Strongly Correlated Materials: the Dawn of the Theoretical Spectroscopy Era ? Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Experimental Spectroscopies Theoretical Spectroscopies Physical Insights into complex Materials Argonne National Laboratories August 18 th

outline Brief intro Exhibit A : actinides Exhibit B : iron pnictides Conclusion

Weakly Correlated Systems : Success of theoretical approaches are based on having a good reference system to describe the relevant materials. 2

“Standard Model of Solids “ Band Theory. Fermi Liquid Theory (Landau 1957). Density Functional Theory (Kohn Sham 1964) energy functional of the density. Reference Frame for Weakly Correlated Systems. Starting point for perturbation theory in the screened Coulomb interactions (Hedin 1965) M. VanSchilfgaarde Phys. Rev. Lett. 93, (2004) + [ - ] Many other properties can be computed, transport, optics, phonons, etc… 3

The “free electron” reference frame does not work for strongly correlated materials. Interesting systems with promising properties under intensive investigation. 4

How to Make Correlated materials ? Put open shell in a cage Oxygen transition metal ion Cage : e.g 6 oxygen atoms (octahedra) or other ligands/geometry Build crystal with this building block or build layers separated by spacers Transition metal (open shell ) Transition metal ions Rare earth ions Actinides Li x CoO2, Na x CoO2 Battery materials Thermoelectrics VO 2 Room temperature MIT La 1-x SrxMnO3 Colossal Magnetoresistance La 1-x Sr x CuO4 High temperature superconductor 5

How to find really interesting correlated materials ? Serendipity An aptitude for making desirable discoveries by accident The Edisonian approach to innovation is characterized by trial and error discovery rather than a systematic theoretical approach. (e.g. carbon microphone, basis of telephone) + E disonian approach 6 The method works ! Resulted in fascinating compounds. Correlated electron materials do “big things “. Large volume collapses, ultra strong magnets, heavy fermions, ………., high temperature superconductivity …… New phenomenal every few years…….. The historical record indicates that Edison's approach was much more complex, that he made use of available theories and resorted to trial and error only when no adequate theory existed But the serendipity part is is a bit slow…. … 6

Spectroscopic tools to “see” the inner working of materials ARPES PES Optics EELS INS IXS Elastic neutron scatttering ……………………. 7

Dynamical Mean Field Theory a reference frame for describing strongly correlated materials 8

Mean Field Theories Replace a many body problem by a single site problem in an effective medium reference frame DMFT A. Georges and G. Kotliar PRB 45, 6479 (1992). DMFT self consistency : medium to reproduce the exact (best ) local spectral function of the problem. Effective medium: quantifieds the notion of “ metallicity” or itineracy 9

LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). Katsenelson and Lichtenstein Sum all “LOCAL “ Feyman graphs on top to a “traditional” mean field self energy for the less correlated orbitals. “LOCAL” refers to a predetermined d or f orbitals (projector) on a given site or “cluster” of sites. Screened Coulomb interactions [ orbitally dependent concept] Total energy functional of density AND spectra. Spectral density functional. Dual description Bands in a frequency dependent “ LOCAL “ potential. atomic shell embedded in a medium. Many advances in implementation by many groups. Reviews : K. Held xxxxx G. Kotliar S. Savrasov K. Haule O Parcollet V. Oudovenko and C. Marianetti RMP (1996) 10

Are the f electrons in An itinerant or localized ? Spin Density Functional Theory Calculations predicts a Moment of 5 bohr magnetons for Pu. No fluctuating or static moment has been found. What is the f-valence of Pu ? Actinide based materials : Nuclear Physics is well understood. Solid State Physics Strong Correlation Problems Remain 11

Localization Delocalization in Elements Mott Transition  Ideal arena for developing DMFT based spectrocopies  Pu  13

DMFT Phonons in fcc  -Pu. Inelastic X ray scattering ( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et.al, Science, 22 August 2003) 14

Angle Integrated Photoemission Angle Integrated Photoemission Havela et. al. Phys. Rev. B 68, (2003) Havela et. al. Phys. Rev. B 68, (2003) Pu is non magnetic – Cm is magnetic TN ~ 150 K. K.Haule J. Shim and GK Nature 446, 513 (2007) 15

18 KSuperconductivity in Pu compound revived the search for Pu moment 16

17

Magnetic form factors are sensitive to valence. So far they theory available has been either in the fully localized and fully atomic limit. Implementation in LDA+DMFT removes this limitation.

Spectroscopical information has to be analyzed in a theoretical context. Experimental Spectroscopies Theoretical Spectroscopies Physical Insights into complex Materials Few parameters many tests. Iterative process. Focus on deviations from dmft results

Desperatly looking for Pu’smoment. Found by C. Marianetti. applying negative pressure to a supercomputer. New algorithm for exact solution of the LDA+DMFT equations. New LDA+DMFT code. Validated earlier approximate treatment.

Paglione and Greene Nature Physics 6, 645(2010) The “space of Fe pnictide/chalcogenide materials” K 1-x Fe 2-2x Se 2 Simple itinerant metals ? Localized Fe d electrons ? Mechanism of superconductivity ? Real time test for methodologies

Fe electrons have a dual character displaying both itinerant and localized, wave and particle like character. Fe electrons have a dual character displaying both itinerant and localized, wave and particle like character. LDA calculation. D. J Singh and M.H. Du Phys. Rev. Lett. 100, (2008), I. Mazin, M. Johannes and collaborators. Itinerant magnets. LDA should be corrected by long wavelength fluctuations effects. LDA overestimate of moment due to proximity to quantum critical point. Include fluctuating twin and antiphase domain boundaries. Haule K, Shim J H and Kotliar G Phys. Rev. Lett. 100, (2008). Parent compounds correlated multiorbital bad semi-metals” U< Uc2, m*/m~3-5). Quite different from LaSrCuO4. Band theory should be supplemented by local correlations to capture local quantum fluctuations. LDA+DMFT +extensions correlations are implemented via F0 F2 F4 applied to a set of orbitals (projector) Localized point of view, magnetic frustration. Q.Si and E.Abrahams Phys. Rev. Lett. 101, (2008). Extension of the t-J model to S=2 multiorbital situation

Early DMFT predictions Importance of correlations Mass enhancement 3-5 Unconventional SC Phonon Tc<1K 23 Generic values of U, 4 ev, and J =.9 ev, orbital built on a 4 ev window

Hunds metals not doped Mott insulators Hunds metals not doped Mott insulators Strength of correlations are due to Fe Hunds rule J not to Hubbard U. K. Haule and G. Kotliar cond-mat arXiv: K. Haule and G. Kotliar cond-mat arXiv: New Journal of Physics 11 (2009) LDA value U=5ev Orbital blocking. In d^6 configuration exponential amplification is regulated by x- fields. Very different than oxides. 24 Later work by other DMFT groups. Liebsch and Ishida, Sangiovanni and Held, Aichorn et. al.

Optical Spectroscopy can be used to determine the mass enhancement relative to the band theory mass (LDA) M. M. Qazilbash et. al. Nature Physics 5, 647 (2009) LDA+DMFT had predicted correlation effects m/m* ~ this WAS seen in OPTICS. 25

2/28/2016 Experiment: W. Z. Hu, et al, PRL 101, (2008). Nakajima, M. et al. Phys. Rev. B 81, (2010). Correct plasma ω p: DMFT ~ 1.6eV Exp ~ 1.6eV LDA ~ 2.6eV 3 peak structure beyond SDW gap beyond SDW gap Good agreement at high energy M. Moment & Optics by LDA+DMFT Z. Yin, KH, G Kotliar, Nature Physics (2011). LDA+DMFT moment: 0.9 LDA moment 2µ B Exp moment: 0.87 µ B 26

Optical anisotropy predicted Good conductivity in x AF (longer) direction Pauli blocking hinders electron hopping In FM(y) direction First two excitations only in AFM x-direction Larger SDW gap in FM y-direction QP want to decrease their kinetic energy by choosing “best path”  orbital ordering DYNAMICAL NEMATICITY Experiment: M. Nakajima, …,S Uchida, PNAS 108, (2011) See also DiGiorgi EPL (2011) Yin, Haule GK Nature Physics Yin, Haule GK Nature Physics

Neutron spectroscopy with LDA+DMFT Theory : H. Park, K. Haule and GK Experiments: L Harriger H. Luo M. Liu T. Perring C Frost H. Ju M. Norman and Pengcheng Dai : arXiv:

Needed absolute intensities Needed absolute intensities f.m. in RPA calculation (U=0.8eV, J=0.2eV) Experiment by Liu …Pengcheng Dai Large fluctuating moment can not be explained by a purely itinerant model- doping dependence different than in localized model. Understandable in the Hunds metal picture. and localized nature. Fluctuating moment by neutrons: 29

Spectroscopical information has to be analyzed in a theoretical context. Experimental Spectroscopies Theoretical Spectroscopies Physical Insights into complex Materials OpticsINSArpesRaman Hunds metals Diffuse hubbard bands Dynamical nematicity Large at low frequency Small occupation imbalance Few parameters many tests. Iterative process. Focus on deviations from dmft results 30

DMFT based Theoretical spectroscopy what have we gained ? Justifies the simplified reference frame. Validates the concepts we use to think about materials. Compare different “states” compare at the same temperature, pressure doping, etc. Definition of “mechanism”. D.J. Scalapino and S.R. White, Phys. Rev. B 58, 8222 (1998) M. Norman M Randeria M. Janko and JC Campuzzano PRB 61, H i rschScience 295, 2226 (2002).

Superexchange Mechanism?. K. Haule and GK Phys. Rev. B 76, (2007). Ex= J ij ( s - n )/t D.J. Scalapino and S.R. White, Phys. Rev. B 58, 8222 (1998). How is the energy distributed in q and w ? Expts; Dai et.al. 18

DMFT based Theoretical spectroscopy what have we gained ? organizes the space of materials. Identifies chemical handles First step towards theory assisted materials design

angledistance Overal trend consistent with Fe-As distance Yin Haule GK nature materials 2011 Hybridization with pnictogen

UC and UN are known metallic nuclear fuels with high melting temperatures. Prediction:U(C.5 N.5 ) should have improved thermal conductivity !!

Summary Acknowledgments Introduction to correlated materials and the case for theoretical spectroscopy. Actinides and mixed valence in Pu, work with K. Haule J. Shim S. Savrasov M. Pezzoli Spectroscopies: Inelastic X ray, PES, XAS, EELS, elastic neutron scattering Iron pnictides and chalcogenides as Hunds metals: optical spectroscopy, ARPES, inelastic neutron scattering. Work with K. Haule, Zhiping Yin, Haeyong Park. Thank you for your attention!!!

arXiv: Theory : arXiv: Theory : arXiv: Magnetism and Charge Dynamics in Iron Pnictides Z. P. Yin, K. Haule, G. KotliarZ. P. YinK. HauleG. Kotliar

2/28/2016Zhiping Yin, Rutgers University 39 X’ Z Y’ X’ Z Y’ arXiv:

QUALITATIVE INSIGHTS: a) Strongly Frequency Dependent Spin and Orbital Exchange Splitting b) Spin splitting large at high frequency. Orbital splitting large at low frequency. c) Qualitative difference between BaFe2As2 and Oxides. QUALITATIVE INSIGHTS: a) Strongly Frequency Dependent Spin and Orbital Exchange Splitting b) Spin splitting large at high frequency. Orbital splitting large at low frequency. c) Qualitative difference between BaFe2As2 and Oxides. Z. P. Yin, KH, G. Kotliar Nature Physics in press.

DOS and valence histogram There is transfer of spectral weight to high energies, spectral weight is conserved. But the DOS is featuresless no satellites, and resembles the LDA! Strong Correlations without Hubbard bands. Big difference between oxides and pnictides important role of As.

Ba122 One particle spectra of Hunds metals Renormalized quasiparticle peak Loses weight with increasing mass No clear Hubbard band, but the spectra is incoherent Renormalized q.p. peak High energy not very different from LDA Incoherent spectra m*~3m B m*~7m B

usually larger, but not when pnictogen height large! Destructive interference leads to kinetic frustration! Effective low energy hoppings Kinetic frustration (not magnetic frustration)

What is the valence in the late actinides ? Plutonium is MIXED VALENCE. DMFT valence histogram. Shim Haule and GK Nature, 446, (2007)

DMFT Concepts Valence Histograms. Describes the history of the “atom” in the solid, multiplets! Weiss Weiss field, collective hybridization function Weiss Weiss field, collective hybridization function Solid = collection of atoms in a medium Functionals of density and spectra give total energies Can be used in combination with GW… Locality Self Energies Correlated Bands Correlated Bands REVIEWS: A. Georges W. Krauth G. Kotliar and M. Rozenberg RMP (1986) G. Kotliar S. Savrasov K. Haule O Parcollet V. Oudovenko and C. Marianetti RMP (1996)

The basic DMFT concepts and techniques were extended to a realistic setting. LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). Sum all “LOCAL “ Feyman graphs on top to a “traditional” mean field self energy. “LOCAL” refers to a predetermied orbital ( d or f ), on a given site or “cluster” of sites. Screened Coulomb interactios and mean field hamiltonian can be computed self cosistently. Total energy functional of densit AND spectra. Spectral density functional. Bands in a frequecy dependent LOCAL potential. “ Self Energy “ Realistic f or d atomic shell embedded in a medium. (Weiss field ). State of the art implementatoin Haule et. al. PRB(2009). Others are in progress. Review: (imitations of early implementations were removed in the third generation of LDA+DMFT methods (see for ex. K. Haule K. Kim and C. Yee Phys. Rev. B 81, (2010) for a detailed description ). Still LDA+DMFT is not a fully ab-initio method, yet. Depends on parameters, F0, F2, F4, F6 (F0 “screened U”) and Edc ( close to that of localized limit ). More important depends on the choice of orbital or projector.[besides one electron stuff + imp solver ]

Desperetly looking for Pu’smoment. Found by C. Marianetti. Pu under (negative ) pressure. C Marianetti, K Haule GK and M. Fluss Phys. Rev. Lett. 101, (2008)

ARPES The origin of several SDW peaks

XAS Branching ratio d 5/2 electron can be excited into either the f 5/2 and f 7/2 level. d 5/2 electron can be excited into either the f 5/2 and f 7/2 level. d 3/2 electron can only be excited into the f 5/2 level. Moore and van der Laan, Ultramicroscopy Thole and Van der Laan. Shim Haule and Kotliar. Typical XAS or EELS spectrum d 3/2 d 5/2 Plutonium cannot be (5f)6 Plutonium cannot be (5f)6 K.T. Moore, M. A.Wall, A. J. Schwartz, B.W. Chung, D. K. Shuh, R. K. Schulze, and J.G. Tobin PRL (2003)

J. Shim K. Haule and G. K. Nature 446, (2007). J. Shim K. Haule and G. K. Nature 446, (2007). Moore, K. T., van der Laan, G., Haire, R. G., Wall, M. A., Schwartz, A. J., and Söderlind, P., 2007, Phys. Rev. Lett. 98, Moore, K. T., van der Laan, G., Wall, M. A., Schwartz, A. J., and Haire, R. G., 2007, Phys. Rev. B 76,

LDA results Finding the f occupancy. Shim Haule and GK. EPL (2008) Curium : correct prediction! PuO strong disagreement with expt.

DMFT concepts: Solids are Made out of Atoms. f shell in a medium. Valence Histogram f shell in a medium. Valence Histogram8 Plutonium has an unusual form of MIXED VALENCE with clear spectral fingerprints. Shim, Khaule Kotliar, Nature, 446, (2007). 13

J. Shim K. Haule and G. K. Nature 446, (2007). J. Shim K. Haule and G. K. Nature 446, (2007). Moore, K. T., van der Laan, G., Haire, R. G., Wall, M. A., Schwartz, A. J., and Söderlind, P., 2007, Phys. Rev. Lett. 98, Moore, K. T., van der Laan, G., Wall, M. A., Schwartz, A. J., and Haire, R. G., 2007, Phys. Rev. B 76,

LDA results Finding the f occupancy. Shim Haule and GK. EPL (2008) Curium : correct prediction! PuO strong disagreement with expt.

Assumptions: t-J Model Hamiltonians and Plaquette DMFT. Question: what is the mechanism for superconductivity? Assumptions: t-J Model Hamiltonians and Plaquette DMFT. Question: what is the mechanism for superconductivity? K. Haule and G. Kotliar (2007) EPL Underdoped Overdoped

Link DMFT. Normal state Real Space Picture. Ferrero et. al. (2010) (similar to plaquette Haule and GK) (2006) Momentum Space Picture: High T Singlet formation. S (singlet),T (triplet) N=2 singlet, triplet E (empty) N=0 1+ states with 1 electron in + orb Underdoped region: arcs shrink as T is reduced. Overdoped region FS sharpens as T is reduced. 17

S. V. Borisenko et al., Phys. Rev. Lett. 105, (2010). LDA DMFT Orbital differentiation->fermi surface

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++

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

GW self energy for Si Beyond 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, (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, , 2006 N Zeyn S. Savrasov and G. K PRL 96, (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.

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, (2008) U is parametrized in terms of Slater integrals F0 F2 F4 …. Savrasov, Kotliar, Abrahams, Nature ( 2001)

Basic Questions Relevant degreens of freedom, effective hamiltonians Strength of the correlations. Localized vs itinerant Fe d electrons Mechanism of the superconductivity and magnetism……. New arena to test the LDA+DMFT methodology [ with and without experimental informantion!]

Critical endpoint Spinodal Uc2 20 P. Limelette et.al. Science 302, 89 (2003) 89 (2003)T=170T=300 M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995) Mo, Denlinger, Kim, Park, Allen, Sekiyama, Yamasaki, Kadono, Suga, Saitoh, Muro, Metcalf, Keller, Held, Eyert, Anisimov, Vollhardt PRL. (2003 ) Mo, Denlinger, Kim, Park, Allen, Sekiyama, Yamasaki, Kadono, Suga, Saitoh, Muro, Metcalf, Keller, Held, Eyert, Anisimov, Vollhardt PRL. (2003 ) High temperature universality and V2O3

Coherence Incoherence Crossover Coherence Incoherence Crossover Hubbard U is not the “relevant” parameter. The Hund’s coupling brings correlations! Specific heat within LDA+DMFT for LaO F 0.1 FeAs at U=4eV LDA value For J=0 there is negligible mass enhancement at U~W! K. Haule and G. Kotliar cond-matK. Haule and G. Kotliar cond-mat arXiv: K. Haule and G. Kotliar cond-mat, LaO F 0.1 FeAs n

Delta Plutonium sits at the edge of the localization delocalization transition. Origin of all the anomalies. The dominant valence is 5f5 but there is strong admixture of 5f6 which quenches the magnetic moment. Mixed valent element. Clear fingerprint in the presence of “quasiparticle multiplets”, low energy peaks which inherit the multiplet structure present at high energy in the atom. Modern Theory of Actinide Physics based on DMFT. Close correspondence and integration of various experimental spectrocopic results Inelastic X ray scattering, XAS, EELS,elastic neutron scattering PES.

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

Photoemission reveals now Z ~.3

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

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

Orbital polarization of the frequency dependent hybridization 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! b)For intermediate  loss in carriers (kinetic energy ) Focus on changes of Neff( , T) at various energy scales  in going to the magnetic state.

Mass enhancement, plasma frequency Optical conductivity PRB 82, (2010) Exp: W.Z. Hu et.al., PRL 101, (2008). Plasma frequency: LDA ~ 2.6eV DMFT ~ 1.6eV Exp ~ 1.6eV Mass enhancement of Fe-5d bands m*/m LDA ~3 Interband peak ~ 0.6eV Drude weight U=5eV, J=0.7eV Theory : Fourth generation of LDA+DMFT methods and codes. Kutepov Haule Savrasov and Kotliar (2010). Mass renormalization without satellites !

Correlation phase diagram and ordered moment of Hunds metals. Yin et al.

Phase diagram :frustrated Hubbard model, integer filling Phase diagram :frustrated Hubbard model, integer filling M. Rozenberg G. Kotliar H. Kajuter G. Thomas PRL75, 105 (1995) T/W 9 Quasiparticles +Hubbard bands Transfer of spectral weight Mott transition Coherence Incoherence Crossover Spectral functions

Critical endpoint Spinodal Uc2 10 P. Limelette et.al. Science 302, 89 (2003) T=170T=300 M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995) Mo, Denlinger, Kim, Park, Allen, Sekiyama, Yamasaki, Kadono, Suga, Saitoh, Muro, Metcalf, Keller, Held, Eyert, Anisimov, Vollhardt PRL. (2003 ) Mo, Denlinger, Kim, Park, Allen, Sekiyama, Yamasaki, Kadono, Suga, Saitoh, Muro, Metcalf, Keller, Held, Eyert, Anisimov, Vollhardt PRL. (2003 ) High temperature universality and V2O3

Pu as a correlated non magnetic f metal. Early LDA + DMFT results. Accounts for the correct volume of delta Pu in a paramagnetic state. [Savrasov Kotliar and Abrahams Nature 410, 793 (2001). Accounts for the correct volume of delta Pu in a paramagnetic state. [Savrasov Kotliar and Abrahams Nature 410, 793 (2001). delta Pu and alpha Pu spectra, QP + Hubbard bands, differ by a subtle redistribution of photoemission spectral weight. delta Pu and alpha Pu spectra, QP + Hubbard bands, differ by a subtle redistribution of photoemission spectral weight. Predicted the phonon spectra of delta Pu [Savrasov et. al. Science (2003) ] in reasonable agreement with subsequent experiments. [Wong et. al. Science (2003)] Predicted the phonon spectra of delta Pu [Savrasov et. al. Science (2003) ] in reasonable agreement with subsequent experiments. [Wong et. al. Science (2003)] Epsilon Pu is stabilized by phonon entropy. Epsilon Pu is stabilized by phonon entropy. Caveats: primitive impurity solvers, no multiplet effects, only PM sates were considered. Caveats: primitive impurity solvers, no multiplet effects, only PM sates were considered. G. Kotliar S. Savrasov K Haule O. Parcollet V. Udovenko C. Marianetti Reviews of Modern Physiscs 78, 865, (2006).