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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)

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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 $upport : NSF -DMR, DOE-Basic Energy Sciences, MURI, NSF materials world network. 1 Collaborators (cuprates) Kristjan Haule Andrey Kutepov S. Savrasov (UC Davis) K. Haule (Rutgers)

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The Great Solid State Dream Machine

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Instrument under constant development/improvement. Power-Limitations-Research Opportunities.

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GW= First order PT in screened Coulomb interactions around LDA 3 Phys. Rev. Lett. 93, 126406 (2004). S. V. Faleev, M. van Schilfgaarde, and T. Kotani,

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Dynamical Mean Field Theory. Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). A( ) 10

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8 atomic levels Quantifying the degree of localization/delocalization Impurity Solver Machine for summing all local diagrams in PT in U to all orders.

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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!!!]

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12 Spectra=- Im G(k, ) 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++

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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 ?

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Conceptual Underpinning Diagrams: PT in W and G. Introduce projector Gloc Wloc : Chitra and Kotliar Phys. Rev. B 62, 12715 (2000) and Phys. Rev.B (2001).

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GW self energy for SiBeyond GW Coordination Sphere Proof of Principle Implementation Full implementation in the context of a a one orbital lattice model. P Sun and G. Kotliar Phys. Rev. B 66, 85120 (2002). Propose GW+DMFT. P.Sun and GK PRL (2004). Test various levels of self consistencyin Gnonloc Pinonloc 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) 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.

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Total energy is derived from a functional of the density and Gloc Total energy is derived from a functional of the density and Gloc CHARGE SELF CONSISTENT LDA+DMFT. S. Savrasov GK (2002) 12 LDA+DMFT as an approximation to the general scheme Recent calculations using B3LYP hybrid + DMFT for Ce2O3. D. Jacob K. Haule and GK EPL 84, 57009 (2008) U is parametrized in terms of Slater integrals F0 F2 F4 …. Savrasov, Kotliar, Abrahams, Nature ( 2001)

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LDA+DMFT Self-Consistency loop [Savrasov Kotliar 2002] Derived from the functional. DMFT U E dc REVIEW : G. Kotliar S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, C.A. Marianetti, RMP 78, 865 (2006).

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Total Energy as a function of volume for Pu W Total Energy as a function of volume for Pu W (ev) vs (a.u. 27.2 ev) Savrasov, Kotliar, Abrahams, Nature ( 2001) Non magnetic correlated state of fcc Pu. N, Zein Following Aryasetiwan Imada Georges Kotliar Bierman and Lichtenstein. PRB 70 195104. (2004) Pu

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DMFT Phonons in fcc -Pu C 11 (GPa) C 44 (GPa) C 12 (GPa) C'(GPa) Theory 34.56 33.03 26.81 3.88 Experiment 36.28 33.59 26.73 4.78 ( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et.al, Science, 22 August 2003)

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Remaining Practical Issues around 2003 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. Physical Picture of Plutonium as a Non Magnetic Strongly Correlated Mixed Valent Metal. Different Phases differ in the Redistribution of Spectral Weight.

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Photoemission Havela et. al. Phys. Rev. B 68, 085101 (2003) Pu is non magnetic – Cm is magnetic TN ~ 150 K. K.Haule J. Shim and GK Nature 446, 513 (2007)

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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…………..

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Iron Pnictides FeSe 1-0.08, (Tc=27K @ 1.48GPa), Mizuguchi et.al., arXiv: 0807.4315

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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.Abrahams Phys. Rev. Lett. 101, 076401. (2008). Localized picture,frustration. t-J model S=3/2 1/2

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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.. LDA+ DMFT for LaF x O 1-x FeAs K. Haule J. Shim and GK. PRL 100, 226402 (2008) The electron phonon coupling is too small to account for superconductivity. ( similar results obtained by D. Singh and Boeri et. al.) 31 “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. “

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FeAs and Mott Hunds materials Hubbard U is not the “relevant” parameter. The Hund’s coupling brings correlations! Specific heat within LDA+DMFT for LaO 1-0.1 F 0.1 FeAs at U=4eV LDA value For J=0 there is negligible mass enhancement at U~W! J~0.35 gives correct order ofMagnitude K. Haule, G. Kotliar, Prediction

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Problem for us: Experimental evidences for weak correlations. NO SATELITES 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) NOT SEEN

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Strong vs Weak Correlation. Transfer of Spectral Weight. Naïve One Band Hubbard Picture When U/W > 1 PT theory fails Z -1 -1 >1 Weight transferred outside W Hubbard Satellites Z <.5 M*/M > 2 Z= 1- b(U/W) 2 Z= 1-(U/Uc) When U/W < 1 PT theory works Z -1 -1 < 1 Spectral weight distributed within W Z>.5 M*/M < 2 YOU CANNOT HAVE THE CAKE [ substantial mass renormalization ] AND EAT IT TOO [not have satellites ]

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Z -1 -1 =.6 M*/M=1.6

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pd model: Z=.5 M*/M=2

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wc=3000cm -1 ~.3 ev Nature Physics 5, 647 (2009) 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 OPTICAL CONDUCTIVITY TO THE RESCUE ?

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Photoemission reveals now Z ~.3

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Eliminate the hybridization to the semicore states included in GW but not in LDA +DMFT by rescaling. Define a projector. Use the same projector in calculating the U’s that you will use in your DMFT calculation 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) Rigorous Definition of the Hubbard U and the Weiss Field Delta in a Solid. Kutepov et.al (2010) Approximate Wloc and Piloc using Self Consistent GW. Kutepov et. al. 2010

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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; yz zx orbitals.

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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

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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!!!

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Theory: Kutepov et.al. Expt Brouet et.al.

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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]

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Magnetic phase of the iron 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.

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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.

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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.

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LDA+DMFT Magnetic moment.95 muB Expt 1 muB L 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)

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Origin of the anisotropy is electronic Optical features sharpen in the polarized spectra. Experimental predictions. Measurements underway ( not easy !)

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

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

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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! b)For intermediate loss in carriers (kinetic energy ) Focus on changes of Neff( , T) at various energy scales in going to the magnetic state.

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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.

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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

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IOWA CMSN network for correlated materials Postdocs –Student-Visitor-Resarch Scientist positions Available!!!! kotliar@physics.rutgers.edu RUTGERS UC DAVIS ARIZONA DOE BES $$$

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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.

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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 !]

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Thanks for your Attention!!

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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

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Challenges Non Equilibrium Longer Scales

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Nanostructures-interfaces-impurities Local self energy approximation

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R. Chitra and Gkotliar Phys.Rev.B62:12715 (2000). S. Savrasov and G. Kotliar PRB Phys. Rev. B 69, 245101 (2004). Practical Route to Total Energies LDA+DMFT functional 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.

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GW self energy for SiBeyond GW Coordination Sphere GW+DMFT Why it should work ? GW+DMFT proposed and fully implmented in the context of a a one orbital lattice model. P Sun and G. Kotliar Phys. Rev. B 66, 85120 (2002). Test various levels of self consistency in Gnonloc Pinonloc P.Sun and GK PRL (2004). See also GW+dc+U Biermann, 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).

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

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Impurity Solver

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Theoretical Spectroscopies.

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Ce In Cerium 115Multiple hybridization gaps 300K eV 10K Larger gap due to hybridization with out of plane In Smaller gap due to hybridization with in-plane In non-f spectra

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

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Conceptual Underpinning : Chitra and Kotliar Phys. Rev. B 62, 12715 (2000) and Phys. Rev.B (2001). Ex. Ir>=|R, > Gloc=G(R , R ’) R,R’ Introduce Notion of Local Greens functions, Wloc, Gloc G=Gloc+Gnonloc. 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 Gloc Gnonloc by keeping perturbative corrections in C-.O.Almbladh, U.von Barth and R.vanLeeuwen Int.J.Mod.Phys. B13, 535 (1999)

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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.

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General impurity problem Diagrammatic expansion in terms of hybridization +Metropolis sampling over the diagrams Exact method: samples all diagrams! Allows correct treatment of multiplets P. Werner et. al. PRL (2006) K.H.aule Phys. Rev. B 75, 155113 (2007) An exact impurity solver, continuous time QMC - expansion in terms of hybridization

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K.Haule J. Shim and GK Nature 446, 513 (2007) Photoemission in Actinides alpa->delta volume collapse transition Curium has large magnetic moment and orders antiferromagnetically Pu does is non magnetic. F0=4,F2=6.1 F0=4.5,F2=7.15 F0=4.5,F2=8.11

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

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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.

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Main DMFT Concepts in electronic structure. 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). Weiss Weiss field, collective hybridization function, quantifies the degree of localization Functionals of density and spectra, total energies: spectral density functional. Local Self Energies and Correlated Bands Orbitally Resolved Spectral Functions Transfer of spectral weight. 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

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

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THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Model Hamiltonians and qualitative considerations in the physics of materials. Or what do we want to.

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