Towards First-Principles Electronic Structure Calculations of Correlated Materials Using Dynamical Mean Field Theory (DMFT). Gabriel Kotliar Physics Department

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: Ti2O3Case study 2: Elemental PuConclusions: Future developments.

THE STATE UNIVERSITY OF NEW JERSEY

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Strongly Correlated Electrons 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.


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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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Dynamical Mean Field Theory 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]


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Phys. Rev. B 45, 6497 A. Georges, G. Kotliar (1992)

†

0 0 0

( )[ ( ')] ( ')o o o oc c U n nb b b

s st m t t tt ¯

¶+ - D - +

¶òò ò

( )wD†

( )( ) ( )L o n o nG c i c iw w D=- á ñ

1( )

1( )

( )[ ][ ]

nk

n kn

G ii

G i

ww e

w

=D - -

D

å

† †

, ,

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

t c c c c U n n

DMFT


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DMFT action and self consistency condition

1

10

1( ) ( )

( )n nn k nk

G i ii t i

w ww m w

-

-é ùê ú= +Sê ú- + - Së ûå

†

0 0

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

s st t t t d t t ¯+òò

10 ( )nG iw-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)

( )niwS

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


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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 .

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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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, [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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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.

Observable: Local Greens function Gii ().

Exact functional [Gii () DMFT Approximation to the functional.(Muller

Hartman 89)

[ , ] log[ ] ( ) ( ) [ ]DMFT DMFTij ii iin n niG Tr i t Tr i G i Gw w w-G S =- - S - S +Få

[ ] Sum of 2PI graphs with local UDMFT atom ii

i

GF = Få


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LDA+DMFT (II)

G0 G

Im puritySolver

S .C .C .

0( ) ( , , ) i

i

r T G r r i e w

w

r w+

= å

2| ( ) | ( )k xc k LMTOV H ka ac r c- Ñ + =

DMFT

U

Edc

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å


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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

Correlation Functions Total Energies etc.

Model Hamiltonian

Crystal structure +Atomic positions


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Interfacing DMFT with band theory

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


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LDA+DMFT (I) 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).


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Application to Ti2O3


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Metal to insulator transition in Ti2O3

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?


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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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Ti2O3 vs V2O3

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.


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Earlier work.

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).


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Ti2O3 LDA-DOS

LTS HTS


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Methodology:1 and 2 site CDMFT

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.


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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 phase (A. Lawson LANL)


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Elastic Deformations

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

Uniform compression:p=-B V/V Volume conserving deformations:

F/A=c44 x/L F/A=c’ x/L


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DFT studies

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 30-Is 30-35% lower than experiment35% lower than experiment

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


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DFT Studies

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

Experimentally 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

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


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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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Other problems with LDA+U Predicts plutonium to be magnetic. Different theories of alpha and delta.


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DMFT - Technical details [spectra and energy]

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).


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DMFT- Phonon Spectra 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.


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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.

Summary

LDA

LDA+U

DMFT

Spectra Method E vs V


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Phonon Spectra

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.


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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 - EfQ =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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Shear anisotropy. Expt. vs Theory

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


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The delta –epsilon transition

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)


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Phonon frequency (Thz ) vs q in epsilon Pu.


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Phonon entropy drives the epsilon delta phase transition

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.


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Phonons epsilon


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Conclusion

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.


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Conclusion 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 Kotliar, S.Savrasov, in New Theoretical approaches to strongly New Theoretical approaches to strongly correlated systemscorrelated systems, Edited by A. Tsvelik, Kluwer Publishers, (2001). , Edited by A. Tsvelik, Kluwer Publishers, (2001). Cond-matt 0308053, S. Biermann F. Aeryasetiwan, A. Georgs PRL Cond-matt 0308053, S. Biermann F. Aeryasetiwan, A. Georgs PRL 2003]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.


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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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A partial list of applications of DMFT to materials.

Colossal Magneto-resistance LaSrMnO3 Double PerovskitesChattopadhyay:2001:PRB} 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


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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


RUTGERS

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 Chattopadhyay:2001Magnetic Semiconductors 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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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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LDA+DMFT functional2 *log[ / 2 ( ) ( )]

( ) ( ) ( ) ( )

1 ( ) ( ')( ) ( ) ' [ ]

2 | ' |

[ ]

R R

n

n KS

KS n n

i

LDAext xc

DC

R

Tr i V r r

V r r dr Tr i G i

r rV r r dr drdr E

r r

G

a b ba

w

w c c

r w w

r rr r

- +Ñ - - S -

- S +

+ + +-

F - F

åò

ò òå

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

1[ ] ( 1)

2DC G Un nF = - ( )0( ) iab

abi

n T G i ew

w+

= å

KS ab [ ( ) G V ( ) ]LDA DMFT a br r


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Epsilon Plutonium.


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Lattice and cluster self energies


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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 o o oc Go c n n U n nb 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

( , ')Do t t+


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Outer loop relax

0( ) ( , , ) i

i

r T G r r i e w

w

r w+

= å


U

Edc

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å

ff &

Impurity Solver

SCC

G,G0

DMFTLDA+U

Imp. Solver: Hartree-Fock


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LDA+DMFT and LDA+U • Static limit of the LDA+DMFT functional , • with atom HF 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

®


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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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Very Partial list of application of realistic DMFT to materials QP bands in ruthenides: A. Liebsch et al (PRL 2000) 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) 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) ………………..


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LDA+DMFT References

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

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


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Spectral Density Functional : effective action construction

Introduce local orbitals, R(r-R), and local GF G(R,R)(i ) =

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

' ( )* ( , ')( ) ( ')R Rdr dr r G r r i r


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LDA+DMFT Self-Consistency loop

G0 G

Im puritySo lver

S .C .C .

0( ) ( , , ) i

i

r T G r r i e w

w

r w+

= å


DMFT

U

E

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å


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Comments on LDA+DMFT• Static limit of the LDA+DMFT functional , with

= HF 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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References

LDA+DMFT: V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and

G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997). ALichtensteinandM.KatsenelsonPhys.Rev.B57,6884(1988).

S.SavrasovG.Kotliarfuncionalformulationforfullselfconsistentimplementationofaspectraldensityfunctional.

ApplicationtoPuS.SavrasovG.KotliarandE.Abrahams(Nature2001).


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References

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.


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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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Dynamical Mean Field Theory 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]


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

Introduce local orbitals, R(r-R)orbitals, and local GF G(R,R)(i ) =

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

Approximate functional using DMFT insights.

' ( )* ( , ')( ) ( ')R Rdr dr r G r r i r


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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+ ExactfunctionalAii ,Ai i+ helatticeselfenergyandothernonlocalquantitiesextendingbeyondtheclusterareOUTSIDEtheformalismandneedtobeinferred.


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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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Dynamical Mean Field Theory 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]


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

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

ALichtensteinandM.KatsenelsonPhys.Rev.B57,6884(1988).

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Reviews:Heldet.al., Psi-k Newsletter \#{\bf 56} (April 2003), p. 65LichtensteinKatsnelsonandKotliarcond-mat/0211076:


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


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Dependence on structure


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Mapping onto impurity models. The local Greens function A, and the auxilliary

quantity canbecomputedfromthesolutionofanimpuritymodelthatdescribestheeffectsoftherestofthelatticeontheonaselectedcentralsite.

Onecanarriveatthesameconceptviathecavityconstruction.


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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

Documents

Towards First-Principles Electronic Structure Calculations of Correlated Materials Using Dynamical Mean Field Theory (DMFT). Gabriel Kotliar Physics Department