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# Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University - PowerPoint PPT Presentation

Towards First-Principles Electronic Structure Calculations of Correlated Materials Using Dynamical Mean Field Theory (DMFT). Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University. CMSN-Workshop on Predictive Capabilities for Strongly Correlated Systems

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Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University

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## Towards First-Principles Electronic Structure Calculations of Correlated Materials Using Dynamical Mean Field Theory (DMFT).

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

CMSN-Workshop on Predictive Capabilities for Strongly Correlated Systems

UT November 7-9 2003

## Outline , Collaborators, References

A. Poteryaev, A. Lichtenstein and G. Kotliar (preprint) (2003)

S.Savrasov G. Kotliar and E. Abrahams, Nature 410,793 (2001).

X. Dai,S. Savrasov, G. Kotliar,A. Migliori, H. Ledbetter, E. Abrahams  Science,  Vol300, 954 (2003)

Funding: Basic Energy Sciences DOE..

DMFT and electronic structure calculations

Case study 1: Ti2O3

Case study 2: Elemental Pu

Conclusions: Future developments.

Two limits of the electronic structure problem are well under control.

Band limit, (LDA or GGA)+ GW, gives good spectra and total energy. Physical properties are accessible in perturbation theory in the screened Coulomb interactions

Well separated atoms, in the presence of spin orbital long range order, expansion around the atomic limit, unrestricted HF, and LDA+U work well for ordered Mott insulators.

Challenge ahead: materials that are not in either one of these regimes. Requires combination of many body theory and one electron theory.

### Strongly Correlated Electrons

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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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Reduce the quantum many body problem to a one site or a cluster of sites, in a medium of non interacting electrons obeying a self consistency condition.

Instead of using functionals of the density, use more sensitive functionals of the one electron spectral function. [density of states for adding or removing particles in a solid, measured in photoemission]

### Dynamical Mean Field Theory

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DMFT

A. Georges, G. Kotliar (1992)

Phys. Rev. B 45, 6497

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### DMFT action and self consistency condition

In general tk is large matrix H[k] , U is a matrix

In the case of cluster is a matrix and is not the self energy, (but can be used to estimate the lattice self energy by projection)

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### Solving the DMFT equations

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

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### DMFT: Effective Action point of view.R. Chitra and G. K Phys Rev. B.62 12715(2000), 63 115110(2001) S Savrasov and G. K. cond-matt 0308053

• Identify observable, A. Construct an exact functional of <A>=a, G [a] which is stationary at the physical value of a.

• Example, density in DFT theory. (Fukuda et. al.)

• When a is local, it gives an exact mapping onto a local problem, defines a Weiss field.

• The method is useful when practical and accurate approximations to the exact functional exist. Example: LDA, GGA, in DFT.

• DMFT, build functionals of the LOCAL spectral function.

• Exact functionals exist. We also have good approximations!

• Extension to an ab initio method. Functional of greens function of electric field and electronic field, functional of the density and the local greens function.

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Observable: Local Greens function Gii (w).

Exact functional G [Gii (w) ].

DMFT Approximation to the functional.(Muller Hartman 89)

DMFT as an approximation to the exact functional of the Greens function, DMFT as a truncation of the BK functional of the full Greens function.

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### LDA+DMFT (II)

Edc

U

DMFT

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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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### 1d Hubbard U/t=4 exact diag 2+6.Capone Civelli and GK

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

Crystal structure +Atomic positions

Model Hamiltonian

Correlation Functions Total Energies etc.

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ROAD 1: Derive model Hamiltonians, solve by DMFT

(or cluster extensions).

V.Anisimov A Poteryaev V.Korotin V.Anokin andG Kotliar J. Phys. Cond. Mat. 35, 7359 (1997).A.Lichtenstein and M Katsnelson PRB (1998).

ROAD 2: Define a functional of the density and of the local Greens function and extremize the functional to get coupled equations for the density and the spectral function and compute total energies.

G. Kotliar, S.Savrasov, in New Theoretical approaches to strongly correlated systems, Edited by A. Tsvelik, Kluwer Publishers, (2001).

### Interfacing DMFT with band theory

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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) . This defines H. The U matrix can be estimated from first principles (Gunnarson and Anisimov, McMahan et.al. Hybertsen et.al) of viewed as parameters.

Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997).

### LDA+DMFT (I)

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### Application to Ti2O3

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Isostructural to V2O3. All the qualitative physics of the high temperature part of the phase diagram of V2O3 can be understood within single site DMFT. Computations with a realistic density of states, and multiorbital impurity model (K. Held and D. Vollhardt ) substantial quantiative improvement.

Is the same thing true in Ti2O3?

### Metal to insulator transition in Ti2O3

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### Ti2O3 V2O3 : Resistivities

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### Ti2O3 Structure

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### Relevant Orbitals: Goodenough picture

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As a function of temperature, there is no magnetic transition in Ti2O3, unlike V2O3

As a function of temperature, there is no structural change, unlike V2O3 which becomes monoclinic at low temperatures.

In V2O3 the distance between the Vanadium pairs incrases as the temperature decreases. In Ti2O3 the distance between the Vanadium pairs decreases as one lowers the temperature.

LTS 250 K, HTS 750 K.

### Ti2O3 vs V2O3

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Band Structure Calculations always produce a good metal. L.F. Mattheiss, J. Phys.: Condens. Matter 8, 5987 (1996)

Unrestricted Hartree Fock calculations produce large antiferromagnetic gap. M. Cati, G. Sandrone, and R. Dovesi, Phys. Rev. B. f55 , 16122 (1997).

### Earlier work.

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### Ti2O3 LDA-DOS

HTS

LTS

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Impurity solver. Multiband QMC.

Derivation of the effective Hamiltonian. Massive downfolding with O Andersen’s new Nth order LMTOS. Coulomb interactions estimated using dielectric constant W=.5 ev. U on titanium 2 ev. J= .2 ev.

### Methodology:1 and 2 site CDMFT

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### Single site DMFT fails. LTS

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Two-site CDMFT for beta=20, and beta=10

(T=500,1000)

Poteryaev Lichtenstein and GK

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### Important role played by the Coulomb nn repulsion.

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### Application to Plutonium

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### Small amounts of Ga stabilize the d phase (A. Lawson LANL)

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### Elastic Deformations

Uniform compression:Dp=-B DV/V

Volume conserving deformations:

F/A=c44Dx/L

F/A=c’ Dx/L

In most cubic materials the shear does not depend strongly on crystal orientation,fcc Al, c44/c’=1.2, in Pu C44/C’ ~ 7largest shear anisotropy of any element.

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

Many studies (Freeman, Koelling 1972)APW methods

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

an equilibrium volume of the d phaseIs 30-35% lower than experiment

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

### DFT studies

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LSDA predicts magnetic long range (Solovyev et.al.)

Experimentally d Pu is not magnetic.

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

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

### DFT Studies

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### Lda vs Exp Spectra

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### Energy vs Volume [GGA+U=4 ev]

EXPT:

Bcc 14.7

Fcc 15.01

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### GGA+U spectra

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Predicts plutonium to be magnetic.

Different theories of alpha and delta.

### Other problems with LDA+U

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Atomic sphere approximation.

Ignore crystal field splittings in the self energies.

Fully relativistic non perturbative treatment of the spin orbit interactions.

Impurity solver: interpolative scheme using slave bosons (low frequency ) and eqn of motion (high frequency).

### DMFT - Technical details [spectra and energy]

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Full potential LMTO with two kappas.

Linear response method in LMTO’s (S. Savrasov)

Impurity solver: lowest order projection (Roth method) in the equations of motion.

### DMFT- Phonon Spectra

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### Pu: DMFT total energy vs Volume (Savrasov Kotliar and Abrahams 2001)

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### Pu Spectra DMFT(Savrasov) EXP (Arko Joyce Morales Wills Jashley PRB 62, 1773 (2000)

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### Alpha and delta Pu

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Alpha phase is also a correlated metal.

It differs from delta in the relative weight of the resonance and the Hubbard band.

Consistent with resistivity and specific heat measurements.

Similar conclusions A. Mc Mahan K. Held and R. Scalettar, for the alpha to gamma transition in Cerium.

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Spectra

Method

E vs V

## Summary

LDA

LDA+U

DMFT

Electrons are the glue that hold the atoms together. Vibration spectra (phonons) probe the electronic structure.

Phonon spectra reveals instablities, via soft modes.

Phonon spectrum of Pu had not been measured.

Short distance behavior of the elastic moduli.

### Phonon Spectra

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### Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003

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### Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev.

E = Ei - Ef

Q =ki - kf

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### Expt. Wong et. al.

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### Expts’ Wong et. al. Science 301. 1078 (2003) Theory Dai et. al. Science 300, 953, (2003)

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C’=(C11-C12)/2 = 4.78 GPa C’=3.9 GPa

C44= 33.59 GPa C44=33.0 GPa

C44/C’ ~ 7 Largest shear anisotropy in any element!

C44/C’ ~ 8.4

### Shear anisotropy. Expt. vs Theory

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The high temperature phase, (epsilon) is body centered cubic, and has a smaller volume than the (fcc) delta phase.

What drives this phase transition?

Having a functional, that computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002)

### The delta –epsilon transition

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### Phonon frequency (Thz ) vs q in epsilon Pu.

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Epsilon is slightly more delocalized than delta, has SMALLER volume and lies at HIGHER energy than delta at T=0. But it has a much larger phonon entropy than delta.

At the phase transition the volume shrinks but the phonon entropy increases.

Estimates of the phase transition following Drumont and Ackland et. al. PRB.65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.

### Phonon entropy drives the epsilon delta phase transition

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### Phonons epsilon

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Develop new methods for treating realistic (system specific) strongly correlated electrons.

The DMFT machinery is in a very primitive state.

Study interesting materials science problems, develop some qualitative understanding of materials properties. Perform quantitative calculations.

DMFT- in its current state of the art, allows us to do both.

### Conclusion

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Serious bottle neck of current interface of DMFT and band theory: U as a frequency independent parameter. Solution: E-DMFT +GW. [G. Kotliar, S.Savrasov, in New Theoretical approaches to strongly correlated systems, Edited by A. Tsvelik, Kluwer Publishers, (2001). Cond-matt 0308053, S. Biermann F. Aeryasetiwan, A. Georgs PRL 2003]

Fully implemented at the level of model Hamiltonian [Ping Sun’s talk]. Needs to be carried over to electronic structure.

Need further improvements of both electronic structure and many body tools. Illustrated compromises [Ti2O3 cluster, single site QMC +downfolding, Pu spectra and energy IPT+ ASA, Pu Phonons single site DMFT full potential+very primitive solver.

### Conclusion

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Careful comparison with experiments. What do we need to reproduce the softening of the 111 phonon ? Better solver at the single site level? Or cluster treatment of fcc Pu.

Need further developments in linear response dynamics to accommodate better solvers.

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Colossal Magneto-resistance LaSrMnO3

LaSrTiO3 doping driven Mott transition

Itinerant Magnetism: Iron Nickel

Half Metals

Pressure Driven Mott Transition V2O3

Presssure Driven Metal to Charge Transfer Insulator NiSeS

Kappa Organics

### A partial list of applications of DMFT to materials.

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Cerium : alpha to gamma transition

Plutonium: delta and epsilon phase

Mott insulators, phonons and spectra, NiO, MnO

Bandwith control CaSrVO3, CaVO3 SrVO3

Heavy fermion without f eleLiV$_{2}$O$_{4}$ctrons

Fullerines K$_{n}$C$_{60}$}

Bechgaard Salts

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Biermann:2001

Quantum criticality of CeCuAu Si et.al.

Heavy Fermion Insulators. Saso et.al.

CrO$_2$. Laad et.al.

FlNaV$_{2}$O$_{5}$ Fluctuating charge order

Strongly Inhomogenous systems, surfaces and surface phase transitions.

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Perfetti:2003,

Liebsch:2003}.

{Ruthenates} Sr$_{2}$RuO$_{4}$ Orbital differentiation.

Ti2O3 Metal to insulator transition

VO2 Metal to Insulator Transition.

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### Rewrite the TOE as an electron boson problem.

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

• r=R+r

• R unit cell vector

• r position within the unit cell. Ir>=|R, r>

• Couple sources to

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• Legendre transfor the sources, eliminating the field f,

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

and G (r R,r’ R)

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### “Kohn Sham “ decomposition.

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### (E)DMFT pproximation to

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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### LDA+DMFT functional

F Sum of local 2PI graphs with local U matrix and local G

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### Epsilon Plutonium.

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### Lattice and cluster self energies

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### DMFT Impurity cavity construction

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### Outer loop relax

Edc

G0

Impurity Solver

Imp. Solver: Hartree-Fock

G,S

U

SCC

DMFT

LDA+U

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Static limit of the LDA+DMFT functional ,

with FatomFHF reduces to the LDA+U functional

of Anisimov Andersen and Zaanen.

Crude approximation. Reasonable in ordered Mott insulators.

Total energy in DMFT can be approximated by LDA+U with an effective U . Extra screening processes in DMFT produce smaller Ueff.

ULDA+U < UDMFT

### LDA+DMFT and LDA+U

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

Gap vs U, Exact solution Lieb and Wu, Ovshinikov

Nc=2 CDMFT

vs Nc=1

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QP bands in ruthenides: A. Liebsch et al (PRL 2000)

N phase of Pu: S. Savrasov G. Kotliar and E. Abrahams (Nature 2001)

MIT in V2O3: K. Held et al (PRL 2001)

Magnetism of Fe, Ni: A. Lichtenstein M. Katsenelson and G. Kotliar et al PRL (2001)

J-G transition in Ce: K. Held A. Mc Mahan R. Scalettar (PRL 2000); M. Zolfl T. et al PRL (2000).

3d doped Mott insulator La1-xSrxTiO3 Anisimov et.al 1997, Nekrasov et.al. 1999, Udovenko et.al 2002)

………………..

### Very Partial list of application of realistic DMFT to materials

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Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997).

Lichtenstein and Katsenelson PRB (1998).

Reviews:

Held Nekrasov Blumer Anisimov and Vollhardt et.al. Int. Jour. of Mod PhysB15, 2611 (2001).

A. Lichtenstein M. Katsnelson and G. Kotliar (2002)

### LDA+DMFT References

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Introduce local orbitals, caR(r-R), and local GF

G(R,R)(i w) =

The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for r(r) and G and performing a Legendre transformation, G[r(r),G(R,R)(iw)]

### Spectral Density Functional : effective action construction

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### LDA+DMFT Self-Consistency loop

E

U

DMFT

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Static limit of the LDA+DMFT functional , with F= FHF reduces to LDA+U

Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing.

Luttinger theorem is obeyed.

Functional formulation is essential for computations of total energies, opens the way to phonon calculations.

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LDA+DMFT:

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 G.Kotliar funcional formulation for full self consistent implementation of a spectral density functional.

Application to Pu S. Savrasov G. Kotliar and E. Abrahams (Nature 2001).

### References

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Long range Coulomb interactios, E-DMFT. R. Chitra and G. Kotliar

Combining E-DMFT and GW, GW-U , G. Kotliar and S. Savrasov

Implementation of E-DMFT , GW at the model level. P Sun and G. Kotliar.

Also S. Biermann et. al.

### References

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Put in the loop of dmft +lda and the functional

And chitra. Put in the effective action perspective. Put in the coupling constant integration.

Put in the cluster.

Think of formula for simga-lattice.

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Basic idea: reduce the quantum many body problem to a one site or a cluster of sites, in a medium of non interacting electrons obeying a self consistency condition.

Basic idea: instead of using functionals of the density, use more sensitive functionals of the one electron spectral function. [density of states for adding or removing particles in a solid, measured in photoemission]

### Dynamical Mean Field Theory

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Master plan. 1) Fix titanite section by putting meet.

2) Fix plutonium section by transporting and putting meet. From Berkeley.

3) Put the ideology. Overview of how really DMFT is used. Models + non models. And within models two pictures. Including the effective action perspective. 1] Coupling constant integration formula for DMFT models.

4) Conclusion. EDMFT in r,r’ and non local corrections around it. Indirect evidence, Ping successes . Indirect evidence, from local GW, that it gives the U’s we need for DMFT……….

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Correlated electrons.

Model Hamiltonians.

DMFT-two perspectives-models and functionals.

cavity.-mention cluster.

How good the local approximation is.

Functional perspective-effective action

DMFT as an exact functional-DMFT as an approximation.

Interface with electronic structure-Anisimov.

Interace with a functional.

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Introduce local orbitals, caR(r-R)orbitals, and local GF

G(R,R)(i w) =

The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for r(r) and G and performing a Legendre transformation, G[r(r),G(R,R)(iw)]

Approximate functional using DMFT insights.

### Spectral Density Functional : effective action construction (Chitra and GK).

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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.

Introduce auxiliary field

Exact “local self energy”

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• One can derive a coupling constant integration formulae (Harris Jones formula) for

• Generate approximations.

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

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Atoms as a reference point. Expansion in t.

Locality does not necessarily mean a single point. Extension to clusters.

Jii --- Jii Ji i+d

Aii --- Ai i+d

S ii --- S i i+d

Exact functional G[Aii ,Ai i+d]

The 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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• 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)

• 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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Basic idea: reduce the quantum many body problem to a one site or a cluster of sites, in a medium of non interacting electrons obeying a self consistency condition.

Basic idea: instead of using functionals of the density, use more sensitive functionals of the one electron spectral function. [density of states for adding or removing particles in a solid, measured in photoemission]

### Dynamical Mean Field Theory

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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 and Abrahams funcional formulation for full self consistent Nature {410}, 793(2001).

Reviews: Held et.al. , Psi-k Newsletter \#{\bf 56} (April 2003), p. 65 Lichtenstein Katsnelson and Kotliar cond-mat/0211076:

### Realistic applications of DMFT References: combinations of DMFT with band theory.

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### Comparaison with the Hartree Fock static limit: LDA+U.

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### Dependence on structure

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The local Greens function A, and the auxilliary quantity S, 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.

### Mapping onto impurity models.

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

Crystal structure +Atomic positions

Model Hamiltonian

Correlation Functions Total Energies etc.

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