Electronic Correlations in Solids: From simple models to real materials

Supported by Deutsche Forschungsgemeinschaft through SFB 484

Hands-on Course: LDA+DMFT University of Hamburg; May 18, 2005

Dieter Vollhardt

Outline:

• "Correlations"

• Correlated Electron Materials

• Models approach/Single impurity physics

• Dynamical Mean-Field Theory (DMFT)

• LDA+DMFT

• Application to real material: (Sr,Ca)VO3, V2O3, Ni

• Current developments / Perspectives

"Correlations"

Correlation [lat.]: con + relatio ("with relation")

Wechselwirkung (-beziehung)Gegenseitige Abhängigkeit

Grammar ("Correlate"): either ... or

Mathematics, natural sciences:

AB A B≠

( ) ( ') ( ) ( ')ρ ρ ρ ρ≠r r r r

e.g., densities:

Beyond (standard) mean-field theory [Weiss/Hartree-Fock,...]

Short-range spatial correlations in everyday life

Time average insufficient

Correlationsvs.

long-range order

(Sempe)

Correlated Electron Materials

Partially filled 3d bands

Partially filled 4f bands

Narrow 3d, 4f orbitals strong electronic correlations

Insu-lator

Solid NeNaCl

Localizedelectrons

Atomic levels n σi

Na, AlExtendedwavesBroad

bandsSimple metal n σk

Tran-sition +rareearthmetals(Ni,V2O3, Ce)

Narrowbands

Corre-lated metal

n nσ σ↔i k

ExampleCharacter,Repre-sentation

Energy levels

PropertyElectronic Bands in Solids

overlap of wave functions: matrix element t

band overlap band width t Wε ∝k ∼

1 aW∼1 lattice spacing: v

average time spent on atom: aε

τ∇ = =k k k W

τ⇒ ∼

Small W: Strong electronic correlations

UBe13,CeCu2Si2Stewart et al. (1983,1984)

Independent electrons2

00lim V

T

c ATT

γ→

= +

1. "Heavy Fermi liquid"

* 1000mm∼

Simple metal: Potassium

0lim ,*V

T

cT

mm

γ→

= ∝ v*F

F mk

=

1.Fermi gas: Ground state

kx

ky

kz

1.Fermi gas: Excited states (T>0)

kx

kz

ky

k-(Eigen)states with infinite life time

Switch on interaction adiabatically (non-perturbative)

1.Landau Fermi liquid

kx

ky

kz

Well-defined k-states (mean-field theory) with finite life time

2.-6 eV satellite in Nickel

Guillot,..., Falicov (1977)

Not reproducible by Density Functional Theory/Local Density Approximation

Microscopic explanation of the -6 eV satellite?

Metal-insulator transition in V2O3

3.

•PI PM: 1. order transition without lattice symmetry change

• Anomalous slope of P(T)

Pomeranchuk effect in 3He

Rice, McWhan (1970); McWhan, Menth, Remeika,

Brinkman, Rice (1973)

heating

Metal-insulator transition in V2O3

3.

3He

Metal-insulator transition in V2O3

3.

Interaction U

3He

Interaction U

V2O3: metal-insulator transition

Interaction

metal

insulator

3He:liquid-solid transition

Interaction

ln 2BS k=S Tγ=

S Tγ=ln 2BS k=

Fermionic correlation effect

3.

Localization – delocalization transition

Microscopic explanation?

•large resistivity changes•huge volume changes•high Tc superconductivity•strong thermoelectric response

•gigantic non-linear optical effects•colossal magnetoresistance

Correlated electron materials

Fascinating topics for fundamental research

Technological applications:• catalyzers• sensors• cables• spintronics• magnets/magnetic storage,...

with

"Complexity"

Model Approach

t U

Gutzwiller, 1963Hubbard, 1963Kanamori, 1963

Hubbard model

Local Hubbard physics:

,nUH n nσ

σ

ε ↑ ↓= +∑ ∑k i ik

ki

†

, ,

c c U n nt σ σσ

↑ ↓= − +∑ ∑i j i ii j i

time

t U

Gutzwiller, 1963Hubbard, 1963Kanamori, 1963

Hubbard model

Local Hubbard physics:

,nUH n nσ

σ

ε ↑ ↓= +∑ ∑k i ik

ki

†

, ,

c c U n nt σ σσ

↑ ↓= − +∑ ∑i j i ii j i

time

t U

Gutzwiller, 1963Hubbard, 1963Kanamori, 1963

Hubbard model

Local Hubbard physics:

n n n n↑ ↓ ↑ ↓≠i i i i

Hartree-(Fock) mean-field theory generally insufficient

Correlation phenomena:Metal-insulator transitionFerromagnetisms,...

,nUH n nσ

σ

ε ↑ ↓= +∑ ∑k i ik

ki

†

, ,

c c U n nt σ σσ

↑ ↓= − +∑ ∑i j i ii j i

Dynamical Mean-Field Theory(DMFT)

Theory of strongly correlated electrons

,nUH n nσ

σ

ε ↑ ↓= +∑ ∑k i ik

ki

Coordination number Z:Z=6 (simple cubic)

Theory of strongly correlated electrons

,nUH n nσ

σ

ε ↑ ↓= +∑ ∑k i ik

ki

Coordination number Z:Z=8 (body-centered cubic)

Theory of strongly correlated electrons

,nUH n nσ

σ

ε ↑ ↓= +∑ ∑k i ik

ki

Coordination number Z:Z=12 (face-centered cubic)

d →∞

Metzner + Vollhardt (1989) Σ(ω)

G( )ω

i

"Single-site" mean-field theorywith full many-body dynamics

Müller-Hartmann (1989)Janis (1991)Janis, Vollhardt (1992)

Theory of strongly correlated electrons

,nUH n nσ

σ

ε ↑ ↓= +∑ ∑k i ik

ki

Georges and Kotliar (1992)Jarrell (1992)

Hubbard model single-impurity Anderson model+ self- consistency

d →∞

Σ(ω)

G( )ω

i

Excursion: Single-impurity Anderson modelt

Non-interacting conduction (s-) electrons

s,d-hybridization+

V2/W

N(E)

EW

single d-orbital ("impurity")+

Ed

Excursion: Single-impurity Anderson modelt

Non-interacting conduction (s-) electrons

N(E)

EW

single d-orbital ("impurity")+

Ed

Excursion: Single-impurity Anderson model

N(E)

E

tNon-interacting conduction (s-) electrons

U+

WEd+UEd-U

U

single d-orbital ("impurity")with interaction U

Ed

Excursion: Single-impurity Anderson modelt

Non-interacting conduction (s-) electrons

U+

single d-orbital ("impurity")with interaction U

s,d-hybridization+

Characteristic 3-peak structurewith non-perturbative energy scale("Kondo physics")

Σ(ω)

G( )ω

iZ⇒∞: connection

with DMFT

Dynamical Mean-Field Theory (DMFT)

Proper time resolved treatment of local electronic interactions:

Physics Today (March 2004) Kotliar, Vollhardt

DMFT study of Mott-Hubbard metal-insulator transition

Hubbard model, n=1

1, *m

Z m− = →∞

Quasiparticles

Quasiparticlerenormalization

DMFT study of Mott-Hubbard metal-insulator transition

Hubbard model, n=1

Transfer of spectral weight: genuine correlation feature

lowerHubbard band

upperHubbard band

1, *m

Z m− = →∞

Quasiparticles

Quasiparticlerenormalization

( )N E ( )N E ( )N E

( )N E ( )N E ( )N E

DMFT study of Mott-Hubbard metal-insulator transition

( )N E ( )N E

Coherent (k-) states: Quasiparticles

Transfer of spectral weight: genuine correlation feature

Incoherentstates(Hubbard bands)

Onlyincoherentstates

Quasiparticles

lowerHubbard band(incoherent)

upperHubbard band(incoherent)

Z<1

DMFT study of Mott-Hubbard metal-insulator transition

Single-impurityAnderson model

Characteristic three-peak structure

Only one type of electron Two types of electrons

Hubbard model (n=1): DMFT phase diagram

Strongly correlatedelectron materials

V2O3NiSe2-xSxκ-organics, ...

Helium-3

Universality due to Fermi statistics

Correlated Electron Materials:LDA+DMFT

Material specific electronic structure calculations

Total energy as functional of )(rρ

Basic quantity: local density )(rρ

Local density approximation (LDA):

3[ ( )] ( ( ))xLDA

c xcE d rEρ ρ→ ∫r r

⇓LDA

t U

t U

How to combine?

Material specific electronic structure (Density functional theory: LDA,GW,...)

Computational scheme for correlated electron materials:

+Local electronic correlations

(Many-body theory: DMFT)

LDA+DMFT

Anisimov, Poteryaev, Korotin, Anokhin, Kotliar (1997)Lichtenstein, Katsnelson (1998)Nekrasov, Held, Blümer, Poteryaev, Anisimov, Vollhardt (2000)

Physics Today, March 2004; Kotliar, Vollhardt

Material specific electronic structure (Density functional theory: LDA,GW,...)

Computational scheme for correlated electron materials:

+Local electronic correlations

(Many-body theory: DMFT)

LDA+U

Long-range order ⇒ energy gap ⇒ additional stiffness

Mimics correlations

LDA+DMFT (simplest version)1) Calculate LDA band structure: ' '

ˆ( )lml m LDAk Hε →

,

ˆd

d

i mi i

dm

n σσ

ε==

− ∆∑∑

double counting correction''mmU σσ

local Coulomb interaction

3) Solve model by DMFT with, e.g., QMC: LDA+DMFT(QMC)

Solve self-consistently:

(i) Effective single impurity problem

(ii) k-integrated Dyson equ.

Σ(ω)

G( )ω

i

3) Solve model by DMFT with, e.g., QMC: LDA+DMFT(QMC)

Solve self-consistently:

(i) Effective single impurity problem

(ii) k-integrated Dyson equ. (orbital degeneracy)

Σ(ω)

G( )ω

i

LDA( )LDAN ε

Application of LDA+DMFT to specific materials

AugsburgG. KellerM. KollarI. LeonovX. RenV. Eyert

DV----------------------------K. Held (MPI Stuttgart)T. Pruschke (Göttingen)

V. I. AnisimovI. A. Nekrasov

Z. Pchelkina...

Ekaterinburg

Ann ArborJ. W. Allen et al.

OsakaS. Suga et al.

3d1 system: (Sr,Ca)VO3

3d1 system: (Sr,Ca)VO3

Inoue et al., PRL (1995)

Photoemission spectroscopy (PES)

Excursion: Spectroscopy

1. Photoemission Spectroscopy (PES)

Angular Resolved PES = ARPES

Measures occupied states of electronic spectral function

PES

Ideal spectral function of a material

PES

Ideal spectral function of a material

PES

Occupied states (ideal)

PES

Occupied states (measured)

2. Inverse Photoemission Spectroscopy (IPES)

Measures unoccupied states of electronic spectral function

Information also available by:

X-ray absorption spectroscopy (XAS)

IPES/XAS

Ideal spectral function of a material

IPES/XAS

Ideal spectral function of a material

IPES/XAS

Unoccupied states (ideal)

IPES/XAS

Unoccupied states (measured)

3d1 system: (Sr,Ca)VO3Experiment vs. DMFT model theory

• One-band, Bethe DOS• Symmetric around EF• U value fitted

Inoue et al., PRL 74 2539 (1995)

Rozenberg et al., PRL 74, 4781 (1995)

3d1 system: (Sr,Ca)VO3Experiment vs. DMFT model theory

Inoue et al., PRL 74 2539 (1995) • One-band, Bethe DOS• Symmetric around EF• U value fitted

Rozenberg et al., PRL 74, 4781 (1995)

Experiment

Photoemission spectra at high photon energies

SrVO3 CaVO3

Osaka – Augsburg – Ekaterinburg collaboration: Sekiyama et al., PRL (2004)

Spectra after subtraction of estimated surface contribution

Osaka – Augsburg – Ekaterinburg collaboration: Sekiyama et al., PRL (2004)

Theory

10% reduction in V-O-V angle

Electronic structure

SrVO3 CaVO3

only 4% bandwidth reduction

detailed structure unimportant10% reduction in V-O-V angle

Electronic structure

Theory

LDA+DMFT results constrained LDA:U=5.55 eV, J=1.0 eV

k-integrated spectral function

1( ) Im ( )A Gω ωπ

= −

Osaka – Augsburg – Ekaterinburg collaboration: Sekiyama et al., PRL (2004)

Osaka – Augsburg –Ekaterinburg collaboration, PRL (2004) + preprint (2005)

Measurement at O K-edge:no symmetry breaking of V 2p shell in final state (XAS IPES)≈

40 years „Kondo effect“

Single-impurityAnderson model

Bulk system

One-bandHubbardmodel (DMFT)

(Ca,Sr)VO3:Experimentand theory(LDA+DMFT)

k- resolved spectra (ARPES) in DMFT

0 1( )( , ) [ ( )]k kLDAω ω ω −= − −G HΣ

matrices in orbital space

k-resolved spectral function1( , ) I ( , )k kmA T rω ωπ

= − G

→

NMTO downfolded vs. LDA+DMFT bandsEkaterinburg – Augsburg – O. K. Andersen – collaboration

Renormalization of LDA bands by LDA+DMFT self-energy;1/Z=m*/m=1.9

LMTO: N=1

Kinks?

Explanation of kink structure

Re '1( ) :

1 ( )dEZ kdk ω

= =Σ−

Z≈1 bei ωmax ≈ N(0)-1Zt;SrVO3: ωmax ≈ 200 meV

NMTO downfolded vs. LDA+DMFT bandsEkaterinburg – Augsburg – O. K. Andersen – collaboration

Renormalization of LDA bands by LDA+DMFT self-energy;1/Z=m*/m=1.9

LMTO: N=1

Kinks!

High-resolution photoemission results on SrVO3

Fujimori et al., cond-mat 0504576

LDA

Extension of LDA+DMFT scheme

LDA Hilbert transform( )LDA

mN ε

Wannier function formalism

“Full Hamiltonian“ approach

•LDA+DMFT(QMC) with full-orbital self-energy (O-2p + V-3d states)•self-consistent merging of LDA and DMFT possible

Andersen, Saha-Dasgupta (2000)Pavarini, Biermann, Poteryaev, Lichtenstein, Georges, Andersen (2004)Ekaterinburg – Augsburg – Ann Arbor – Osaka – collaboration (2005)

Full-orbital DMFT scheme with Wannier functions

Ekaterinburg – Augsburg – Ann Arbor – Osaka – collaboration, PRB (2005)

3d2 system: V2O3

3d2 system: V2O3

Interaction U

x x

isotropic cubic trigonal

V2O3: LDA Spectra

U=5.0 eV, J=0.93 eV

Held, Keller, Eyert, Vollhardt, and Anisimov, PRL (2001)

metallic:

insulating:

V2O3: LDA+DMFT Spectra

U=5.0 eV, J=0.93 eV

Keller, Held, Eyert, Vollhardt, Anisimov; PRB (2004)

Metallic V2O3: Photoemission Spectra

Metallic V2O3: Photoemission Spectra in Theory and Experiment

U=5.0 eV, J=0.93 eV

2 ggt eσ+

↓

Ann Arbor – Osaka – Augsburg – Ekaterinburg collaboration; Mo et al., PRL (2003)

Metallic V2O3: Photoemission and XAS Spectra in Theory and Experiment

U=5.0 eV, J=0.93 eV

2 ggt eσ+

↓

Ann Arbor – Osaka – Augsburg – Ekaterinburg collaboration; Mo et al., PRL (2003)

Insulating V2O3: Filling of the Mott gap

Filling of gap with increasing temperaturegenuine feature of Mott-Hubbard MIT

Mo et al., PRL (2004)Held et al. (2004)

Ferromagnetic Materials

Generalized fcc lattice ( )Z →∞

DMFT: Ferromagnetism in the one-band Hubbard modelUlmke (1998)

Microscopic conditions for ferromagnetism:Wahle, Blümer, Schlipf, Held, Vollhardt (1998)

DMFT: Ferromagnetism in the one-band Hubbard modelUlmke (1998)

LDA+DMFT for ferromagnetic Ni Lichtenstein, Katsnelson, Kotliar (2004)

-6 eVsatellite

LDSA

Beyond DMFT: Cluster Extensions

Dynamical cluster approx. (DCA) Jarrell et al. (2000)Cluster DMFT (CDMFT) Kotliar et al. (2001)Self-energy functional theory Potthoff (2003)

Σ(ω)

G( )ω

i ⇒

Antiferromagnetic d-wave 2 × 2 periodically repeated cluster

hole doping

Lichtenstein, Katsnelson (2000)

Comparison with Experiments in Cuprates:Spectral Function ( , 0) vs. A ω →k k

hole doped

Shen et al. (2004)

2x2 CDMFT

Civelli, Capone, Kancharla,Parcollet, Kotliar;cond-mat/0411696

Tc ≈ 0.025t

8A16B

16A

Zd=1 Zd=2 Zd=3No. of independent neighboring d-wave plaquettes: • d-wave order non-local

(4 sites)

• Expect large size and geometry effects in small clusters

• Tc is suppressed when the number of adjacent plaquettes <4

• Tc≈0.025t when the

number of adjacent plaquettes is complete

d-wave pairing in the 2D Hubbard model (U=4t; n=0.90)

Cluster Zd

8A 112A 216A 320A 424A 426A 4

Maier, Jarrell, Schulthess, Kent, White; cond-mat/0504529

Conclusion• Formulation of DMFT, LDA+DMFT

• Applications:

e.g., V-3d1 system SrVO3 CaVO3 - explains bulk sensitive experiments

why are the surfaces so different?

Current Developments/Perspectives•Self-consistent merging of LDA/GW and DMFT

•Cluster generalizations, faster impurity solver

•Applications to complex/low-dimensional systems