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

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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

- 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).

A(w)

10

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. 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

- 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.

Introduce projector GlocWloc

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

Proof of Principle Implementation

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

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 [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 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.

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

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 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)

- 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…………..

Broad Range of Viewpoints

- 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

.

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

J~0.35 gives correct order ofMagnituder.

FeAs and Mott Hunds materialsHubbard 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

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 . 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 ]

M*/M=1.6

OPTICAL CONDUCTIVITY TO THE RESCUE ?

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)

Go back to basics: U’s for DMFT. ( 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 rescalingRigorous 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

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

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

DOS [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!!!

Possible tests of these ideas and methods

- 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 ironarsenide 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.

- 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 ?

- 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 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

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

Spin polarization of the frequency dependent selfenergy (real part). Frequency dependent exchange splitting. Large at high energies.

Orbital polarization of the frequency dependenthybridization Weiss field. Lives only at very low energies.

Magnetic Stripe Phase of the 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) 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.

- 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 materialsPostdocs –Student-Visitor-Resarch Scientist positions Available!!!! kotliar@physics.rutgers.edu

IOWA

RUTGERS

DOE

BES

$$$

ARIZONA

UC DAVIS

When to use DMFT

- 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

- 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 !]

Challenges

- 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

Challenges

- Non Equilibrium
- Longer Scales

Nanostructures-interfaces-impurities

Local self energy approximation

Practical Route to Total Energies LDA+DMFT 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 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 Zeyn S. Savrasov and G. K PRL 96, 226403 (2006)

Cutoff Radius R

Solver

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

Conceptual Underpinning : 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

- 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.

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

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 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)

- 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.

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).

13

Second problem: soft underbelly of the interface between electronic structure and DMFT .

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