1. Gabriel Kotliar and Center for Materials Theory $upport : NSF -DMR DOE-Basic Energy Sciences Collaborators: K. Haule and J. Shim Ref: Nature 446, 513, (2007) PT Colloquium LANL May 3 rd 2007 1 Strong Correlation Effects in the Actinide Series

2. Band Theory: electrons as waves. Landau Fermi Liquid Theory. Electrons in a Solid:the Standard Model Quantitative Tools. Density Functional Theory Kohn Sham (1964) Rigid bands, optical transitions, thermodynamics, transport……… Static Mean Field Theory. 2 Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965) n band index, e.g. s, p, d,,f M. VanSchilfgaarde

3. Strong Correlation Problem:where the standard model fails Fermi Liquid Theory works but parameters can’t be computed in perturbation theory. Fermi Liquid Theory does NOT work. Need new concepts to replace of rigid bands ! Partially filled d and f shells. Competition between kinetic and Coulomb interactions. Breakdown of the wave picture. Need to incorporate a real space perspective (Mott). Non perturbative problem. 4

4. 5f elements: actinide series 5f elements: actinide series s/cAFFM Deocalisation Localization 1.4K 0.4 K 0.9K0.8K52K25K52K

5. Delocalization Localization in Actinides after G. Lander, Science (2003). Mott Transition    Pu 

6. Basic Questions How does the electron go from being localized to itinerant. How do the physical properties evolve. How to bridge between the microscopic information (atomic positions) and experimental measurements. New concepts, new techniques….. DMFT simplest approach to meet this challenge

7. Phases of Pu A. Lawson. Los Alamos Science

8. Small amounts of impurities stabilize  phase. A. Lawson Los Alamos Science

9. Anomalous Resistivity Maximum metallic resistivity

10. Specific heat and susceptibility. Pu is non magnetic J. Lashley

11. Standard model FAILS in the late actinides Predicts Pu and Am to be magnetic, with a large moment. (about 5  B) Paramagnetic DFT understimates volume of delta Pu by 25 % Many proposals to explain why Pu is non magnetic. Mixed level model Zwicknagl and Fulde, Erickson Balatzki Wills, (5f) 4 conf. LDA+U (Shick, Anisimov) (5f) 6 conf Cannot account for anomalous transport and thermodynamics

12. DMFT Spectral Function Photoemission and correlations Probability of removing an electron and transfering energy  =Ei-Ef, and momentum k f(  ) A(  ) M 2 e Angle integrated spectral function 8 a)Weak Correlation b)Strong Correlation

13. DMFT cavity construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). Happy marriage of atomic and band physics. Extremize functional of A(  ) DMFT cavity construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). Happy marriage of atomic and band physics. Extremize functional of A(  ) Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68, 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004). G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti (RMP 2006). Extremize a functional of the local spectra. Local self energy.

14. Dynamical Mean Field Theory Weiss field is a function. Multiple scales in strongly correlated materials. Exact large coordination (Metzner and Vollhardt 89). Not restricted to single site-CDMFT. Extension to real materials DFT+DMFT. Input slater integrals. Functionals of density and spectra. Review Kotliar et. al. RMP (2006) 12

15. Total Energy as a function of volume for Pu 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. Nick Zein Following Aryasetiwan et. al. PRB 70 195104. (2004) Moment is first reduced by orbital spin moment compensation. The remaining moment is screened by the spd and f electrons

16. Double well structure and  Pu Qualitative explanation of negative thermal expansion[ Lawson, A. C., Roberts J. A., Martinez, B., and Richardson, J. W., Jr. Phil. Mag. B, 82, 1837,(2002). G. Kotliar J.Low Temp. Phys vol.126, 1009 27. (2002)] Natural consequence of the conclusions on the model Hamiltonian level. We had two solutions at the same U, one metallic and one insulating. Relaxing the volume expands the insulator and contract the metal. F(T,V)=Fphonons+ Finvar

17. 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) 21

18. The “DMFT- valence” in the late actinides. Time scale of the fluctuations. Ef*

19. Photoemission Gouder, Havela PRB 2002, 2003

20. Photoemission Spectra[ Shim. Haule,GK Nature (2007)] alpa->delta volume collapse transition F0=4,F2=6.1 F0=4.5,F2=7.15

21. Photoemission and Mixed valence in Pu [Ground State> =a[f 5 (spd) 3 > +b [f 6 (spd) 2 > <f5 ]  ----<f6] <f4 ]  ----<f5] <f6 ]----.<f7]

22. Approach the Mott point from the right Am under pressure Experimental Equation of State (after Heathman et.al, PRL 2000) Mott Transition? “Soft” “Hard”

23. Am equation of state. LDA+DMFT.New acceleration technique for solving DMFT equations S. Savrasov K. Haule G. Kotliar cond-mat. 0507552 (2005) Superconductivity ambient pressure J. L. Smith and R. G. Haire, Science 200, 535 (1978).

24. Mott transition in open (right) and closed (left) shell systems. Superconductivity ? Application to Am ? S S U U  T Log[2J+1] Uc  ~1/(Uc-U) J=0 ??? Tc

26. Resistivity of Am under pressure. J. C. Griveau et.al. PRL 94, 097002 (2005).

27. Photoemission spectra using Hubbard I solver and Sunca. [Savrasov Haule and Kotliar cond-mat 0507552 PRL (2006)] Hubbard bands width is determined by multiplet splittings. Expt Negele, Theory Savrasov Haule

28. Photomission Spectra of Am under pressure. Sunca. Onset of mixed valence. Savrasov Haule Kotliar (2005) PRL (2006)

29. Conclusions Pu and Am are unique strongly correlated elements. Unique mixed valence. They require, new concepts, new computational methods, new algorithms, DMFT ! Interplay of theory and experiment. Many extensions of DMFT are possible, many strongly correlated compounds, research opportunity in correlated materials.

32. Prospects for Extensions and Applications to More Complex Heavy Fermion Systems More complicated crystal structures, more atoms per unit cell. 115’s, alpha Pu…… Non local physics. Heavy fermion quantum criticality. a) Local Quantum Criticality scenario of Q. Si and collaborators. Nature 413 (2001) 804. Single site EDMFT b) Cluster Quantum Multicriticality. L. DeLeo and GK. Requires 2 impurity Kondo model for its description. Better interface with electronic structure

36. Conclusion Am Americium undergoes Mott transition under pressure. [AmIII-AmIV] boundary. Unusual superconductivity and resistivities. Theoretical clue mixed valent due to admixture of (5f) upon application of pressure. Realizes Mott transition from the insulating side, towards a close shell configuration..

38. W 110 =2/3 and banching ratio Moore and van der Laan, Ultramicroscopy 2007.

39. X Ray Absortion and Branching ratio:theory ShimHaule and Kotliar Expt. K. Moore G. Van der Laan G. Haire M. Wall and A. Schartz

41. . Mott transition in the open shell case. Heathman et. al. Science 309,110 (2006) Approach the Mott transition from the right.

42. LS coupling L=0 S=7  jj coupling J=7/2   =2S+L Expt monent. is closer to L S coupling Curium is magnetic Hurray et.al. Physica. B (1980) 217

43. K.Haule and J. Shim Trends in Actinides alpa->delta volume collapse transition Curium has large magnetic moment and orders antif Pu does is non magnetic. F0=4,F2=6.1 F0=4.5,F2=7.15 F0=4.5,F2=8.11

44. Conclusion A Few References …… A.Georges, G. K., W. Krauth and M. J. Rozenberg, Reviews of. Modern Physics 68, 13 (1996). G. K, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, C.A. Marianetti, RMP 78, 865-951, (2006). G. K and D. Vollhardt Physics Today, Vol 57, 53 (2004). 29

45. W 110 =2/3 and banching ratio Moore and van der Laan, Ultramicroscopy 2007.

46. J. Tobin et.al. PRB 72,085109 (2005) K. Moore et.al. XAS white lines branching ratio and EELS: Pu is closer to jj coupling

47. 2/3 in the late actinides [DMFT results: K. Haule and J. Shim ] See the expt. work of K. Moore G. Van der Laan G. Haire M. Wall and A. Schartz Am H2

48. The “DMFT- valence” in the late actinides. Fluctuation time scale Ef *-1

52. 2/3 in the late actinides [DMFT results: K. Haule and J. Shim ] See the expt. work of K. Moore G. Van der Laan G. Haire M. Wall and A. Schartz Am H2

53. W 110 =2/3 and banching ratio See the expt. work of K. Moore G. Van der Laan G. Haire M. Wall and A. Schartz Am H2

54. U/t=4. Two Site Cellular DMFTin the 1D Hubbard model Two Site Cellular DMFT ( G.. Kotliar et.al. PRL (2001)) in the 1D Hubbard model M.Capone M.Civelli V. Kancharla C.Castellani and GK PRB 69,195105 (2004)T. D Stanescu and GK PRB (2006)24

56. “ Invar model “ for Pu-Ga. Lawson et. al.Phil. Mag. (2006) Data fits only if the excited state has zero stiffness.

57. Conclusions Constant interplay between theory and experiment has lead to new advances. General anomalies of correlated electrons and anomalous system specific studies, need for a flexible approach. (DMFT). New understanding of Pu. Methodology applicable to a large number of other problems, involving correlated electrions, thermoelectrics, batteries, optical devices, memories, high temperature superconductors, ……..

58. Conclusions DMFT produces non magnetic state, around a fluctuating (5f)^5 configuraton with correct volume the qualitative features of the photoemission spectra, and a double minima structure in the E vs V curve. Correlated view of the alpha and delta phases of Pu. Interplay of correlations and electron phonon interactions (delta-epsilon). Calculations can be refined in many ways, electronic structure calculations for correlated electrons research program, MINDLAB, ….

59. What do we want from materials theory? New concepts, qualitative ideas Understanding, explanation of existent experiments, and predictions of new ones. Quantitative capabilities with predictive power. Notoriously difficult to achieve in strongly correlated materials.

60. Some new insights into the funny properties of Pu Physical anomalies, are the result of the unique position of Pu in the periodic table, where the f electrons are near a localization delocalization transition. We learned how to think about this unusual situation using spectral functions. Delta and Alpha Pu are both strongly correlated, the DMFT mean field free energy has a double well structure, for the same value of U. One where the f electron is a bit more localized (delta) than in the other (alpha). Negative thermal expansion, multitude of phases.

61. Quantitative calculations Photoemission spectra,equilibrium volume, and vibration spectra of delta. Good agreement with experiments given the approximations made.Many systematic improvements are needed. Work is at the early stages, only a few quantities in one phase have been considered. Other phases? Metastability ? Effects of impurities? What else, do electrons at the edge of a localization localization do ? [ See epsilon Pu spectra ]

62. Collaborators, Acknowledgements References Los Alamos Science,26, (2000) S. Savrasov and G. Kotliar Phys. Rev. Lett. 84, 3670-3673, (2000). S.Savrasov G. Kotliar and E. Abrahams, Nature 410, 793 (2001).Phys. Rev. Lett. 84, 3670-3673, (2000)Nature 410, 793 (2001). X. Dai,S. Savrasov, G. Kotliar,A. Migliori, H. Ledbetter, E. Abrahams Science, Vol300, 954 (2003). Collaborators: S. Savrasov ( Rutgers-NJIT) X. Dai ( Rutgers), E. Abrahams (Rutgers), A. Migliori (LANL),H Ledbeter(LANL). Acknowledgements: G Lander (ITU) J Thompson(LANL) Funding: NSF, DOE, LANL.

63. Cluster DMFTlimitations of single site DMFT Cluster DMFT: removes limitations of single site DMFT No k dependence of the self energy. No d-wave superconductivity. No Peierls dimerization. No (R)valence bonds. Reviews: Reviews: Georges et.al. RMP(1996). Th. Maier et. al. RMP (2005); Kotliar et..al. RMP (2006). 23

64. U/t=4. Two Site Cellular DMFTin the 1D Hubbard model Two Site Cellular DMFT ( G.. Kotliar et.al. PRL (2001)) in the 1D Hubbard model M.Capone M.Civelli V. Kancharla C.Castellani and GK PRB 69,195105 (2004)T. D Stanescu and GK PRB (2006)24

65. Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965) Self Energy VanShilfgaarde (2005) VanShilfgaarde (2005) 3

68. Conclusions Unique properties of Pu and Am under pressure result from a proximity of a localization delocalization transition. Rare form of mixed valence. DMFT provides a good start. Qualitative insights, some quantitative predictions into delta Pu. Other Pu phases. Meaningful interplay of theory and experiment. Key in condensed matter physics.

70. Smith Kmeko Phase diagram. Minimum in melting curve and divergence of the compressibility at the Mott endpoint

71. The enhancement of the specific heat, further evidence for an open shell configuration, presence of electronic entropy. J. Lashley et.al. PRB(2005)

72. Double well structure and  Pu Qualitative explanation of negative thermal expansion[ Lawson, A. C., Roberts J. A., Martinez, B., and Richardson, J. W., Jr. Phil. Mag. B, 82, 1837,(2002). G. Kotliar J.Low Temp. Phys vol.126, 1009 27. (2002)] Natural consequence of the conclusions on the model Hamiltonian level. We had two solutions at the same U, one metallic and one insulating. Relaxing the volume expands the insulator and contract the metal. F(T,V)=Fphonons+ Finvar

73. “ Invar model “ for Pu-Ga. (Data fits if the excited state has zero stiffness.

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

75. A. Georges, G. Kotliar (1992) A(  ) 11

76. Expt. Wong et. al.

77. Elastic Deformations In most cubic materials the shear does not depend strongly on crystal orientation,fcc Al, c 44 /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=c 44  x/L F/A=c’  x/L

79. Localization Delocalization in Actinides after G. Lander, Science (2003). Mott Transition   Modern understanding of this phenomena using functional approach toDMFT. K Haule S.Savrasov J Shim  Pu  18

80. = n7/2 – 4/3 n5/2 nf = n7/2 + n5/2

81. Spectral Function and Photoemission Probability of removing an electron and transfering energy  =Ei-Ef, and momentum k f(  ) A(  ) M 2 e Angle integrated spectral function 8

82. Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965) Self Energy Succesful description of the total energy and the excitation spectra of a large number of simple metals semiconductors and insulators. Succesful description of the total energy and the excitation spectra of a large number of simple metals semiconductors and insulators. Succesfully predicts semiconducting gaps, phonon frequencies, resistivities, of countless materials. 3 a)Weak Correlation b)Strong Correlation

83. T/W Phase diagram of a Hubbard model with partial frustration at integer filling. [Rozenberg et. al. PRL 1995] Evolution of the Local Spectra as a function of U,and T. Mott transition driven by transfer of spectral weight Zhang Rozenberg Kotliar PRL (1993)...

84. OUTLINE The challenge of strongly correlated electron systems. Heavy Fermions and Late actinides: experimental overview Introduction to Dynamical Mean Field Theory (DMFT). Theory of delta Pu Theory of Am and Cm Conclusions

85. Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev. E = E i - E f Q = k i - k f

86. W 110 =2/3 and banching ratio Moore and van der Laan, Ultramicroscopy 2007.

87. 2/3 in the late actinides [DMFT results: K. Haule and J. Shim ] See the expt. work of K. Moore G. Van der Laan G. Haire M. Wall and A. Schartz Am H2

88. W 110 =2/3 and banching ratio See the expt. work of K. Moore G. Van der Laan G. Haire M. Wall and A. Schartz Am H2

89. Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003

91. Band Theory: electrons as waves. Landau Fermi Liquid Theory. Electrons in a Solid:the Standard Model Quantitative Tools. Density Functional Theory Kohn Sham (1964) Rigid bands, optical transitions, thermodynamics, transport……… Static Mean Field Theory. 2 Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965) n band index, e.g. s, p, d,,f