1. Electronic Structure of Elemental Plutonium: A Dynamical Mean Field Perspective (DMFT) Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University MRS Boston 2003

2. Collaborators, References S. Savrasov and G. Kotliar PRL 84 3670 (2000). S.Savrasov G. Kotliar and E. Abrahams, Nature 410,793 (2001). X. Dai,S. Savrasov, G. Kotliar,A. Migliori, H. Ledbetter, E. Abrahams Science, Vol300, 954 (2003). S. Murthy Rutgers Ph.D Thesis (2004). Introduction: basic questions and alternative theories. Dynamical Mean Field Theory (DMFT). Alpha and Delta Pu. Both phases are strongly correlated and differ in only in the distribution of one electron spectral weight. Delta and Epsilon Pu differ dramatically in their phonon spectra. Epsilon Pu is strongly anharmonic. Outlook. Support: Dynamical mean field method NSF. Actinides DOE Basic Energy Sciences

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu phases Small amounts of Ga stabilize the  phase (A. Lawson LANL) Los Alamos Science,26, (2000 ).

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Americium under pressure (Lindbaum et. al. PRB 2003)

5. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS More conventional electronic structure approaches 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, 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 30% lower than experiment o This is the largest discrepancy ever known in DFT based calculations.

6. 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 Alternative approach to delta Pu, Wills et. al. (5f) 4 core+ 1f(5f)in conduction band. [SIC-LDA]

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

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field Theory (Reviews) A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] Lichtenstein Katsnelson and Kotliar cond-mat- 0211076 K. Held et.al., Psi-k Newsletter 56 (April 2003).

9. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field Theory Basic idea: reduce the quantum many body problem to a one site or a cluster of sites, in a medium of non interacting electrons obeying a self consistency condition.[A. Georges and GK Phys. Rev. B 45, 6497, 1992]. Atom in a medium = Quantum impurity model. Solid in a frequency dependent potential. Basic idea: instead of using functionals of the density, use more sensitive functionals of the one electron spectral function. [density of states for adding or removing particles in a solid, measured in photoemission] [GK R. Chitra GKPhys. Rev. B62, 12715 (2000). and S. Savrasov cond-matt 0308053]. Allows computation of total energy AND one electron spectra.

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field Theory

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Approximations Electronic structure. LMTO’s ASA, LMTO full potential. Crystal field splitting in the self energies is neglected. W(r,r’) (w) replaced by U on the f electrons. 4 ev. No multiplet splittings. Non perturbative treatment of spin orbit coupling. Approximate Impurity Solver. Interpolative Perturbation Theory and Hubbard I.

12. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS W (ev) vs (a.u. 27.2 ev) N.Zein G. Kotliar and S. Savrasov

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS What is the dominant atomic configuration? 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 More realistic calculations, Mtot 0 is quenched by crystal Fields and Kondo effect. Moment lives at high q  Contrast Am:(5f) 6

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu: DMFT total energy vs volume Savrasov Kotliar and Abrahams Nature 410,793 (2001)

15. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Double well structure and  Pu Qualitative explanation of negative thermal expansion

16. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT calculations for fcc Americium S. Murthy and G. K(2003)

17. 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 earlier studies of the Mott transition phase diagram once electronic structure is about to vary.

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Alpha and delta Pu Photoemission Spectra DMFT(Savrasov et.al.) EXP ( Arko Joyce Morales Wills Jashley PRB 62, 1773 (2000))

19. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS  Pu and delta Pu differ electronically by the distribution of spectral weight in the resonance and the Hubbard band. 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 and near the Mott limit.

20. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity PRL 91,061401 (2003)

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu is NOT MAGNETIC, alpha and delta have comparable susceptibility and specifi heat.

22. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Important Physics Proximity to the Mott Transition. Redistribution of spectral weight. Simultaneous description of band physics and atomic physics. All captured by DMFT in the approximations used.!

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu phases Small amounts of Ga stabilize the  phase (A. Lawson LANL) Los Alamos Science,26, (2000 ).

24. 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? LDA+DMFT functional computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002). Combine linear response and DMFT.

25. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003

26. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Expts’ Wong et. al. Science 301. 1078 (2003) Theory Dai et. al. Science 300, 953, (2003)

27. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comparison of theory and experiment. Good agreement over the majority of the Brillouin zone, is significant. The phonon frequencies depend on the forces acting on the atoms as a result of their displacement. Ability to compute forces, is a first step to derive potentials, and do molecular dynamics. Discrepancies along (111) are significant. Role of temperature ? Improve the impurity solver ? Non local corrections, and deviations from DMFT.

28. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Elastic constants theory (LDA+DMFT with a Hubbard1 solver, Dai et. al. and experiments,( Letbetter and Moment ). Large c44/c’ ratio. C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa) Theory 34.56 33.03 26.81 3.88 Expt 36.28 33.59 26.73 4.78

29. 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)

30. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Epsilon Plutonium.

31. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonon entropy drives the epsilon delta phase transition Epsilon is slightly more delocalized than delta, has SMALLER volume and lies at HIGHER energy than delta at T=0. 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 following Drumont and G. Ackland et. al. PRB.65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.

32. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Transverse Phonon along (0,1,1) in epsilon Pu in self consistent Born approximation.

33. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Negative thermal expansion of Pu revisited. The distortion described by C' is very soft, nearly like a liquid,. C' measures the rigidity against the volume conserving tetragonal deformation. This is in fact the deformation from fcc towards a bcc along a Bain path. Previous LDA+ U study [Bouchet et. al. ] and our DMFT study show that the total energy difference between  phase and  phases is quite small and is around 1000K. Soft behavior along the Bain path.  Pu can sample the bcc structure, which has lower volume by the thermal fluctuation along Bain path.

34. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights into the anomalous 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. The Mott transition across the actinide series [ B. Johansson Phil Mag. 30,469 (1974)] concept has finally been worked out!.We learned how to think about this unusual situation using DMFT, Weiss fields, local spectral functions etc.

35. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusions DMFT produces non magnetic state, around a fluctuating (5f)^5 configuration with correct volume the qualitative features of the photoemission spectra, quasiparticle resonance and Hubbard band, 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 account for delta-epsilon transition. Anomalous phonons in epsilon Pu. Calculations can be refined, include multiplets, better impurity solvers, frequency dependent U’s, electronic entropy. User friendly interfaces.

36. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Experiments and Theory are Needed to test the different pictures of the elctronic structure of PU Model of Wills et al. : 4 (5f) electrons are core-like and 1 is delocalized. DMFT picture: all the 5 (5f) electrons are equivalent, they are localized over short time scales and itinerant over long time scales resulting in Hubbard band and quasiparticle resonance in the spectra. Both pictures require strong correlations in the delta phase but how to differentiate between them experimentally ? Alpha phase. Resonant Photoemission (J. Tobin et. al. ) Probe unoccupied states. Upper Hubbard band, BIS. Optics. X ray absortion. Etc.. Fermi Surface Probes. Is Luttinger theorem obeyed ?

37. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Open questions ? Is the softening along the 110 direction in delta Pu, temperature dependent ? Is the discrepancy between theory and experiments the result of not including the resonance in the phonon calculation or the result of not including non local corrections ?

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS J. Tobin et. al. PHYSICAL REVIEW B 68, 155109,2003

43. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS A. Arko et. al. PRB 15. (2000), 1773.

44. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

45. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Acknowledgements: Development of DMFT Collaborators: E. Abrahams,V. Anisimov, R. Chitra, V. Dobrosavlevic, X. Dai, D. Fisher, A. Georges, K. Haule H. Kajueter, W.Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, G. Palsson, A. Poteryaev, 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

46. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)

47. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS More recent work, organics, Limelette et. al. PRL 91,061401 (2003)

48. 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).

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

50. 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

51. 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)

52. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS X.Zhang M. Rozenberg G. Kotliar (PRL 1993) Concepts : three peak structure and transfer of spectral weigth. Evolution at T=0 half filling full frustration

53. 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)

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

55. 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

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

57. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Cerium

58. 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.

59. 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.

60. 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.

61. 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)

62. 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  )]

63. 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

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

65. 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

66. 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.

67. 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).

68. 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)

69. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+U bands. (Savrasov GK, PRL 2000). Similar work Bouchet et. al. 2000

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

71. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS S. Murthy Rutgers Ph.D Thesis P vs V for fcc Am

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

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

74. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Am photoemission spectra. Expt (Negele ) DMFT Theory (S. Murthy)

75. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Plutonium and 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.

76. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phases of Pu (A. Lawson LANL) Los Alamos Science 26, (2000)

77. Summary LDA LDA+U  LRO DMFT Spectra Method E vs V

78. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT: elastic constants

79. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phases of Pu (A. Lawson LANL) Los Alamos Science 26, (2000)