1. Dynamical Mean Field Theory or Metallic Plutonium Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University IWOSMA Berkeley October 2002 Collaborators: S. Savrasov (NJIT) and Xi Dai (Rutgers)

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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.

3. 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).

4. 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

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 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.

8. 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

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

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity

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

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

13. 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.

14. 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

15. 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).

16. 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.

17. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Example: DMFT for lattice model (e.g. single band Hubbard). Observable: Local Greens function G ii (  ). Exact functional  [G ii (  )  DMFT Approximation to the functional.

18. 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

19. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic DMFT phase diagram one band Hubbard model (half filling, semicircular DOS, partial frustration) Rozenberg et.al PRL (1995)

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

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phase Diagrams :V 2 O 3, Ni Se 2 -x S x Mc Whan et. Al 1971,. Czek et. al. J. Mag. Mag. Mat. 3, 58 (1976),

22. 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

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Cerium

24. 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

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

26. 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.

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

28. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Cerium

29. 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

30. 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)

31. 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

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

33. 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

34. 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. 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

35. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Very Partial list of application of realistic DMFT to materials QP bands in ruthenides: A. Liebsch et al (PRL 2000) N phase of Pu: S. Savrasov G. Kotliar and E. Abrahams (Nature 2001) MIT in V 2 O 3 : K. Held et al (PRL 2001) Magnetism of Fe, Ni: A. Lichtenstein M. Katsenelson and G. Kotliar et al PRL (2001) J-G transition in Ce: K. Held A. Mc Mahan R. Scalettar (PRL 2000); M. Zolfl T. et al PRL (2000). 3d doped Mott insulator La1-xSrxTiO3 Anisimov et.al 1997, Nekrasov et.al. 1999, Udovenko et.al 2002) ………………..

36. 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)

37. 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  )]

38. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT Self-Consistency loop DMFT U E

39. 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.

40. 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).

41. 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.

42. 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

43. 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

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

45. 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).

46. 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)

47. 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.

48. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu: DMFT total energy vs Volume (Savrasov 00)

49. 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.

50. 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.

51. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comparaison with the Hartree Fock static limit: LDA+U.

52. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dependence on structure

53. 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)

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

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

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

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

58. 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

59. 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. Calculations can and should be refined and extended.

60. 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

61. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT EXPERIMENTS

62. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu: Anomalous thermal expansion ( Smith and Boring )

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

64. 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)

65. 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)

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

67. 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

68. 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).

69. 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.

70. 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

71. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Combining LDA and DMFT The light, SP electrons well described by LDA. The heavier D electrons treat by DMFT. LDA already contains an average interaction of the heavy electrons, subtract this out by shifting the heavy level (double counting term, Edc, review Anismov Aersetiwan and Lichtenstein ) Atomic physics parameters. U=F0 cost of double occupancy irrespectively of spin, J=F2+F4, Hunds energy favoring spin polarization, F2/F4=.6,….. Calculations of U, Edc, (Gunnarson and Anisimov, McMahan et.al. Hybertsen et.al) or viewed as parameters

72. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT MODELS RESULTS

73. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS QMC calculationof n vs  (Kotliar Murthy Rozenberg PRL 2002, 2 band, U=3.0)  diverges at generic Mott endpoints

74. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Compressibilty divergence

75. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Cerium

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

77. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT functional  Atom =Sum of all local 2PI graphs build with local Coulomb interaction matrix, parametrized by Slater integrals F0, F2 and F4 and local G.Express  in terms of AIM model.

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

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

80. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Solving the DMFT equations Wide variety of computational tools (QMC,ED….)Analytical Methods Reviews: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996). Prushke T. Jarrell M. and Freericks J. Adv. Phys. 44,187 (1995)

81. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Density functional theory and Dynamical Mean Field Theory DFT: Static mean field, electrons in an effective potential. Functional of the density. DMFT: Promote the local (or a few cluster Greens functions ) as the basic quantities of the theory. Express the free energy as a functional of these local quantities and the density. Provide useful approximations to the functional.

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

83. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT Connection with atomic limit Weiss field

84. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Double counting term (Lichtenstein et.al) Problem : What is the LDA+U functional, a functional of? What is n ab ?

85. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Plutonium

86. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS PU: (“cubic ALPHA” AND DELTA