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Towards an ab -initio theory of correlated materials . The challenge of the iron pnictides . Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University. Work with K. Haule Z. Ying A. Kutepov (Rutgers) S. Savrasov (U.C. Davis). Outline.

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Work with k haule z ying a kutepov rutgers s savrasov u c davis

Towards an ab-initio theory of correlated materials . The challenge of the iron pnictides.Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University

Work with K. Haule Z. Ying A. Kutepov (Rutgers) S. Savrasov (U.C. Davis)


Outline
Outline

  • Quantifying the strength of the correlations.

    First Principles DMFT –framework.

  • Case study : Iron Pnictides, do correlations matter ? Early DMFT results.

  • Recent results for the optics.

  • Outlook

Collaborators (cuprates) Kristjan Haule AndreyKutepovS. Savrasov (UC Davis) K. Haule (Rutgers)

$upport : NSF -DMR , DOE-Basic Energy Sciences, MURI, NSF materials world network.

1



Instrument under constant development/improvement.

Power-Limitations-Research Opportunities.


GW= First order PT in screened Coulomb interactions around LDA

S. V. Faleev, M. van Schilfgaarde, and T. Kotani,

3

Phys. Rev. Lett. 93, 126406 (2004).


Dynamical mean field theory cavity construction a georges and g kotliar prb 45 6479 1992
Dynamical Mean Field Theory. Cavity Construction. LDAA. Georges and G. Kotliar PRB 45, 6479 (1992).

A(w)

10


Impurity solver

atomic levels LDA

Impurity Solver

Machine for summing all local diagrams in PT in U to all orders.

Quantifying the degree of

localization/delocalization

8


The power of the method increased tremendously fueled by advances in impurity solvers.

Extensions to cluster of sites, CDMFT capture short range correlations, k dependent self

Over the past few years evidence for the accuracy of the method for Hubbard models accumulated very rapidly from: a) comparison of different cluster sizes,

b) comparison of dmft predictions with experiments

c) experiments in cold atom traps realizing the Hubbard model and compared with DMFT

d) It’s now possible to compare with exact numerical solutions of Hubbard model at high temperatures. (agreement within a few percent) [Gull et. al.]

Just as with LDA, we know how to interpret the results and where and how ( in which direction ) it is biased. Local approximation is very accurate in many regions of parameter space.

Non locality in time is essential. [ Warning for other methods!!!]


LDA+DMFT. V. advances in impurity solvers. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). Lichtenstein and Katsnelson (1998) LDA++

Spectra=- Im G(k,w)

12


Pros issues
Pros------------------- Issues advances in impurity solvers.

  • Freed Model Hamiltonian treatments from many parameters.

  • Brought DMFT to the attention of the electronic stuture community.

  • Serious improvement over LDA+U Multiplets!

  • Triggered numerous collaborations between electronic structure groups and many body theorist

  • Proof of principle of a realistic implementations of multiorbital DMFT. Spectra of LaSr TiO3

  • H[k] is not affected by correlations. No feedback on the density.

  • Parameters (i.e. U, Edc,)

  • Not clear how to compute total Energies.

  • Primitive Impurity Solvers

    (IPT)

  • Embeddings in general basis sets. Beyond LMTO-ASA

  • Projectors to define Gloc.

  • Foundation ?


Conceptual underpinning diagrams pt in w and g
Conceptual Underpinning Diagrams: PT in W and G. advances in impurity solvers.

Introduce projector GlocWloc

: Chitra and Kotliar Phys. Rev. B 62, 12715 (2000) and Phys. Rev.B (2001).


Proof of Principle Implementation advances in impurity solvers.

Full implementation in the context of a a one orbital lattice model.

P Sun and G. KotliarPhys. Rev. B 66, 85120 (2002). Propose GW+DMFT .

P.Sun and GK PRL (2004). Test various levels of self consistencyinGnonlocPinonloc

Test notion of locality in LMTO basis set in various materials.

N. Zeyn S. Savrasov and G. Kotliar PRL 96, 226403, 2006

N Zeyn S. Savrasov and G. K PRL 96, 226403 (2006)

GW self energy for Si

Beyond GW

Still, summing all diagramas with dynamical U and obtaining the GW starting point is extremely expensive. So this is still

a point of principle rather than a practical tool.

Coordination Sphere

Coordination Sphere


LDA+DMFT as an approximation to the general scheme advances in impurity solvers.

U is parametrized in terms of Slater integrals F0 F2 F4 ….

Recent calculations using B3LYP hybrid + DMFT for Ce2O3. D. Jacob K. Haule and GK EPL 84, 57009 (2008)

Total energy is derived from a functional of the density and Gloc

CHARGE SELF CONSISTENT LDA+DMFT. S. Savrasov GK (2002)

Savrasov, Kotliar, Abrahams, Nature ( 2001)

12


Lda dmft self consistency loop savrasov kotliar 2002 derived from the functional
LDA+DMFT Self-Consistency advances in impurity solvers. loop [SavrasovKotliar 2002]Derived from the functional.

Edc

U

DMFT

  • REVIEW : G. Kotliar S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, C.A. Marianetti, RMP 78, 865 (2006).


Total Energy as a function of volume for advances in impurity solvers. Pu W(ev) vs (a.u. 27.2 ev)

Pu

N, ZeinFollowing AryasetiwanImada Georges KotliarBierman and Lichtenstein. PRB 70 195104. (2004)

Savrasov, Kotliar, Abrahams, Nature ( 2001)

Non magnetic correlated state of fccPu.


C advances in impurity solvers. 11 (GPa)

C44 (GPa)

C12 (GPa)

C'(GPa)

Theory

34.56

33.03

26.81

3.88

Experiment

36.28

33.59

26.73

4.78

DMFT Phonons in fcc d-Pu

( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)

(experiments from Wong et.al, Science, 22 August 2003)


Remaining practical issues around 2003
Remaining Practical Issues around 2003 advances in impurity solvers.

Physical Picture of Plutonium as a Non Magnetic Strongly Correlated Mixed Valent Metal.

Different Phases differ in the Redistribution of Spectral Weight.

  • Spectra: no multiplet structure due to solver limitations.

  • Test quality and role of projectors / U’s

  • Need Practical Solvers for Electronic Structure.

  • Need good criteria for good projectors.

  • Problems largely solved in a series of works by K. Haule.


Havela advances in impurity solvers. et. al. Phys. Rev. B 68, 085101 (2003)

Photoemission

Pu is non magnetic – Cm is magnetic TN ~ 150 K.

K.Haule J. Shim and GK Nature 446, 513 (2007)


Unresolved issues around 2008
Unresolved issues ( around 2008) advances in impurity solvers.

  • STILL NEEDED Non empirical reliable determinations of Uabcd and Edc

    together with optimal projector, given finite computational resources [ Wanniers, projective LMTO’s etc ]

    STILL NEEDED practical solvers with retarded interactions.

  • But then came the iron pnictides…………..


Iron pnictides
Iron advances in impurity solvers. Pnictides

FeSe1-0.08, (Tc=27K @ 1.48GPa),

Mizuguchi et.al., arXiv: 0807.4315


Broad range of viewpoints
Broad Range of Viewpoints advances in impurity solvers.

  • D. J Singh and M.H. Du Phys. Rev. Lett. 100, 237003 (2008). Itinerant magnetism.

  • LDA+Spin Fluctuations.

  • Haule K, Shim J H and Kotliar G Phys. Rev. Lett. 100, 226402 (2008)Correlated “Bad Semi-Metal” (U< Uc2) Multi-orbital model. Z ~0.2–0.3.

  • LDA+DMFT +extensions

  • Q.Si and E.AbrahamsPhys. Rev. Lett. 101, 076401. (2008). Localized picture,frustration.

  • t-J model S=3/2 1/2


LDA+ DMFT for advances in impurity solvers. LaFxO1-xFeAs

.

Correlations are important: LaOFeAs is on the itinerant side but not too far away from the metal-insulator transition . Non trivial renormalizations Z~ .2 -.3

Typical parameters U=4eV, J=0.7eV. Bad semimetal at high temperatures. Large scattering rate.

ALL 5 d bands important (for J>0.3) relatively small crystal fields

For J=0.7eV, Uc2 =4.5 Correlated insulator.

“Since the Coulomb correlation is below the critical U for this system to undergo a Mott transition, an itinerant approach might be adequate. This will involve a generalization of the spin-fluctuation exchange mechanism to a multiorbital situation. In adition the pairing interaction might involve not just spin but also orbital fluctuations within the d subshell. “

The electron phonon coupling is too small to account for superconductivity. ( similar results obtained by D. Singh and

Boeri et. al.)

K. Haule J. Shim and GK . PRL

100, 226402 (2008)

31


Feas and mott hunds materials

LDA value advances in impurity solvers.

J~0.35 gives correct order ofMagnituder.

FeAs and Mott Hunds materials

Hubbard U is not the “relevant” parameter.

The Hund’s coupling brings correlations!

Specific heat within LDA+DMFT

for LaO1-0.1F0.1FeAs at U=4eV

Prediction

For J=0 there is negligible mass enhancement at U~W!

K. Haule, G. Kotliar,


Problem for us experimental evidences for weak correlations no satelites
Problem for us: Experimental evidences for weak correlations advances in impurity solvers. . NO SATELITES

NOT SEEN

  • XES: no lower Hubbard band or sharp quasiparticle peak

  • XAS: XAS and RIXS spectra are each qualitatively similar to Fe metal

  • XPS: itinerant character of Fe 3d electrons

  • V. I. Anisimov, et al, PhysicaC 469, 442 (2009)

  • W. L. Yang, et al, PRB 80, 014508 (2009)


Strong vs weak correlation transfer of spectral weight na ve one band hubbard picture
Strong advances in impurity solvers. vs Weak Correlation. Transfer of Spectral Weight . Naïve One Band Hubbard Picture

Z= 1- b(U/W)2

Z= 1-(U/Uc)

Z> .5

M*/M < 2

When U/W < 1 PT theory works

Z-1 -1 < 1

Spectral weight distributed within W

When U/W > 1 PT theory fails

Z-1-1 >1

Weight transferred outside W

Hubbard Satellites

Z < .5

M*/M > 2

YOU CANNOT HAVE THE CAKE [ substantial mass renormalization ] AND EAT IT TOO

[not have satellites ]


Z advances in impurity solvers. -1 -1 =.6

M*/M=1.6


Z=.5 advances in impurity solvers.

M*/M=2

pd model:


OPTICAL CONDUCTIVITY TO THE RESCUE ? advances in impurity solvers.

wc=3000cm-1 ~ .3 ev

M. M. Qazilbash,1,, J. J. Hamlin,1 R. E. Baumbach,1 Lijun Zhang,2 D. J. Singh,2 M. B. Maple,1 and D. N. Basov1

Nature Physics 5, 647 (2009)


Photoemission reveals now Z ~ .3 advances in impurity solvers.


Eliminate the hybridization to the semicore states included in gw but not in lda dmft by rescaling

Go back to basics: U’s for DMFT. ( advances in impurity solvers. Kutepov et. al. building on the PT in G and W by R. Chitra and G. K)

Eliminate the hybridization to the semicore states included in GW but not in LDA +DMFT by rescaling

Rigorous Definition of the Hubbard U and the Weiss Field Delta in a Solid. Kutepov et.al (2010)

.

Define a projector. Use the same projector in calculating the U’s that you will use in your DMFT calculation

Approximate Wloc and Piloc using Self Consistent GW. Kutepov et. al. 2010


Freq dep u matrix well parametrized by f0 f2 f4
Freq. dep. U matrix well advances in impurity solvers. parametrized by F0 F2 F4

F0 = 4:9 eV, F2 = 6:4 eV and F4 = 4:3 eV., nc=6.2

Z =1/2 for x2- y2 and z2 , Z =1/3 f xz; yzzxorbitals.


Theory kutepove et al 2010 expt quazilbash et al
Theory: advances in impurity solvers. Kutepove et.al. (2010) Expt:. Quazilbash,et.al

DMFT F0 = 4:9 eV, F2 = 6:4 eV and F4 = 4:3 eV., nc=6.2


Dos kutepov et al 2010
DOS [ advances in impurity solvers. kutepov et.al. 2010]

There is transfer of spectral weight to high energies, spectral weight is conserved. But the DOS is featuresless no satellites, and resembles the LDA!

Big difference between oxides and pnictides. Pnictides can have substantial mass renormalizations and no sharp satellites!!!


Theory kutepov et al expt brouet et al
Theory: advances in impurity solvers. Kutepov et.al. ExptBrouet et.al.


Possible tests of these ideas and methods
Possible tests of these ideas and methods advances in impurity solvers.

  • Coherent spectral weight in dispersive Fe and As like features would only account for a fraction of the total weight.

  • Compute the properties in the ordered state with the same parameters used for the PM states.

  • LDA has difficulties getting the right moment, does LDA+DMFT solve that problem ? Comparison across families. [ Yin et. al. Work in progress]


Magnetic phase of the iron arsenide compounds important issues
Magnetic phase of the iron advances in impurity solvers. arsenide compounds: important issues.

  • LDA predicted the correct magnetic

    structure [ (0, pi) state [ stripe ]

    Predicted a strong anisotropy of the magnetic excitation spectra. J. Dong et al PRL 83, 27006 (2008). Z. P, Yin, et al, PRL 101, 047001 (2008).

  • PROBLEMS WITH ITINERANT PICTURE

  • LSDA predicts a moment of 2 muB . Expt ~ 1mub. Origin of failure of LSDA ?

  • Needs ad hoc adjustments i.e. negative U, still the spectra does not fit.


  • Anisotropy observed dc transport and in STM. advances in impurity solvers.

  • Chu, J.-H. et al. Evidence for an electron nematic phase

    transition in underdoped iron pnictide superconductors.

    arXiv:1002.3364.

    Chuang, T.-M., et al. Nematic electronic structure in the "parent" state of the iron-based superconductor

    Ca(Fe1xCox)2As2. Science 327, 181 (2010).

  • Is the origin of the anisotropy structural or electronic ?

  • Does the nematic order drive the magnetic order or visceversa ?

    PROBLEMS WITH LOCALIZED PICTURE

  • Short bond ferromagnetic, long bond antiferromangnetic.

  • Short bond is less conductive than the long bond.


Is the magnetism itinerant or localized
Is the magnetism itinerant or localized ? advances in impurity solvers.

  • What drives the magnetism. Itinerant picture [ reduction in double occupancy, cost in kinetic energy ]. Localized picture [ gain in superexchange , pay double occupancy ].

  • What selects the stripe state ?

  • Conductivity measurments + theory should help elucidate these issues.


Lda dmft magnetic moment 95 mub expt 1 mub
LDA+DMFT Magnetic moment .95 advances in impurity solvers. muBExpt 1 muB

EXPT: Hu, W. Z. et al. Phys. Rev. Lett. 101, 257005. (2008).

EXPT: Nakajima, M. et al. Phys. Rev. B 81, 104528 (2010)

Theory Yin et. al. (2010)

L


Origin of the anisotropy is electronic
Origin of the anisotropy is electronic advances in impurity solvers.

Optical features sharpen in the polarized spectra. Experimental predictions. Measurements underway ( not easy !)


Spin polarization of the frequency dependent self advances in impurity solvers. energy (real part). Frequency dependent exchange splitting. Large at high energies.


Orbital polarization of the frequency dependent advances in impurity solvers. hybridization Weiss field. Lives only at very low energies.


Magnetic Stripe Phase of the advances in impurity solvers. FeAs materials: new insights from LDA+DMFT Z. Yin K. Haule and GK [ in preparation]

a) At low energies conductivity goes up. Rapid coherence crossover from an incoherent normal state compensates for a loss of carriers. Gain kinetic energy at very low energies!

  • For intermediate L, loss in carriers (kinetic energy )

Focus on changes of Neff(L, T) at various energy scales L, in going to the magnetic state.


  • c) advances in impurity solvers. Spin polarization is a high energy phenomena, the exchange splitting is three times bigger at infinity than at zero. LSDA will miss that. [ Even with negative U, will not get both spectra and moment right ! ]

  • d) The cost in kinetic energy is compensated by a gain in Hunds rule energy! Mott Hunds metal. [ Hunds is in, Hubbard is out !]

  • e) Validates our earlier view that the Hunds rule coupling and its renormalization governs the physics of this material. At high energies J is strong .7 ev, and induces the magnetic moment.

  • f) At low energies the Hunds rule coupling renormalizes down to zero and good quasiparticles emerge. Coherence incohrence crossover.


Does this matter for superconductivity the next step
Does This Matter for Superconductivity ?? advances in impurity solvers. The next step.

  • Orbital polarization is a LOW energy phenomena. [ Spin and orbital densities behave in complete opposite ways ]. The most favorable form is orbital polarization is ferromagnetic arrange so as to occupy the orbital which can move better along the AF direction. [ i.e. xz moves better along x direction ]

END OF THE EDUCATIONAL PART


CMSN network for correlated materials advances in impurity solvers. Postdocs –Student-Visitor-Resarch Scientist positions Available!!!! [email protected]

IOWA

RUTGERS

DOE

BES

$$$

ARIZONA

UC DAVIS


When to use dmft
When to use DMFT advances in impurity solvers.

  • Designed to treat strongly correlated electron materials in [ for example Mott transition problem in 3d transition metal oxides]

  • Designed to compute one electron spectral functions, photoemission and BIS

  • Designed to treat finite electronic temperature

  • Combines ideas of physics (bands ) and chemistry (local CI). Cluster schemse become expensive!

  • STILL ONLY GAME IN TOWN FOR A SMALL CALLS OF SYSTEMS. AS A GENERAL PURPOSE TOOL AND IN THE “GREY AREAS “ CONSIDER:

  • Relatively new method. Still in rapid developing.


Dmft the middle way
DMFT : the middle way advances in impurity solvers.

  • More expensive than density functional theory ( because it targets spectral properties)

  • Less expensive than direct application of QMC or CI (because it only uses these tools locally )

  • Utilizes advances in electronic structure [ DMFT can be built on top of LDA, hybrid-DFT, GW ] and techniques such as QMC or CI, and its various levels of approx to solve the impurity problem.

  • Greens function method, based on a judicious use of the local approximation. Proved its mettle in the Mott transition problem in the context of the model Hamiltonians.

  • Realistic DMFT Goal, combine those ideas with technology from electronic structure methods to understand and predict properties of correlated materials.

  • Testing methods: “simple” models, experiments, predictive power ? [ lots of fun !]


Thanks for your attention
Thanks for your Attention!! advances in impurity solvers.


Challenges
Challenges advances in impurity solvers.

  • Optimal choice of projectors.

  • Basis sets [LMTO, LAPW, plane waves+PAW’s…..]

  • Optimal description of the “weakly correlated sector” [ dft , GW, hybrids ]

  • Cluster DMFT

  • Determination of the screened F0, F2, F4


Challenges1
Challenges advances in impurity solvers.

  • Non Equilibrium

  • Longer Scales


Nanostructures-interfaces-impurities advances in impurity solvers.

Local self energy approximation


Practical route to total energies lda dmft functional
Practical Route to Total Energies LDA+DMFT advances in impurity solvers. functional

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

Notice Explicit Dependence on : U , DC, and Projectors

[ Orbitals ], and Independence of basis set.

R. Chitraand Gkotliar Phys.Rev.B62:12715 (2000). S. Savrasovand G. Kotliar PRB Phys. Rev. B 69, 245101 (2004).


GW+DMFT Why it should work ? advances in impurity solvers.

GW+DMFT proposed and fully implmented in the context of a a one orbital lattice model.

P Sun and G. KotliarPhys. Rev. B 66, 85120 (2002).

Test various levels of self consistency in GnonlocPinonlocP.Sun and GK PRL (2004). See also GW+dc+UBiermann, F.Aryasetiawan. and A. Georges, PRL 90, 86402 (2003)

Test notion of locality in LMTO basis set in various materials. N. Zeyn S. Savrasov and G. Kotliar PRL 96, 226403, (2006). Include higher order graphs, first implementation of GW+DMFT (with a perturbative impurity solver).

GW self energy for Si

Beyond GW

Coordination Sphere

Coordination Sphere


N advances in impurity solvers. Zeyn S. Savrasov and G. K PRL 96, 226403 (2006)

Cutoff Radius R


Impurity advances in impurity solvers.

Solver


Theoretical spectroscopies
Theoretical advances in impurity solvers. Spectroscopies.


10K advances in impurity solvers.

In

eV

Ce

In

Cerium 115Multiple hybridization gaps

non-f spectra

300K

  • Larger gap due to hybridization with out of plane In

  • Smaller gap due to hybridization with in-plane In


C. Yee G. advances in impurity solvers. Kotliar K. Haule Phys. Rev. B 81, 035105 (2010)


Conceptual underpinning chitra and kotliar phys rev b 62 12715 2000 and phys rev b 2001
Conceptual Underpinning : advances in impurity solvers. Chitra and Kotliar Phys. Rev. B 62, 12715 (2000) and Phys. Rev.B (2001).

C-.O.Almbladh, U.von Barth and R.vanLeeuwenInt.J.Mod.Phys. B13, 535 (1999)

Introduce Notion of Local Greens functions, Wloc, Gloc G=Gloc+Gnonloc .

Ex. Ir>=|R, r> Gloc=G(Rr, R r’) dR,R’

Sum of 2PI graphs

One can also view as an approximation to an exact Spectral Density Functional of Gloc and Wloc. One can do further PT in GlocGnonloc by keeping perturbative corrections in


Pros remaining issues
Pros-------------------Remaining Issues advances in impurity solvers.

  • Freed Model Hamiltonian treatments from many parameters.

  • Brought DMFT to the attention of the electronic stuture community.

  • Serious improvement over LDA+U

  • Triggered numerous collaborations between electronic structure groups and many body theorist

  • Implementation for Spectra of LaSr TiO3

  • H[k] is not affected by correlations. No feedback on the density.

  • Parameters (i.e. U)

  • Not clear how to compute total Energies.

  • Primitive Impurity Solvers

    (IPT)

  • Not very accurate basis sets to write H[k] LMTO-ASA

  • Very primitive projectors to define Gloc.


An exact impurity solver, advances in impurity solvers.

continuous time QMC - expansion in terms of hybridization

P. Werner et. al. PRL (2006) K.H.aule Phys. Rev. B 75, 155113 (2007)

General impurity problem

Diagrammatic expansion in terms of hybridization D

+Metropolis sampling over the diagrams

  • Exact method: samples all diagrams!

  • Allows correct treatment of multiplets


K haule j shim and gk nature 446 513 2007 photoemission in actinides
K.Haule advances in impurity solvers. J. Shim and GK Nature 446, 513 (2007)Photoemission in Actinides

alpa->delta volume collapse transition

F0=4,F2=6.1

F0=4.5,F2=7.15

F0=4.5,F2=8.11

Curium has large magnetic moment and orders antiferromagneticallyPu does is non magnetic.


Keep all bands, no advances in impurity solvers. downfolding, apply correlations to Fe orbitals. Use realistic parameters CAN HAVE MASS RENORMALIZATIONS AND NOT SATELLITES!!!


K haule j shim and g k prl 100 226402 2008
K Haule advances in impurity solvers. J.Shim and G. K PRL 100, 226402 (2008)

  • Since the Coulomb correlation is below the critical U for this system to undergo a Mott transition, an itinerant approach might be adequate. This will involve a generalization of the spin-fluctuation exchange mechanism to a multiorbital situation. In adition the pairing interaction might involve not just spin but also orbital fluctuations within the d subshell.

  • The DMFT approach predicts a renormalized low energy band with a

    fraction of the original width (Z 0.2 – 0.3) . At high temperatures, bad (semi-)metal with large scattering rate. Coherence incoherence crossover.


Main dmft concepts in electronic structure
Main DMFT Concepts in electronic structure advances in impurity solvers. .

Local Self Energies and Correlated Bands

Orbitally Resolved Spectral Functions

Transfer of spectral weight.

Weiss Weiss field, collective

hybridization function, quantifies the degree of localization

Valence Histograms. Describes the

Probability of finding the correlated

state site in the solid in a given configuration

J. H. Shim, K. Haule, and G. Kotliar, Nature London 446, 513 (2007).

Functionals of density and

spectra, total energies: spectral

density functional.

Review: Realistic DMFT. Rev. Mod. Phys. G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti Rev. Mod. Phys. 78,865 (2006).

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