Localized Electrons with Wien2kLDA+U, EECE, MLWF, DMFT

Elias AssmannVienna University of Technology, Institute for Solid State Physics

WIEN2013@PSU, Aug 14

LDA vs. Localized Electrons

❼ LDA/GGA has problems withstrongly correlatedsystems❼ localizedorbitals (3 d, 4f )

~ explicitly add“Coulomb interaction”

1 LDA+U, EECE: addorbital-dependent potentialsto LDA❼ insulators

2 DMFT: build many-body theoryon top of LDA❼ correlated metals❼ partially-filled bands❼ Wannier functionsas basis

The Hubbard Model

t

U

single-band Hubbard model:.

⟨i j ⟩σ

+iσ jσ iH = − t c c + U

. ni

ni

ˆ ↑ ˆ ↓

❼ “hopping” t (kinetic energy)❼ interaction U

The Hubbard Model

t

U

single-band Hubbard model:.

⟨i j ⟩σ

+iσ jσ iH = − t c c + U

. ni

ni

ˆ ↑ ˆ ↓

❼ “hopping” t (kinetic energy)❼ interaction U

multi-bandgeneralization:

U nˆ↑nˆ↓ →

.

ijkl

+ +Uijklci cj clck

often parametrized with: intra-orbital U, inter-orbital V, Hund exchange J

Strontium Vanadate SrVO3

❼ correlated metal❼ cubic perovskite

❼ VO octahedra

62g❼ isolated t1 manifold

-5 0 5 10

DO

S

ω [eV]

Outline

1 Preliminaries

2 LDA+U, EECE — Orbital-Dependent Potentials LDA + Coulomb Repulsion U

On-Site Hybrid/Hartree-Fock

3 Wannier Functions and DMFT

LDA + Coulomb Repulsion U (-orb)

❼ split states into❼ “delocalized” ~ LDA❼ “localized” ~ LDA+U (usually, d or f states)

❼ augment LDA with local orbital-dependent energy:

LDA 12.

i/=j

E = E + U n n

i j12− UN(N −

1)

mean-field Hubbardterm (multi-band)

nˆi → ni = ⟨nˆi ⟩

double-countingcorrection (LDA contains part of U)

[Anisimov et al., Phys. Rev. B 48, 16929 (1993)]

LDA+U: Effects

LDA 12.

i/=j

i j12E = E + U n n − UN(N −

1)

∂E

∂ni

LDAi

1 i❼ orbital energies ε i = = ε + U ( /2 − n ) ⇒ unoccupied states ↑, occupied states ↓

❼ creates / enlargesgaps( ~ Mott insulators)

❼ breaks symmetries( ~ spin order, orbital order, . . . )

LDA+U: Practicalities

❼ conceptuallysimple, computationallycheap

❼ ambiguitiesin practice:

[Madsen and Novák, wien2k.at]❼ U values?

~ constrained LDA❼ enforce occupation of target orbital❼ U ∼ ∆Etot

❼ double-countingcorrection12 ¯0“around mean-field”, UN(N −

n)[Czyz yk and Sawatzky, PRB 1994]

metallic or less strongly correlated121“self-interaction correction”, UN(N −

1)[Anisimov et al., PRB 1993]

strongly correlated systems 2“HMF” [Anisimov et al., PRB 1991]

LDA+U: Program Flow

❼ must be spin-polarized

lapwdmdensity matrix nij = ⟨nˆinˆj ⟩

(case.dmatup,dn)orbLDA+ U potential

(case.vorbup,dn)lapw1 -orbincludes LDA+ U potential

runsp -orb

up,dn

up,dn-up,dn up,dn-up,dn

lapw0 orb-lapw1-orb- lapw2 lapwdm- lcore mixer-orb

LDA+U: Program Flow

❼ must be spin-polarized runsp -orb

up,dn

up,dn-up,dn up,dn-up,dn

lapw0 orb-lapw1-orb- lapw2 lapwdm- lcore mixer-orb

-9. Emin [Ry]1 #atoms2 1 2 atom, #l, l0 0 mode

lapwdmdensity matrix nij = ⟨nˆinˆj ⟩(case.dmatup,dn)

orbLDA+ U potential (case.vorbup,dn)

lapw1 -orbincludes LDA+ U potential

case.indm[c]case.inorb

110mode,#atoms,iprPRATT 1.0

2 1 21

0.26 0.00

mixing atom, #l, ldouble-counting U,J [Ry] (Ueff=U-J)

LDA+U: SrVO3

-5 5 10

DOS

0ω [eV]

LDA

LDA+Ut2g

On-Site Hybrid Functionals (-eece)

❼ split states as in LDA+U → ρse l , Ψsel

❼ What’s in a name?❼ “exact exchange for correlated electrons”❼ on-site hybrid/Hartree-Fock vs. full hybrid

❼ augment LDA with local orbital-dependent energy:❼ on-site hybrid

E = ELDA[ρ] + α.EHF[Ψse l] − ELDA[ρsel])

x

❼ on-site Hartree-Fockx x

E = ELDA[ρ] +

α(EHF[Ψsel ] − ELDA[

xcρ sel])

❼ must choose α; Exc[ρ] is not linear[Novák et al. Phys. Stat. Sol. B 243, 563 (2006)]

[Tran, Blaha, Schwarz, and Novák, Phys. Rev. B 74, 155108 (2006)]

EECE: Program Flow

❼ must be spin-polarized

lapwdmdensity matrix nij = ⟨nˆinˆj ⟩

(case.dmatup,dn)lapw2, lapw0, orbEECEpotential

(case.vorbup,dn)lapw1 -orbincludes EECE potential

runsp -eece

-up,dn-up,dn-up,dn-up,dn-up,dn

-up,dn

lapw0 lapw1-orb lapw2 lcore lapwdm lapw2-eece lapw0-eece orbmixer-eece

EECE: Program Flow

❼ must be spin-polarized

lapwdmdensity matrix nij = ⟨nˆinˆj ⟩

(case.dmatup,dn)lapw2, lapw0, orbEECEpotential

(case.vorbup,dn)lapw1 -orbincludes EECE potential

runsp -eece

-up,dn-up,dn-up,dn-up,dn-up,dn

-up,dn

lapw0 lapw1-orb lapw2 lcore lapwdm lapw2-eece lapw0-eece orbmixer-eece

case.ineece-9.0 1 Emin [Ry], #atoms2 1 2 iatom nlorb lorb HYBR HYBR/EECE mode0.25 α

case.inorb, case.indm[c](generated automatically)

EECE: SrVO3

-5 5 10

DO

S

0

ω [eV]

LDA

EECE

t2g

Outline

1 Preliminaries

2 LDA+U, EECE — Orbital-Dependent Potentials

3 Wannier Functions and DMFTMaximally Localized Wannier Functions Wannier90Wien2WannierDynamical Mean-Field Theory at a Glance

Wannier Functions

from Marzari et al.

❼ Fourier transformsof Blochfunctions:

, V ikR|w nR ⟩ = (2π)3 dk e |ψ nk⟩

BZ

Wannier Functions

from Marzari et al.

❼ Fourier transformsof Blochfunctions:

, V ikR|w nR ⟩ = (2π)3 dk e |ψ nk⟩

BZ❼ “gauge” freedom:

V (2π )

|w nR ⟩=

,

BZ

.ikR

dk e Umn(k) |ψ mk⟩

m~ choose U(k) to

minimize spread ⟨∆r2⟩

Maximally Localized Wannier Functions

❼ choose U(k) to minimize spread ~ MLWF

❼ total spread Ω = ΩI + Ω~ can be split into gauge-invariantpart andrest

~ minimize Ω~

❼ wannier90 computes U(k) in this way[Marzari et al., Rev. Mod. Phys. 84, 1419 (2012)]

http://wannier.org

❼ wien2wannier provides interface to Wien2k[Kunes , Wissgott et al., Comp. Phys. Commun. 181, 1888]

http://www.wien2k.at/reg_user/unsupported/wien2wannier/

Disentanglement

from Marzari et al.

fcc-Cu,5 d-like WF,2 interstitial s-like WFnum_bands = 12num_wann = 7

❼ other bands may cross target manifold

~ must select bands to Wannierize

❼ V(k) . J(k) × N.

❼ selection determines ΩI

~ minimize also ΩI

MLWF: Applications

❼ analysis of chemicalbonding

❼ electricpolarizationand orbital magnetization~ Oleg Rubel’s talk (tomorrow, 10:30)

❼ Wannierinterpolation K → GF F − 1

H(k)|K − → H(R)|K−1 − → H(k)|G

❼ Wannier functions asbasis functions❼ tight-bindingmodel H(k) = U+(k) ε(k) U(k)

~ realistic dynamical mean-field theory(DMFT)

wannier90

case.winnum_bandsnum_wann num_iter

= 3= 3= 1000

num_print_cycles =100dis_froz_min dis_froz_max

= 7.= 9.

bands_plot_project = 1

case.woutFinal StateWF centre and spread WF centre and spread WF centre and spread

1 ...2 ...3 ...

Sum of centres and spread ...

[Marzari et al., Rev. Mod. Phys. 84, 1419 (2012)]http://wannier.org

wien2wanniercase.w2winBOTH21 23 min band, max band

233LJMAX, #Wannier functions

#terms

2

2

2 2 -2 0.00000000 0.70710677 atom, L, M, coeff2 2 2 0.00000000 -0.707106772 2 -1 0.00000000 0.707106772 2 1 0.00000000 0.707106772 2 -1 0.70710677 0.000000002 2 1 -0.70710677 0.00000000

[Kunes , Wissgott et al., Comp. Phys. Commun. 181, 1888]http://www.wien2k.at/reg_

user/unsupported/wien2wannier/

MLWF: Program FLow

0 normal SCF run − → converged density, band structure

1 prepare_w2wdir.sh, init_w2w: prepare input files

2 xlapw1 -options − → eigenvectors onfull k-mesh

3 w2w case − → overlap ⟨umk |unk ⟩

4 shift_energy case

5 wannier90.x case − → U(k)

wien2wannier Features❼ spin-polarized cases, spin-orbit coupling❼ any functional, LAPW, APW+LO basis❼ disentanglement❼ plotting:interface toXCrysDen / VESTA❼ woptic: optical conductivity with Wannier functions

σ(Ω) ∼

.

k,ω

f (ω)−f (ω+Ω) Ω

tr .VA(k, ω + Ω)VA(k, ω ) .

❼ adaptive k-integration❼ includes self-energy

X(ω) (DMFT)

http://www.wien2k.at/reg_user/unsupported/wien2wannier/

[Wissgott, Kunes et al. Phys. Rev. B 85, 205133]

LDA+DMFT

LDA❼ realistic calculations❼ fails for strong correlations

model Hamiltonians❼ simplified, abstract model❼ full correlations

LDA+DMFT

LDA❼ realistic calculations❼ fails for strong correlations

model Hamiltonians❼ simplified, abstract model❼ full correlations

LDA+DMFT ⇒ realisticcalculation including most importantcorrelationeffects

Held, Adv. Phys. 56, 829 (2007)

Kotliar et al., Rev. Mod. Phys. 78, 865 (2006)

Correlation Regimes

U: screened localinteraction W: bandwidth, ∼ t

From lattice models to DMFT

tU

U

U

UU

U U

UU

lattice model❼ e− hopbetween sites❼ localrepulsion(screened)

From lattice models to DMFT

tU

U

U

UU

U U

UU

U

lattice model❼ e− hopbetween sites❼ localrepulsion(screened)

Σ

impurity model❼ oneinteractingsite❼ non-interacting“bath”

From lattice models to DMFT

tDMFT

U

U

U

UU

U U

UUΣ

U

lattice model❼ e− hopbetween sites❼ localrepulsion(screened)

impurity model❼ oneinteractingsite❼ non-interacting“bath”

dynamical mean-field theory❼ lattice model ›→ impurity model❼ self-energy X(ω)

Georges et al., RMP 1996,Kotliar & Vollhardt, Phys. Today 2004

LDA+DMFT for SrVO3

❼ DMFT basis: 3 degenerate t2g Wannier functions❼ U = 5.05 eV, T = 1160 K

❼ DMFT(QMC) has finite T❼ low T is hard (“sign problem”)

experiment: Makino et al., PRB 58, 4348 (1997)DMFT: Philipp Wissgott

LDA+U, EECE, DMFT comparedLDA+U, EECE

❼ density functionals❼ single-particle model

❼ good for insulators (U W)❼ work inside MT sphere

basis: Ym

❼ double counting❼ semi-empirical U, α

l❼ computationally cheap

LDA+DMFT❼ “spectral density functional theory”

❼ many-body physics❼ basis: Wannier functions

(or similar)❼ whole U/ W range❼ sophisticated, expensive

❼ double counting❼ semi-empirical U