Dynamical Mean Field Theory in Electronic Structure

Calculations:Applications to solids with f and d electrons

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

Joint work with Sergej Savrasov

X International Workshop on computational Material Science Total

energy and Force Methods 11-13 January 2001 Trieste

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Outline Problems posed by the

electronic structure of strongly correlated electrons

Dynamical Mean Field Theory Concepts, DMFT and DFT

A case study of system specific properties: f electrons in Pu (S. Savrasov , GK)

A case study involving d electrons La1-xSrxTiO3

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Mott : Correlations localize the electron

Low densities, electron behaves as a particle,use atomic physics, real space

One particle excitations: Hubbard Atoms: sharp excitation lines corresponding to adding or removing electrons. In solids they broaden by their incoherent motion, Hubbard bands (eg. bandsNiO, CoO MnO….)

High T, local moments

Magnetic and Orbital Ordering at low T

Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics.

1

T

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The Strong Correlation Problem

Two limiting cases of the electronic structure of solids are understood:the high density limit and the limit of well separated atoms.

High densities, the is electron be a wave, use band theory, k-space

One particle excitations: quasi-particle,quasi-hole bands, collective modes.

Density Functional Theory with approximations suggested by the Kohn Sham formulation, (LDA GGA) is a successful computational tool for the total energy, and a good starting point for perturbative calculation of spectra, such as GW.……………………

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Localization vs Delocalization Strong Correlation Problem

•A large number of compounds with electrons which are not close to the well understood limits (localized or itinerant).•These systems display anomalous behavior (departure from the standard model of solids).•Neither LDA or LDA+U or Hartree Fock works well•Dynamical Mean Field Theory: Simplest approach to the electronic structure, which interpolates correctly between atoms and bands

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La1-xSrxTiO3 photoemission

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DMFT for lattice hamiltonians

k independent k dependent G, Local Approximation Treglia et. al 1980

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How to compute

View locally the lattice problem as a (multiorbital) Anderson impurity model

The local site is now embedded in a medium characterized by

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How to determine the medium Use the impurity model to

compute and the impurity local Greens function. Require that impurity local Greens function equal to the lattice local Greens function.

Weiss field

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Solving the DMFT equations

G0 G

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

S .C .C .

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

•Analytical Methods

G0 G

Im puritySo lver

S .C .C .

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DMFT

Construction is easily extended to states with broken translational spin and orbital order.

Large number of techniques for solving DMFT equations for a review see

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

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DMFT : effective action construction (Fukuda, Valiev Fernando , Chitra GK).

Select a set of local orbitals. Define a frequency dependent, local

Greens function (spectral function) The energy can be expressed as a

functional of the local Greens function (R. Chitra and G. Kotliar PRB 2000)

The functional can be built in perturbation theory in the interaction (well defined diagrammatic rules )

The functional can also be constructed from the atomic limit.

A useful approximation to the exact functional can be constructed, this

Is the DMFT functional.

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Effective action Fukuda, Valiev and Ferndando, Chitra GK

dynamical analog of kohn Sham potential

Sum of all local graphs which are 2PI

Reduces to self energy when self energy is local, approximation.

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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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LDA+DMFT

The light, SP (or SPD) electrons are extended, well described by LDA

The heavy, D (or F) electrons are localized,treat by DMFT.

LDA already contains an average interaction of the

heavy electrons, substract this out by shifting the heavy level (double counting term)

The U matrix can be estimated from first principles

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LDA+DMFT loop (in a tight binding basis, say LMTO’s )given interaction matrix U 0) Guess (r), G(i 1) Form Vxc , Solve AIM to

get and local Greens function of heavy orbitals.

Form LMTO Matrix , overlap matrix and heavy level shift E , form G(k, i

3) Recompute the density and Weiss function G(i to go back to 1.

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LDA+DMFT

To implement step 3 we use

Notice the Weiss field,E and self energies use

only heavy block, while H is full.

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LDA+DMFTAn exact functional by expresses the total energy as a functional of the local spectral function (or local Greens function) of the heavy electrons, and of the total density.

•The construction proceeds by Legendre transformation (G. Kotliar and S. Savrasov 2001).

•LDA+DMFT useful approximation for this functional.

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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 GK full self consistent implementation (2001)

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Delocalization Localization across the actinide series

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Pu: Complex Phase Diagram (J. Smith LANL)

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Small amounts of Ga stabilize the phase

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Problems with LDA

o DFT in the LDA or GGA is a well established tool for the calculation of ground state properties.

o Many studies (Freeman, Koelling 1972)APW methods

o ASA and FP-LMTO Soderlind et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give

o an equilibrium volume of the an equilibrium volume of the phasephaseIs 35% lower than Is 35% lower than experimentexperiment

o This is the largest discrepancy ever known in DFT based calculations.

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Problems with LDA LSDA predicts magnetic long range

order which is not observed experimentally (Solovyev et.al.)

If one treats the f electrons as part of the core LDA overestimates the volume by 30%

LDA predicts correctly the volume of the phase of Pu, when full potential LMTO (Soderlind and Wills). This is usually taken as an indication that Pu is a weakly correlated system.

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Other Methods

LDA+ U (Savrasov and Kotliar 2000, Bouchet et. Al 2000) predicts correct volume of Pu with the constrained LDA estimate of U=4 ev. However, it predicts spurious magnetic long range order and a spectra which is very different from experiments.

Ideal system to try realistic DMFT

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Pu: DMFT total energy vs Volume (Savrasov 00)

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Lda vs Exp SpectraD

OS

, st./

[eV

*cel

l]

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Pu Spectra DMFT(Savrasov) EXP (Arko et. Al)

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Pu Specific Heat

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Anomalous ResistivityJ. Smith LANL

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(Tokura et. Al. 1993)A doped Mott insulator:LaxSr1-xO3

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DMFT calculation, U near the Mott transition, Rozenberg et.al 94

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Hall Coefficient, electron like.

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Conclusions DMFT is a many body approach which is

able to describe on the same footing the atomic limit (open shell atoms) and bands. Works even in the localization

Delocalization crossover. [Improvement in Eq.Volume and spectra and other quantities (relative to LDA GGA) in Pu and titanites,prototype systems]

Progress in many body theory results in better understanding of materials: Pu, LaSrTiO3……….

Future work: improve treatment of local Hamiltonian multiplets.

Future work: improve treatment of spd electrons (E-DMFT) Coulomb screening, connection with GW

Extension to multiple site clusters(DCA M. Jarrell et. Al., C-DMFT G.Kotliar

et.al, Two impurity DMFT Schiller, Ingersent Georges Kotliar, …..)