1. Elemental Plutonium: a strongly correlated metal Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Collaborators: S. Savrasov (NJIT) X. Dai( Rutgers )

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Physics of Pu The Problem: This?Or this?

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS For me the problem is :THIS. The Mott Phenomena Evolution of the electronic structure between the atomic limit and the band limit in an open shell situation. The “”in between regime” is ubiquitous central them in strongly correlated systems, gives rise to interesting physics. Example Mott transition across the actinide series [ B. Johansson Phil Mag. 30,469 (1974)] Revisit the problem using a new insights and new techniques from the solution of the Mott transition problem within dynamical mean field theory in the model Hamiltonian context. Use the ideas and concepts that resulted from this development to give physical qualitative insights into real materials. Turn the technology developed to solve simple models into a practical quantitative electronic structure method.

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction: some Pu puzzles. Results: Minimum of the melting curve, Delta Pu: Most probable valence, size of the local moment Equilibrium Volume. Photoemission Spectral. Stabilization of Epsilon Pu: Conclusions

5. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in the actinide series (Smith Kmetko phase diagram)

6. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Small amounts of Ga stabilize the  phase (A. Lawson LANL)

7. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Shear anisotropy. C’=(C11-C12)/2 4.78 C44= 33.59 19.70 C44/C’ ~ 8 Largest shear anisotropy in any element! LDA Calculations (Bouchet) C’= -48

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Plutonium Puzzles o DFT in the LDA or GGA is a well established tool for the calculation of ground state properties. o Many studies (Freeman, Koelling 1972)APW methods o ASA and FP-LMTO Soderlind et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give o an equilibrium volume of the  phase  Is 35% lower than experiment o This is the largest discrepancy ever known in DFT based calculations.

9. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DFT Studies LSDA predicts magnetic long range (Solovyev et.al.) Experimentally  Pu is not magnetic. If one treats the f electrons as part of the core LDA overestimates the volume by 30% DFT in GGA predicts correctly the volume of the  phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills) is used. This is usually taken as an indication that  Pu is a weakly correlated system Alterantive approach Wills et. al. (5f)4 core+ 1f(5f)in conduction band.

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu Specific Heat

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity

12. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu is NOT MAGNETIC

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Specific heat and susceptibility.

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Problems with the conventional viewpoint of  Pu U/W is not so different in alpha and delta The specific heat of delta Pu, is only twice as big as that of alpha Pu. The susceptibility of alpha Pu is in fact larger than that of delta Pu. The resistivity of alpha Pu is comparable to that of delta Pu.

15. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction: some Pu puzzles. DMFT, qualitative aspects of the Mott transition from model Hamiltonians DMFT as an electronic structure method. DMFT results for delta Pu, and some qualitative insights. Conclusions

16. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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. We have solved “the hydrogen atom problem” of strongly correlated electron systems.

17. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of the Spectral Function with Temperature Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange and Rozenberg Phys. Rev. Lett. 84, 5180 (2000)

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Generalized phase diagram T U/W Structure, bands, orbitals

19. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative phase diagram in the U, T,  plane (two band Kotliar Murthy Rozenberg PRL (2002). Coexistence regions between localized and delocalized spectral functions. k diverges at generic Mott endpoints

20. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in layered organic conductors S Lefebvre et al. Ito et.al, Kanoda’s talk Bourbonnais talk Magnetic Frustration

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ultrasound study of Fournier et. al. (2002)

22. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Minimum in melting curve and divergence of the compressibility at the Mott endpoint

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Minimum of the melting point Divergence of the compressibility at the Mott transition endpoint. Rapid variation of the density of the solid as a function of pressure, in the localization delocalization crossover region. Slow variation of the volume as a function of pressure in the liquid phase Elastic anomalies, more pronounced with orbital degeneracy.

24. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Minimum in melting curve and divergence of the compressibility at the Mott endpoint

25. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Cerium

26. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction: some Pu puzzles. DMFT, qualitative aspects of the Mott transition in model Hamiltonians. DMFT as an electronic structure method. DMFT results for delta Pu, and some qualitative insights. Conclusions

27. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Solving the DMFT equations Wide variety of computational tools (QMC,ED….)Analytical Methods Extension to ordered states. Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

28. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Realistic DMFT loop

29. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT-outer loop relax DMFT U E dc

30. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outer loop relax U E dc Impurity Solver SCC G,  G0G0 DMFT LDA+U Imp. Solver: Hartree-Fock

31. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction: some Pu puzzles. DMFT, qualitative aspects of the Mott transition in model Hamiltonians. DMFT as an electronic structure method. Realistic DMFT and Plutonium Conclusions

32. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS What is the dominant atomic configuration? Local moment? Snapshots of the f electron Dominant configuration:(5f) 5 Naïve view Lz=-3,-2,-1,0,1 ML=-5  B S=5/2 Ms=5  B Mtot=0

33. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+U bands. (Savrasov GK, PRL 2000).

34. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Magnetic moment L=5, S=5/2, J=5/2, Mtot=Ms=  B gJ =.7  B Crystal fields     GGA+U estimate (Savrasov and Kotliar 2000) ML=-3.9 Mtot=1.1 This bit is quenched by Kondo effect of spd electrons [ DMFT treatment] Experimental consequence: neutrons large magnetic field induced form factor (G. Lander).

35. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu: DMFT total energy vs Volume ( Savrasov Kotliar and Abrahams 2001)

36. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Double well structure and  Pu Qualitative explanation of negative thermal expansion Sensitivity to impurities which easily raise the energy of the  -like minimum.

37. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field View of Pu ( Savrasov Kotliar and Abrahams, Nature 2001) 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). Is the natural consequence of the model Hamiltonian phase diagram once electronic structure is about to vary.

38. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on the HF static limit Describes only the Hubbard bands. No QP states. Single well structure in the E vs V curve. (Savrasov and Kotliar PRL)

39. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Lda vs Exp Spectra

40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS X.Zhang M. Rozenberg G. Kotliar (PRL 1993) Spectral Evolution at T=0 half filling full frustration

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu Spectra DMFT(Savrasov) EXP ( Arko Joyce Morales Wills Jashley PRB 62, 1773 (2000)

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comparaison with LDA+U

43. Summary LDA LDA+U DMFT Spectra Method E vs V

44. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The delta –epsilon transition The high temperature phase, (epsilon) is body centered cubic, and has a smaller volume than the (fcc) delta phase. What drives this phase transition? Having a functional, that computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002)

45. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Energy vs Volume

46. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Energy vs Volume

47. Success story : Density Functional Linear Response Tremendous progress in ab initio modelling of lattice dynamics & electron-phonon interactions has been achieved ( Review: Baroni et.al, Rev. Mod. Phys, 73, 515, 2001 ) (Savrasov, PRB 1996)

48. Results for NiO: Phonons Solid circles – theory, open circles – exp. ( Roy et.al, 1976 ) DMFT Savrasov and GK PRL 2003

49. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT for Mott insulators

50. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonon freq (THz) vs q in delta Pu (Dai et. al. )

51. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Shear anisotropy. Expt. vs Theory C’=(C11-C12)/2 = 4.78 GPa C’=3.37GPa C44= 33.59 GPa C44=19.7 GPa C44/C’ ~ 8 Largest shear anisotropy in any element! C44/C’ ~ 6

52. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonon frequency (Thz ) vs q in epsilon Pu.

53. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

54. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Temperature stabilizes a very anharmonic phonon mode

55. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonons epsilon

56. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonon entropy drives the epsilon delta phase transition Epsilon is slightly more metallic than delta, but it has a much larger phonon entropy than delta. At the phase transition the volume shrinks but the phonon entropy increases. Estimates of the phase transition neglecting the Electronic entropy: TC 600 K.

57. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction: some Pu puzzles. DMFT, qualitative aspects of the Mott transition in model Hamiltonians. DMFT as an electronic structure method. DMFT results for delta Pu, and some qualitative insights. Conclusions

58. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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.

59. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusions Outsanding question: electronic entropy, lattice dynamics. In the making, new generation of DMFT programs, QMC with multiplets, full potential DMFT, frequency dependent U’s, multiplet effects, combination of DMFT with GW

60. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Acknowledgements: Development of DMFT Collaborators: V. Anisimov, R. Chitra, V. Dobrosavlevic, X. Dai, D. Fisher, A. Georges, H. Kajueter, W.Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, G. Palsson, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko, I. Yang, X.Y. Zhang Support: NSF DMR 0096462 Support: Instrumentation. NSF DMR-0116068 Work on Fe and Ni: ONR4-2650 Work on Pu: DOE DE-FG02-99ER45761 and LANL subcontract No. 03737-001-02

61. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

62. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT MODELS.

63. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mean-Field : Classical vs Quantum Classical case Quantum case Phys. Rev. B 45, 6497 A. Georges, G. Kotliar (1992)

64. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Example: Single site DMFT, functional formulation Express in terms of Weiss field (G. Kotliar EPJB 99) Local self energy (Muller Hartman 89)

65. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Impurity cavity construction

66. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] Weiss field

67. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Case study: IPT half filled Hubbard one band (Uc1) exact = 2.2+_.2 (Exact diag, Rozenberg, Kajueter, Kotliar PRB 1996), confirmed by Noack and Gebhardt (1999) (Uc1) IPT =2.6 (Uc2) exact =2.97+_.05(Projective self consistent method, Moeller Si Rozenberg Kotliar Fisher PRL 1995 ), (Confirmed by R. Bulla 1999) (Uc 2 ) IPT =3.3 (T MIT ) exact =.026+_.004 (QMC Rozenberg Chitra and Kotliar PRL 1999), (T MIT ) IPT =.045 (U MIT ) exact =2.38 +-.03 (QMC Rozenberg Chitra and Kotliar PRL 1999), (U MIT ) IPT =2.5 (Confirmed by Bulla 2001) For realistic studies errors due to other sources (for example the value of U, are at least of the same order of magnitude).

68. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional The exact functional can be built in perturbation theory in the interaction (well defined diagrammatic rules )The functional can also be constructed from the atomic limit, but no explicit expression exists. DFT is useful because good approximations to the exact density functional  DFT  (r)] exist, e.g. LDA, GGA A useful approximation to the exact functional can be constructed, the DMFT +LDA functional.

69. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Interfacing DMFT in calculations of the electronic structure of correlated materials Crystal Structure +atomic positions Correlation functions Total energies etc. Model Hamiltonian

70. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT functional  Sum of local 2PI graphs with local U matrix and local G

71. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT and LDA+U Static limit of the LDA+DMFT functional, with  atom  HF reduces to the LDA+U functional of Anisimov Andersen and Zaanen. Crude approximation. Reasonable in ordered Mott insulators. Short time picture of the systems. Total energy in DMFT can be approximated by LDA+U with an effective U. Extra screening processes in DMFT produce smaller Ueff. U LDA+U < U DMFT

72. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

73. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS E-DMFT +GW P. Sun and G. Kotliar Phys. Rev. B 2002

74. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT and LDA+U Static limit of the LDA+DMFT functional, with  atom  HF reduces to the LDA+U functional of Anisimov Andersen and Zaanen. Crude approximation. Reasonable in ordered Mott insulators. Short time picture of the systems. Total energy in DMFT can be approximated by LDA+U with an effective U.

75. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT References Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997). Lichtenstein and Katsenelson PRB (1998). Kotliar, Savrasov, in New Theoretical approaches to strongly correlated systems, Edited by A. Tsvelik, Kluwer Publishers, (2001). Reviews: Kotliar, Savrasov, in New Theoretical approaches to strongly correlated systems, Edited by A. Tsvelik, Kluwer Publishers, (2001). Held Nekrasov Blumer Anisimov and Vollhardt et.al. Int. Jour. of Mod PhysB15, 2611 (2001). Held Nekrasov Blumer Anisimov and Vollhardt et.al. Int. Jour. of Mod PhysB15, 2611 (2001). A. Lichtenstein M. Katsnelson and G. Kotliar (2002)

76. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on LDA+DMFT Static limit of the LDA+DMFT functional, with  =  HF reduces to LDA+U Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing. Luttinger theorem is obeyed. Functional formulation is essential for computations of total energies, opens the way to phonon calculations.

77. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS References LDA+DMFT: V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997). A Lichtenstein and M. Katsenelson Phys. Rev. B 57, 6884 (1988). S. Savrasov G.Kotliar funcional formulation for full self consistent implementation of a spectral density functional. Application to Pu S. Savrasov G. Kotliar and E. Abrahams (Nature 2001).

78. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT: Effective Action point of view. R. Chitra and G. Kotliar Phys Rev. B. (2000), (2001). Identify observable, A. Construct an exact functional of =a,  [a] which is stationary at the physical value of a. Example, density in DFT theory. (Fukuda et. al.) When a is local, it gives an exact mapping onto a local problem, defines a Weiss field. The method is useful when practical and accurate approximations to the exact functional exist. Example: LDA, GGA, in DFT. It is useful to introduce a Lagrange multiplier  conjugate to a,  [a,  It gives as a byproduct a additional lattice information.

79. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

80. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Interface DMFT with electronic structure. Derive model Hamiltonians, solve by DMFT (or cluster extensions). Total energy?  Full many body aproach, treat light electrons by GW or screened HF, heavy electrons by DMFT [E-DMFT frequency dependent interactionsGK and S. Savrasov, P.Sun and GK cond-matt 0205522]  Treat correlated electrons with DMFT and light electrons with DFT (LDA, GGA +DMFT)

81. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional : effective action construction Introduce local orbitals,   R (r-R), and local GF G(R,R)(i  ) = The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for  (r) and G and performing a Legendre transformation,  (r),G(R,R)(i  )]

82. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT approximate functional The light, SP (or SPD) electrons are extended, well described by LDA The heavy, D (or F) electrons are localized,treat by DMFT. LDA already contains an average interaction of the heavy electrons, substract this out by shifting the heavy level (double counting term) The U matrix can be estimated from first principles (Gunnarson and Anisimov, McMahan et.al. Hybertsen et.al) of viewed as parameters

83. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

84. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS References Long range Coulomb interactios, E-DMFT. R. Chitra and G. Kotliar Combining E-DMFT and GW, GW-U, G. Kotliar and S. Savrasov Implementation of E-DMFT, GW at the model level. P Sun and G. Kotliar. Also S. Biermann et. al.

85. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

86. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Energy difference between epsilon and delta

87. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Connection between local spectra and cohesive energy using Anderson impurity models foreshadowed by J. Allen and R. Martin PRL 49, 1106 (1982) in the context of KVC for cerium. Identificaton of Kondo resonance n Ce, PRB 28, 5347 (1983).

88. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS E-DMFT+GW effective action G= D=

89. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

90. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field Theory(DMFT) Review: A. Georges G. Kotliar W. Krauth M. Rozenberg. Rev Mod Phys 68,1 (1996) Local approximation (Treglia and Ducastelle PRB 21,3729), local self energy, as in CPA. Exact the limit defined by Metzner and Vollhardt prl 62,324(1989) inifinite. Mean field approach to many body systems, maps lattice model onto a quantum impurity model (e.g. Anderson impurity model )in a self consistent medium for which powerful theoretical methods exist. (A. Georges and G. Kotliar prb45,6479 (1992).

91. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Technical details Multiorbital situation and several atoms per unit cell considerably increase the size of the space H (of heavy electrons). QMC scales as [N(N-1)/2]^3 N dimension of H Fast interpolation schemes (Slave Boson at low frequency, Roth method at high frequency, + 1 st mode coupling correction), match at intermediate frequencies. (Savrasov et.al 2001)

92. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Technical details Atomic sphere approximation. Ignore crystal field splittings in the self energies. Fully relativistic non perturbative treatment of the spin orbit interactions.