Dynamical Mean Field Theory DMFT and electronic structure calculations

Dynamical Mean Field Theory DMFT and electronic structure calculations

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

ICTP Trieste August 2003

THE STATE UNIVERSITY OF NEW JERSEY

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Outline Pedagogical Introduction to DMFT for correlated electron

systems, parallel with DFT GW .[R. Chitra , P Sun, S. Savrasov and GK]

How good is the local approximation ? a) a brief look at some recent experiments. b) compare with exact results in one dimension. c) look at corrections. What new effects do cluster corrections

bring on top of single site DMFT ? Some answerson a model of kappa organics (O. Parcollet G. Biroli and GK) Some system specific calculations for materials near the Mott

transition: La1-x Srx TiO3 , , Pu………


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Weakly correlated electrons:band theory. Simple conceptual picture of the ground

state, excitation spectra, transport properties of many systems (simple metals, semiconductors,….).

A methods for performing quantitative calculations. (Density functional theory, in various approximations and GW).


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Start with TOE


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DFT: effective action construction

( )( )

Wr

j r


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DFT: Kohn Sham formulation

=


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Exchange and correlation energy Exact formal expressions can be given in

terms of a coupling constant integration.[Harris-Jones, adiabatic connection]

DFT is useful because practical accurate expressions for Exc, exist.

LDA, GGA, hybrids,


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Kohn Sham reference system

2 / 2 ( ) KS kj kj kjV r y e y- Ñ + =

( ')( )[ ( )] ( ) ' [ ]

| ' | ( )xc

KS ext

ErV r r V r dr

r r r

drr r

dr= + +

-ò

2( ) ( ) | ( ) |kj

kj kjr f rr e y=å


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Kohn Sham Greens function is an good point to compute spectra in perturbation theory in screenedCoulomb interaction GW,G0W0

Practical implementations, introduce a finite basis set.

Division into valence (active ) degrees of freedom and core.


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DFT+GW program has been less succesful in correlated situations.

Strong interactions localize the particles. Atoms with open shells are not easily connected to band theory.

The spectrum in this case, contain Hubbard bands which are NOT simply perturbatively connected to the Kohn Sham orbitals.

Need an alternative reference point for doing perturbation theory!

Need to treat bands and atomic excitations on the same footing.

DMFT!


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Strongly correlated systems are usually treated with model Hamiltonians

Conceptually one wants to restrict the number of degrees of freedom by eliminating high energy degrees of freedom.

In practice other methods (eg constrained LDA are used)


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Two roads for ab-initio calculation of electronic structure of strongly correlated materials

Correlation Functions Total Energies etc.

Model Hamiltonian

Crystal structure +Atomic positions


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DMFT Model Hamiltonian.

Exact functional of the

local Greens function A

+


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DMFT for model Hamiltonians. Kohn Sham formulation.

ij ii ijd=S S

[ , ] log[ ] ( ) ( )

[ ]

ijn n n

xc ii

A Tr i t Tr i A i

A

w w w-GS =- - S - S

+F

Introduce auxiliary field

1( )

( )ii n

xck

n k nii

A ii t i

A

wd

w wd

é ùê úê ú= ê úFê ú- -ê úë û

åExact “local self energy”


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About XC functional. One can derive a coupling constant integration

formulae (Harris Jones formula) for

Generate approximations.

The exact formalism generates the local Greens function and ii is NOT the self energy. However one can use the approach as starting point for computing other quantities.

[ ]xc iiAF

[ ]xcDMFT atom ii

i

AF = Få


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Comments on functional construction

Atoms as a reference point. Expansion in t. Locality does not necessarily mean a single point.

Extension to clusters. Jii --- Jii Ji i+ Aii --- Ai i+ ii --- i i+ Exact functionalAii ,Ai i+ he lattice self energy and other non local

quantities extending beyond the cluster are OUTSIDE the formalism and need to be inferred.


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Comments on funct. construction.

Construction of approximations in the cluster case requires care to maintain causality.

One good approximate construction is the cellular DMFT: a) take a supercell of the desired range,b)

[ ]xcCDMFT scells

scells

AF = Få c) obtain estimate of the lattice self energy by restoring translational symmetry. Many other cluster approximations (eg. DCA, the use of lattice self energy in

self consitency condition, restrictions of BK functional, etc. exist). Causality and classical limit of these methods has recently been clarified [ G Biroli O Parcollet and GK]


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Lattice and cluster self energies


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Mapping onto impurity models.

The local Greens function A, and the auxilliary quantity can be computed from the solution of an impurity model that describes the effects of the rest of the lattice on the on a selected central site.

One can arrive at the same concept via the cavity construction.


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1

10

1( ) ( )

( )n nn k nk

G i ii t i

w ww m w

-

-é ùê ú= +Sê ú- + - Sê úë ûå

DMFT Impurity cavity construction

1

10

1( ) ( )

V ( )n nk nk

D i ii

w ww

-

-é ùê ú= +Pê ú- Pê úë ûå

0

1 † 10 0 ( )( )[ ] ( ) [ ( ) ( ) ]n n n n S Gi G G i c i c ia bw w w w- -S = + á ñ

†

0 0

( ) ( , ') ( ') ( , ') o o o oc Go c n n Ub b

s st t t t d t t ¯+òò

† †

, ,

( )( )ij ij i j j i i ii j i

t c c c c U n n

()

1 100 0 0( )[ ] ( ) [ ( ) ( ) ]n n n n Si G D i n i n iw w w w- -P = + á ñ

,ij i j

i j

V n n

( , ') o oDo n nt t ¯+


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Solving the DMFT equations

G 0 G

I m p u r i t yS o l v e r

S . C .C .

•Wide variety of computational tools (QMC,ED….)Analytical Methods

Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

G0 G

Im puritySo lver

S .C .C .


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Two roads for ab-initio calculation of electronic structure of strongly correlated materials

Correlation Functions Total Energies etc.

Model Hamiltonian

Crystal structure +Atomic positions


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Start with the TOE


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Rewrite the TOE as an electron boson problem.

1 †1( ) ( , ') ( ') ( ) ( ) ( )

2Cx V x x x i x x xff f y y-+ +òò ò


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Build effective action for the local greens functions of the fermion and Bose field

r=R+ R unit cell vector position within the unit cell. Ir>=|R, Couple sources to

† ( ) ( ') R Ry r y r( ') ( )R Rf r f r( )Rf r


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Legendre transfor the sources, eliminating the field

Build exact functional of the correlation functionsW(r R,r’ R)

and G (r R,r’ R)

( ') ( ) ( ') ( )R R R R Wf r f r f r f r< >- < >< >= †( ') ( ')G R Ry r y r=- < >


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“Kohn Sham “ decomposition.

[ ] [ , ]HE xc G Wr y+ +


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(E)DMFT pproximation to [ , ]xc G Wy

Sum over all LOCAL 2PI graphs (integrations are restricted over the unit cell ) built with W and G

Map into impurity model to generate G and W

Go beyond this approximation by returning to many body theory and adding the first non local correction.


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Test on extended Hubbard model V/U=.25, P Sun and GK


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EDMFT functional.


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Returning to many body physics.


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Take the solution of the EDMFT equations as an approximation for the TRUE local self energy, and add the leading NON LOCAL corrections to the self energy G_NL W_NL, as a correction.

Do it self consistently and as a one shot iteration G0_NL W0_NL and compare the results.


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Average Z vs U (P. Sun 2003)


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Z1: K dependent part of QP residue.


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Functional of density and local Greens function. G. Kotliar and S. Savrasov

(see S. Savrasov’s talk )


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LDA+DMFT References

V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997).

A Lichtenstein and M. Katsenelson Phys. Rev. B 57, 6884 (1988).

S. Savrasov and G.Kotliar, funcional formulation for full self consistent implementation. Savrasov Kotliar and Abrahams . Application to delta Pu Nature (2001)


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How good is the LOCAL approximation: Exhibit A


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C-DMFT: test in one dimension. (Bolech, Kancharla GK cond-mat 2002)

Gap vs U, Exact solution Lieb and Wu, Ovshinikov

Nc=2 CDMFT

vs Nc=1


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How good is the local approximation ? Single site DMFT study of the Mott

transition, based on a study of the Hubbard model on frustrated lattices made several interesting qualitative predictions.

New experiments and reexamination of old ones give credence to that the local picture is quite good.


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V2O3 under pressure or


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Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)


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NiSe2-xSx


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Qualitative single site DMFT predictions. Spectra of the strongly correlated metallic

regime contains both quasiparticle-like and Hubbard band-like features.

Mott transition is drive by transfer of spectral weight.


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Schematic DMFT phase diagram Hubbard model (partial frustration) [observation of temperature dependent transfer of spectral weight in optics]

M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)


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Insights from DMFTThe Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phaseControl parameters: doping, temperature,pressure…


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Evolution of the Spectral Function with Temperature

Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000)


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Schematic DMFT phase diagram Hubbard model (partial frustration). Evidence for QP peak in V2O3 from optics.

M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)


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. ARPES measurements on NiS2-xSex

Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998)


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QP in V2O3 was recently found Mo et.al


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Anomalous Resistivity and Mott transition Ni Se2-x Sx

Crossover from Fermi liquid to bad metal to semiconductor to paramagnetic insulator.


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More recent work, Limelette et. al.


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Ising critical endpoint! In V2O3 P. Limelette et.al.


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How good is the local approximation ? Study of the Mott transition within CDMFT.

Are the single site DMFT results robust ?

How are they modified by short range magnetic correlations?

Study a frustrated model.


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organics

ET = BEDT-TTF=Bisethylene dithio tetrathiafulvalene

(ET)2 X

Increasing pressure ----- increasing t’ ------------

X0 X1 X2 X3 (Cu)2CN)3 Cu(NCN)2 Cl Cu(NCN2)2Br Cu(NCS)2 Spin liquid Mott transition


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Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)


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Evolution of the Spectral FunctionU/D=2, U/D=2.25 (Parcollet et.al.)

Uc=2.35+-.05, Tc/D=1/44


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Conjecture (O.Parcollet G. Biroli and GK ) Formation of hot regions is a more general

phenomena due to the proximity to the Mott point.


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Deviations from single site DMFT


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Illustration with some materials [see Savrasov talk] A method which can describe the Mott

transition should have very broad applications to electronic structure.


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Conclusions

DMFT as an EXACT first principles theory. Analogy with density functional theory.

DMFT as an approximation. For many problems the local approximation (local=single site , link or paquette ) is unexpectedely accurate.

Many applications to materials. S. Savrasov’s talk + many other talks in this workshop!

Documents

Dynamical Mean Field Theory DMFT and electronic structure calculations