Merging First-Principles and Model Approaches

Ferdi AryasetiawanResearch Institute for Computational Sciences, AIST,

Tsukuba, Ibaraki 305-8568 – Japan

ISSP 2007.07.25

Collaborators:

Antoine Georges and Silke Biermann (Ecole Polytechnique, France)

Takashi Miyake and Rei Sakuma (RICS-AIST)

Outline

•Motivations: The need to go beyond one-particle picture in correlated materials•Previous works: LDA+U, LDA+DMFT•The GW approximation: success and difficulties•Combining GW and DMFT: A first-principles scheme for correlated materials•A simplified implementation: Application to ferromagnetic nickel

First Part

Second Part (if time permits)

Constrained RPA:Calculating the Hubbard U from first-principles

Fujimori PRL 69, 1796 (1992)

Difficultto treatwithin

one-particletheory

Spectral evolution as a function of U/bandwidth

Georges et al Rev. Mod. Phys.1996

(Dynamical Mean-Field Theory)

Can use LDA

Can use LDA+U

metal

insulator

La/YTiO3

cubicorthorhombic

[1] Solovyev, Hamada, and Terakura, PRB 53, 7158 (1996) (LDA+U: correct magnetic structure of La/YTiO3)[2] Pavarini, Biermann, Poteryaev, Lichtenstein, Georges, and Andersen, PRL 92, 176403 (2004) (LDA+DMFT: consistent description of metal-insulator transition)

These materials sharesimilar electronic structure in LDA

and they are all predicted to be metals

ExperimentallySrVO3 and CaVO3 are metals

LaTiO3 and YTiO3 are insulators

Typical electronic structure of correlated materials:Partially filled narrow 3d or 4f band across the Fermi level

By slight distortion or pressure the ratio of U/bandwidth changes and the materials can undergo, e.g., phase transitions (metal-insulator).

competition between kinetic energy (bandwidth) and U.

The main actiontakes place here

D

U

U=0

U/D>1

Competition between kinetic energy and U(itineracy and localisation)

satellite

QP

LowerHubbard band

UpperHubbard band

U/D>>1

When U is small it is preferable for the electrons to delocalise metal

When U is large it is costly for the electronsto hop localised (Mott insulator)

Electrons prefer to “spread” themselves to lower their kinetic energy.

For intermediate U it is a mixture oflocalised and delocalised electrons.

D

U

i

ij iiij nnUcc

DH

4

U=0

U/D>1

Mapping to a Hubbard model

satellite

QP

LowerHubbard band

UpperHubbard band

U/D>>1

More realistic models take intoaccount the underlying one-electron

band structureLDA+U and LDA+DMFT

To treat systems with strong on-site correlations:Previous works: LDA+U and LDA+DMFT

''

.

'',

'

.

'''

'

.

''

'''',

)(2

1

2

1

miim

correl

im

tingdoublecounmiim

imim

correl

mim

imm

imm

imim

correl

imm

imm

miim

all

im

LDAmiim

ccH

nnJU

nnU

ccHH

Ad-hocdouble-counting

term

Adjustable U

Anisimov et al, J. Phys. Condens. Matter 9, 7359 (1997) (LDA+U, LDA+DMFT)Lichtenstein and Katsnelson, PRB 57, 6884 (1998) (LDA+DMFT)

The Goal

To construct a consistent theoretical scheme that can describethe electronic structure of correlated materials from first-principles.

“Consistent” means that the scheme should be capable ofdescribing the continuous transition from metal to insulator, i.e.,

it can describe both the itinerant and localised characters of the electrons.

One of the main features of correlated electrons is that they haveboth itinerant and localised characters.

First-Principles Methods:•Local Density Approximation (LDA) (ground states)•GW method (excited states)

Model Approaches:•Exact diagonalisation (Lanczos)•Quantum Monte Carlo (QMC)•Dynamical Mean-Field Theory (DMFT)•LDA+U and LDA+DMFT

Can treat strong correlationsbut need adjustable parameters

Insufficient to treatcorrelated materials

Combine First-Principles and Model Approaches: GW+DMFT

Electronic structure ofcorrelated materialsfrom first-principles

spintronicselectronictransport

nanotech.opticaldevices

Possible applications

The GW approximation

Hartree-Fock approximation:

)'()'()(

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

rrvrr

rrvtrriGrrocc

kkk

x

GW approximation:

);',();',();',( trrWtrriGtrrGWxc

- - - - - -v

G

= = = = =

G

W is a screened interaction

Lars Hedin, Phys. Rev. 139, A796 (1965)

vW 1

);',(Re)'()();',(Re *kn

occ

knknknSEX rrWrrrr

'

)';',(Im')'()();',(Re

0

*

knknknknCOH

rrWdrrrr

Screened exchange: from the poles of G

Correlation hole: interaction between an electron and its screening hole

Since W<v, it reduces the Hartree-Fock band gap.

In addition we have another term arising from the poles of W

COHSEXGWxc

0),0;',()'(2

1 knifrrWrr

From Hybertsen and Louie, PRB34, 5390 (1986)

Assessing the GWA using the Polaron Hamiltonian:

)( bbgccbbccH p

iiD

pp

11)(

i

g

DgGdi

p

2

2 )'()'('2

)(

,0

)(!

12)/()(

2

nn

g

n

geA p

n

p

p

pg /2

iG

1)(

Exact solution:

pPlasmons satellites

2

42

pp

QP

ggE

0)(Re

p

p

p

Sat

ggE

22 2

Quasi-particle and satellite energy

Error in QP energy is smallerthan error in satellite energy

i

g

p

2

)(

)(Im)(Re

1)(

iwG

There is no spin dependence in W, only in G: Exchange is taken into account properly, but not correlation between the same spin.

Collective excitations only arise from RPA: (1-Pv)= 0 (plasmons). Satellites arising from local correlations (atomic multiplets) are probably not fully captured.

The Hubbard Hamiltonian cannot be transformed into a polaron-type Hamiltonian for which GW is good.

Usually start from a single Slater determinant: Difficult to treat systems that are inherently dominated by a few Slater determinants.

Self-consistent GW appears to give poor spectra (?)

Some difficulties with the GWA

GW

Band insulator

Mott-Hubbard insulator

Iminfinite

Removes QP and transfers

weight to Hubbard bands

(QP life-time ~ 1/Im)

Imfinite

Shifts and broadens

Quasi-Particle

Band insulator vs Mott-Hubbard insulator

UDMFT

Strongly correlated systems are problematic for LDA: LDA often predicts (anti-ferromagnetic) insulators to be metals.

DMFT: Map the lattice to an impurity embedded in a “bath”.

“bath”

A. Georges et al, Rev. Mod. Phys. 68, 13 (1996)G. Kotliar and D. Vollhardt, Physics Today (March 2004)

Dynamical

Even the GW approximation may not be sufficient.

U

A la Fukuyama

)()()'()'('

nndUccddSeff

-10G

Effective dynamics for an impurity problem

with the dynamical mean-field -10G

bath

)'( 0G

The Coulomb interaction is fully taken into account in one site (impurity), the rest of the sites (medium) is treated as an effective field

Dynamical Mean-Field Theory (DMFT)A. Georges et al, Rev. Mod. Phys. 68, 13 (1996)

)()()( 1 iGii impimp -1

0G

k k impklatticeloc ii

ikGiG)(

1),()(

)()0()()( locSimp GcTcGeff Self-consistency:

effSimp cTcG )0()()( Green’s function

Self-energy

Restore the lattice periodicity

Comparison between GW and DMFT

DMFT(Dynamical Mean-Field Theory)

GW

iGWOnsite, no k-dependence Full k-dependence

Onsite interactions are summed to all orders

Approximate (RPA), good for long-range correlations

Model Hamiltonian

Adjustable parameters

Real systems

No parameters

Can treat systems with strong onsite correlations (3d and 4f)

Good for sp systems but has problems with 3d and 4f systems

Basic physical idea of GW+DMFT

0

R )()( 0000 DMFT

)()( 00 GWRR

The onsite self-energy is taken to be DMFT.

The off-site self-energy is approximated by GW .

P. Sun and G. Kotliar, PRB 66, 85120 (2002)S. Biermann, F.A. and A. Georges, PRL 90, 86402 (2003)

General Framework

Generalized Luttinger-Ward functional*

*Almbladh, von Barth and van Leeuwen, Int. J. Mod. Phys. B 13, 535 (1999)

WPPVW

W

GGG

G

2,0

,0

11

10

1

],[]1/[2

1ln

2

1]1/[ln],[ 0 WGVWtrWtrGGtrGtrWG

][]1/[ln][ 00 GGGtrGtrG

Luttinger-Ward functional

][][]ln[][ 00 GGtrGtrG

Approximation for

)(),(),(),(

)(),(),(),(

iPikPikPikP

iikikik

impk

GWGW

impk

GWGW

PWG

2,

],[],[ '' RRRRsiteonimp

RRRRsiteoffGW WGWG

2

1

2

1],[ trGWGWGGW wwww

Conservingapproximation

Self-Consistency Conditions

impk

loc

kimploc

WkPkVW

GkkGG

11

110

)]()([

)]()([

UUUW

cTcG

impimp

Simp eff

)0()()( 00

The Hubbard U is screened within the impurity model such that the screened U is equal to the local W

k

impimpGW WGk )()(),(

Additionalcondition:

Determines U

11

11

impimp

impimp

impimp

WUP

UUUW

G

G

Self-consistency loop

impq

GWGW

impq

GWGW

PqPkPkP

qkk

)()()(

)()()(

kloc

kloc

kPkVW

kkGG

11

110

)]()([

)]()([

imploc

imploc

PWU

G

11

11G

Impurity: given Weiss field G and U

Combine GW and imp

Check self-consistency: Gloc=Gimp? Wloc=Wimp?

New Weiss field G and U

Simplified GW+DMFT scheme

1

1

)])0(2

1(

)(),(),([)(

onsitexcimp

nimpk

noffsiteGWnHnloc

VTr

iikikGiG

Offsite self-energy: corrected by GW

Onsite self-energy: corrected by DMFT

Non-self-consistent GW + DMFT (static impurity model)

Test: Application to ferromagnetic nickel(Simplified GW+DMFT scheme)

Correlation problems in nickel within LDA:

LDA Exp.

3d band width 4 eV 3 eV

Exchange splitting 0.6 eV 0.3 eV

Satellite none 6 eV

GW

3.2 eV

0.6 eV

Very weak

(if any)

GW+DMFT

(Experimental data fromBuenemann et al, cond-mat/0204142)

majority

minority

majority

minorityexp

----- LDA

Nickel band structure

Too large LDA bandwidth

S. Biermann et al, PRL 90, 86402 (2003)

GW+DMFT density of states of nickel

no 6 eV satellite in LDA

Exchange splitting too large by 0.3 eV in LDA

Next target:

From V. Eyert

metal

insulator

From Miyake

Koethe et al

From Biermann et al, PRL 94, 026404 (2005)

Non-local self-energy is important a good test for GW+DMFT

Single impurity DMFT does notopen up a gap in M1

Preliminary GW result from Sakuma and Miyake

O2p

a_1g

One-shot GW gives too small gap. need DMFT? self-consistency?

Weak satellites

The self-energy around theLDA energy is not linear.

Summary

•GW+DMFT scheme:

-Potentially allows for ab-initio electronic structure calculations of correlated systems with partially filled localized orbitals and describes phase transitions.

-Unlike LDA+DMFT,*The Hubbard U is determined self-consistently*Avoids double counting.

Challenges

•Global self-consistency•Solving the impurity problem with an energy-dependent U•Treat all orbitals on equal footing