 2

( )

00

0

kj

kj

kj

H W W

W P

W P

is exactly reduced to linear set of equations in the extended space


Many Body Electronic Structure MethodMany Body Electronic Structure Method

11

0 1 2 0

1 1

2 2

( ) [ ( ) ]

0

0

i

i i

WH W W H

PW P

W P

The proof lies in mathematical identityGreen function G(r,r’,)

Physical part of the electron is described by the first component of the vector(1)

(2)

(3)

kj

kj

kj

2| |( )

kj

j kj

AG k

Electronic Green function is non interacting like but with more poles:


Many Body Electronic Structure for Many Body Electronic Structure for 77FF00 Americium Americium


Cluster Exact DiagonalizationCluster Exact Diagonalization

[ . .]imp d k k k kd kk k

H d d Ud d dd c c V c d c c

Cluster Hamiltonian

| |impH E

kd

kdV

The cluster Hamiltonian is exact diagonalized

' '

0

0 | | | | 0( )

( )m m

imp

d dG

E E

The Green function is calculated:

Self-energy is extracted:2

1ˆˆ ( ) [ ( )] kdimp d imp

k k

VG


Electronic Structure calculation with LDA+CEDElectronic Structure calculation with LDA+CED

( )

( )

ss ps ds

LDA sp pp dp

sd pd dd d DC

H H H

H H H H

H H H V

LDA+ Hamiltonian is diagonalized

Green Function is calculated2| ( ) |

( )( )

kj

LDAj kj

AG k

Density of states can be visualized

1( ) Im ( )LDA

k

N G k


Eg’s

Illustration: Many Body Bands for NiOIllustration: Many Body Bands for NiO

LDA+cluster exact diagonaization for NiO above TN: 8 6 2 6

2 2[ ]g g pd t e o

8 7d d

8 9d d

6 5o o

(atomic like LHB)

(atomic like UHB)

O-hole coupledto local d moment(Zhang-Rice like)


Generalized Zhang-Rice PhysicsGeneralized Zhang-Rice Physics

NiO(d8) CoO(d7) FeO(d6) MnO(d5)

SNi=1

SO=1/2

SO=1/2 SCu=1/2

CuO2(d9)

Zhang-Rice Singlet (Stot=0)

Doublet (Stot=1/2) Triplet (Stot=1) Quartet (Stot=3/2) Quintet (Stot=2)

SO=1/2 SMn=5/2SO=1/2 SFe=2SO=1/2 SCo=3/2

JAF


NiO: LDA+CED compared with ARPESNiO: LDA+CED compared with ARPES

Dispersion of doublet


CoO: LDA+CED compared with ARPESCoO: LDA+CED compared with ARPES

Dispersion of triplet


LDA+CED for HTSCs: Dispersion of Zhang-Rice singletLDA+CED for HTSCs: Dispersion of Zhang-Rice singlet


Magnons, Exchange Interactions, Tc’sMagnons, Exchange Interactions, Tc’s

Realistic treatment of magnetic exchange interactionsin strongly-correlated systems:

• Spin waves, magnetic ordering temperatures

• Necessary input to Heisenberg, Kondo Hamiltonians

• Spin-phonon interactions, incommensurability, magnetoferroelectricity


Magnetic Force TheoremMagnetic Force Theorem

0'

'

( , ')RR LDAR R

B BJ r r

Exchange Constants via Linear Response (Lichtenstein et. al, 1987)

Spin Wave spectra, Curie temperatures, Spin Dynamics (Antropov et.al, 1995)

After Halilov, et. al, 1998

Fe


Exchange Constants, Spin Waves, Neel Tc’sExchange Constants, Spin Waves, Neel Tc’s

Magnetic force theorem for DMFT has been recently discussed (Katsnelson, Lichtenstein, PRB 2000)

Using rational representation for self-energy, magnetic force theorem can be simplified (X. Wan, Q. Yin, SS, PRL 2006)

Expression for exchange constants looks similar to DFT

However, eigenstates which describe Hubbard bands, quasiparticle bands, multiplet transitions, etc. appear here.

[( ) ( )] [ ]DMFTJ Tr GG Tr BGGB


Spin Wave Spectrum in NiOSpin Wave Spectrum in NiO


Calculated Neel Temperatures Calculated Neel Temperatures


ConclusionConclusion

There are natural extensions of LDA+U method:

LDA+Hubbard 1 is a method where full frequency dependent atomic self-energy is used

LDA+ED is a method where self-energy is extracted from cluster calculations

LDA+DMFT is a general method where correlated orbitals are treated with full frequency resolution

Many new phenomena (atomic multiplets, mixed valence, Kondo effect) canbe studied with the electronic structure calculations

from lda+u to lda+dmft

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