1. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The Mott transition in transition metal oxides and in organic materials: a dynamical mean field theory (DMFT) perspective. Part 1+ 2 Gabriel Kotliar Center for Materials Theory Rutgers University

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The Mott transition in f electron materials:a dynamical mean field theory (DMFT) perspective. Part 3. Gabriel Kotliar Center for Materials Theory Rutgers University

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline  Some comments on the role of DMFT in solid state physics problems and the strong correlation problem. (Part I)  Introduction to DMFT: cavity construction E- DMFT and cluster methods. (Part I) Introduction to DMFT: functional method. (Part III)  Interfaces with electronic structure. DMFT as a first principles method. first principles approach GW-U method. LDA+DMFT. (Part III)

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline  The Mott transition problem. (Part I)  Predictions of single site DMFT, and experimental verification. Phase Diagram, Optics, Photoemission, Transport. (Part I)  Conclusions of Part I. System specific studies of materials. LDA+DMFT. Some case studies.

5. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline  SYSTEM SPECIFIC STUDIES.  The Mott transition in kappa organics. A CDMFT study. (Part II)  The metal to insulator transition in Ti2O3.A CMDFT study. (Part II)  Itinerant Magnetism in Fe and Ni. (Part II).  Conclusions of Part II.

6. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline  The Mott transition in f electron systems. The role of the spd band. The coupling to the structure, volume collapse transitions. (Part III)  Case study: alpha-gamma transition in Cerium. Photoemission and Optical Spectroscopy. (Part III)  Case study: the phases of plutonium, photoemission, total energy and lattice vibrations. (Part III)  Case study: Americium under pressure. (Part III).

7. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Weakly correlated electrons:band theory. Simple conceptual picture of the ground state, excitation spectra, transport properties of many systems (simple metals, semiconductors,….). A methods for performing quantitative calculations. (Density functional theory, in various approximations+ perturbation theory in the Coulomb interactions).

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Momentum Space, bands, k in Brillouin zone is good quantum number. Landau Fermi liquid theory interactions renormalize away at low energy, simple band picture in effective field holds. The electron in a solid: wave picture Maximum metallic resistivity 200  ohm cm

9. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Standard Model of Solids  Qualitative predictions: low temperature dependence of thermodynamics and transport.  Optical response, transition between the bands.  Qualitative predictions: filled bands give rise to insulting behavior. Compounds with odd number of electrons are metals.  Quantitative tools: Density Functional Theory with approximations suggested by the Kohn Sham formulation, (LDA GGA) is a successful computational tool for the total energy. Good starting point for perturbative calculation of spectra,eg. GW. Kinetic equations yield transport coefficients.

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Kohn Sham reference system Excellent starting point for computation of spectra in perturbation theory in screened Coulomb interaction GW.

11. 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)

12. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+GW: semiconducting gaps

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The electron in a solid: particle picture. Array of hydrogen atoms is insulating if a>>a B. Mott: correlations localize the electron e_ e_ e_ e_ Superexchange Think in real space, solid collection of atoms High T : local moments, Low T spin-orbital order,RVB.

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott : Correlations localize the electron Low densities, electron behaves as a particle,use atomic physics, real space One particle excitations: Hubbard Atoms: sharp excitation lines corresponding to adding or removing electrons. In solids they broaden by their incoherent motion, Hubbard bands (eg. bandsNiO, CoO MnO….) H H H + H H H motion of H+ forms the lower Hubbard band H H H H - H H motion of H_ forms the upper Hubbard band Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics.

15. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Localization vs Delocalization Strong Correlation Problem A large number of compounds with electrons in partially filled shells, are not close to the well understood limits (localized or itinerant). Non perturbative problem. These systems display anomalous behavior (departure from the standard model of solids). Neither LDA or LDA+U or Hartree Fock work well. Dynamical Mean Field Theory: Simplest approach to electronic structure, which interpolates correctly between atoms and bands. Treats QP bands and Hubbard bands.

16. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strong correlation anomalies Metals with resistivities which exceed the Mott Ioffe Reggel limit. Transfer of spectral weight which is non local in frequency. Dramatic failure of DFT based approximations (say DFT-GW) in predicting physical properties.

17. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Failure of the standard model : Anomalous Resistivity :LiV 2 O 4 Takagi et.al. PRL 2000

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Failure of the Standard Model: Anomalous Spectral Weight Transfer Optical Conductivity Schlesinger et.al (1993) Neff depends on T

19. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The competition of kinetic energy and Coulomb interactions, is a central issue that needs to be resolved. One needs a tool that treats quasiparticle bands and Hubbard bands on the same footing to contain the band and atomic limit. The approach should allow to incorporate material specific information. When the neither the band or the atomic description applies, a new reference point for thinking about correlated electrons is needed. DMFT!

20. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline  Some comments on the role of DMFT in solid state physics problems and the strong correlation problem.  Introduction to DMFT: cavity construction E-DMFT and cluster methods. Introduction to DMFT: functional method.  Interfaces with electronic structure. A truly first principles approach GW-U method. LDA+DMFT.

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT

22. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Single site DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)] Weiss field

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Extension to clusters. Cellular DMFT. C-DMFT. G. Kotliar,S.Y. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) tˆ(K) is the hopping expressed in the superlattice notations. Other cluster extensions (DCA, nested cluster schemes, PCMDFT ), causality issues, O. Parcollet, G. Biroli and GK cond-matt 0307587 (2003)

24. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS More general DMFT loop

25. 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)

26. 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)]

27. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Methods of solution : some examples  Iterative perturbation theory. A Georges and G Kotliar PRB 45, 6479 (1992). H Kajueter and G. Kotliar PRL (1996). Interpolative schemes (Oudovenko et.al.)  Exact diag schemes Rozenberg et. al. PRL 72, 2761 (1994)Krauth and Caffarel. PRL 72, 1545 (1994)  Projective method G Moeller et. al. PRL 74 2082 (1995).  NRG R. Bulla PRL 83, 136 (1999)

28. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS QMC M. Jarrell, PRL 69 (1992) 168, Rozenberg Zhang Kotliar PRL 69, 1236 (1992),A Georges and W Krauth PRL 69, 1240 (1992) M. Rozenberg PRB 55, 4855 (1987). NCA Prushke et. al. (1993). SUNCA K. Haule (2003). Analytic approaches, slave bosons. Analytic treatment near special points.

29. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS How good is DMFT ?

30. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Single site DMFT is exact in the Limit of large lattice coordination Metzner Vollhardt, 89 Muller-Hartmann 89

31. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS C-DMFT: test in one dimension. (Bolech, Kancharla GK PRB 2003) Gap vs U, Exact solution Lieb and Wu, Ovshinikov Nc=2 CDMFT vs Nc=1

32. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS N vs mu in one dimensional Hubbard model. Compare 2 site cluster (in exact diag with Nb=8) vs exact Bethe Anzats, [M. Capone C. Castellani M.Civelli and GK (2003)]

33. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on DMFT. Review of DMFT, technical tools for solving DMFT eqs. A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] CDMFT, instead of studying finite systems with open or periodic boundary conditions, study a system in a medium. Connection with DMRG, infer the density matrix by using a Gaussian anzats, and the periodicity of the system.

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

35. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline  The Mott transition problem.  Predictions of DMFT, and experimental verification. Phase Diagram, Optics, Photoemission, Transport. System specific studies of materials. LDA+DMFT. Some case studies.

36. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The Mott transition problem Universal and non universal aspects. Frustration and the success of DMFT. In the phases without long range order, DMFT is valid if T > Jeff. Need frustration to supress it. When T < Jeff LRO sets in. If Tneel is to high it oblitarates the Mott phenomena. t vs U fundamental competition and secondary instabilities.

37. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS V 2 O 3 under pressure or

38. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS NiSe 2-x S x

39. 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)

40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Single site DMFT and expt. Single site DMFT study of the Mott transition, based on a study of the Hubbard model on frustrated lattices made several interesting qualitative predictions. New experiments and reexamination of old ones give credence to that the local picture is quite good.

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pressure Driven Mott transition

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights from DMFT  Low temperature Ordered phases. Stability depends on chemistry and crystal structure  High temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT. Role of magnetic frustration.

43. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic DMFT phase diagram of a partially frustrated integered filled Hubbard model.

44. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative single site DMFT predictions. Spectra of the strongly correlated metallic regime contains both quasiparticle-like and Hubbard band-like features. Mott transition is drive by transfer of spectral weight.

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

46. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Parallel development: Fujimori et.al

47. 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 nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000)

48. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative single site DMFT predictions: Optics Spectra of the strongly correlated metallic regime contains both quasiparticle-like and Hubbard band-like features. Mott transition is drive by transfer of spectral weight. Consequences for optics.

49. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic DMFT phase diagram Hubbard model (partial frustration). Evidence for QP peak in V2O3 from optics. M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

50. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous transfer of spectral weight in v2O3

51. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous transfer of optical spectral weight V2O3 :M Rozenberg G. Kotliar and H. Kajuter Phys. Rev. B 54, 8452 (1996). M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

52. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous transfer of optical spectral weight, NiSeS. [Miyasaka and Takagi 2000]

53. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Spectral Weight Transfer: Optics Schlesinger et.al (FeSi) PRL 71,1748, (1993) B Bucher et.al. Ce 2 Bi 4 Pt 3 PRL 72, 522 (1994), Rozenberg et.al. PRB 54, 8452, (1996). ApreciableT dependence found. Below energy

54. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Searching for a quasiparticle peak

55. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS. ARPES measurements on NiS 2-x Se x Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998)

56. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 1998); S.-K. Mo et al., Phys. Rev. Lett. 90, 186403 (2003). 16. K. Maiti et al., Europhys. Lett. 55, 246 (2001); A. Sekiyama et al., http://arXiv.org/abs/con d-mat/0312429. 17. P. Limelette et al., Science 302, 89 (2003); F. Kagawa et al., http://arXiv.org/abs/cond- mat/0307304.

57. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS S.-K. Mo et al., Phys. Rev. Lett. 90, 186403 (2003)..

58. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic DMFT phase Implications for transport.

59. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity and Mott transition Ni Se 2-x S x Crossover from Fermi liquid to bad metal to semiconductor to paramagnetic insulator.

60. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

61. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ising endpoint finally found P. Limelette et al., Science 302, 89 (2003) F. Kagawa et al.,http://arXiv.org/abs/cond- mat/0307304

62. 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 nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000)

63. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS V 2-x Cr x O 3

64. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ising critical endpoint in (VCr) 2 O 3 Limlette et. al. Science 302, 89 (2003)

65. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion G. Kotliar Science 89 (2003) An electronic model accounts for all the qualitative features of the finite temperature of a frustrated system at integer occupancy. The observation of the spinodal lines and the wide classical critical region where mean field holds indicate the coupling to the lattice is quantitatively very important.

66. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion Single site DMFT describes the main features of the experiments at high temperatures using a simple model. Made non trivial predictions. Finite temperature conclusions are robust. At low temperatures clusters will bring refinements of this picture. System specific studies including electron phonon coupling needed to obtain quantitative estimates of say T_MIT.

67. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The Mott transition in transition metal oxides and in organic materials: a dynamical mean field theory (DMFT) perspective. Part 1+ 2 Gabriel Kotliar Center for Materials Theory Rutgers University

68. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic Single Site DMFT phase diagram.

69. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Single site DMFT has provides a first glimpse of what happens to the electronic structure as a function of correlation strength, and temperature. Neglects the effects of magnetic exchange J on the quasiparticle properties in the paramagnetic phase. To take these into account we need to go beyond single site DMFT. Test and refine the single site DMFT picture.

70. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS O. Parcollet G. Biroli and G. Kotliar (kappa organics) cond-mat 0308577 A. Poteryaev A. Lichtenstein and G. Kotliar (Ti2O3 ) cond-matt 0311319 Fe and Ni. A. Lichtenstein M. Katsnelson and G. Kotliar. Phys Rev. Lett 87, 67205, (2001 ). Cerium. K. Haule et. al. unpublished.

71. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Two roads for ab-initio calculation of electronic structure of strongly correlated materials Correlation Functions Total Energies etc. Model Hamiltonian Crystal structure +Atomic positions

72. 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)

73. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS  organics ET = BEDT-TTF=Bisethylene dithio tetrathiafulvalene  organics = (ET)2 X Increasing pressure -----  increasing t’  ------------ X0 X1 X2 X3 (Cu)(2CN)3 Cu(NCN)2 Cl Cu(NCN2)2Br Cu(NCS)2 Spin liquid Mott transition

74. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Theoretical issue: is there a Mott transition in the integer filled Hubbard model, and is it well described by the single site DMFT ?

75. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Controversy on the unfrustrated case. Comment on "Absence of a Slater Transition in the Two- Dimensional Hubbard Model" B. KyungB. Kyung, J.S. Landry, D. Poulin, A.- M.S. Tremblay Phys. Rev. Lett. 90, 099702-1 (2003)J.S. LandryD. PoulinA.- M.S. Tremblay

76. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Double Occupancy vs U CDMFT Parcollet, Biroli GK (2003) Study frustrated t t’ model t’/t=.9

77. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Compare with single site results Rozenberg Chitra Kotliar PRL 2002

78. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in cluster (QMC)

79. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of the spectral function at low frequency. If the k dependence of the self energy is weak, we expect to see contour lines corresponding to Ek = const and a height increasing as we approach the Fermi surface.

80. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of the k resolved Spectral Function at zero frequency. (Parcollet Biroli and GK) Uc=2.35+-.05, Tc/D=1/44 U/D=2 U/D=2.25

81. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Near the transition k dependence is strong. Qualitative effect, formation of hot regions! D wave gapping of the single particle spectra as the Mott transition is approached. Square symmetry is restored as we approched the insulator.

82. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Deviations from single site DMFT

83. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Lattice and cluster self energies

84. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mechanism for hot spot formation

85. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion. Mott transition survives in the cluster setting. Role of magnetic frustration. Surprising result: formation of hot and cold regions as a result of an approach to the Mott transition. General result ? Unexpected role of the next nearest neighbor self energy. CDMFT a new window to extend DMFT to lower temperatures.

86. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline  SYSTEM SPECIFIC STUDIES.  The Mott transition in kappa organics. A CDMFT study.  The metal to insulator transition in Ti2O3.A CMDFT study.  Itinerant Magnetism in Fe and Ni.  Cerium.

87. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Metal to insulator transition in Ti2O3 Isostructural to V2O3. All the qualitative physics of the high temperature part of the phase diagram of V2O3 can be understood within single site DMFT. Computations with a realistic density of states, and multiorbital impurity model (Mo et. al. ) quantitative improvement. Is the same thing true in Ti2O3?

88. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ti2O3 V2O3 : Resistivities

89. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ti2O3 Structure

90. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Relevant Orbitals: Goodenough picture

91. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Physical parameters U= 2ev. Nearest neighbor Coulomb repulsions, V, W.5 =.5 ev.

92. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ti2O3 vs V2O3 As a function of temperature, there is no magnetic transition in Ti2O3, unlike V2O3 As a function of temperature, there is no structural change, unlike V2O3 which becomes monoclinic at low temperatures. In V2O3 the distance between the Vanadium pairs increases as the temperature decreases. In Ti2O3 the distance between the Vanadium pairs decreases as one lowers the temperature. LTS 250 K, HTS 750 K.

93. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Earlier work. Band Structure Calculations always produce a good metal. L.F. Mattheiss, J. Phys.: Condens. Matter 8, 5987 (1996) Unrestricted Hartree Fock calculations produce large antiferromagnetic gap. M. Cati, G. Sandrone, and R. Dovesi, Phys. Rev. B. f55, 16122 (1997).

94. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ti2O3 LDA-DOS LTS HTS

95. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Methodology:1 and 2 site CDMFT Impurity solver. Multiband QMC. Derivation of the effective Hamiltonian. Massive downfolding with O Andersen’s new Nth order LMTOS. Coulomb interactions estimated using dielectric constant W=.5 ev. U on titanium 2 ev. J=.2 ev.

96. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Single site DMFT LTS

97. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Two-site CDMFT for beta=20, and beta=10 (T=500,1000) Poteryaev Lichtenstein and GK

98. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Single site DMFT vs CDMFT

99. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Important role played by the Coulomb nn repulsion.

100. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Physical Origin of the renormalization of the hopping. Generalized Anderson impurity model. A Coulomb repulsion between the local orbital and the bath of conduction electrons Renormalizes down the hybridization between the local orbital and the conduction electron. D. Haldane PRL 40, 416 (1978) and Cambridge Unive. Ph.D thesis. Q. Si and G. Kotliar PRL 70, 3143 (1993).

101. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS CDMFT Single site DMFT, Basic Unit single site. Titanium Oxide Basic Unit: Titanium pair. Kappa Organics, Basic Unit 4 site plaquette. Minimal reference frames.

102. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Case study Fe and Ni Archetypical itinerant ferromagnets LSDA predicts correct low T moment Band picture holds at low T Main puzzle: at high temperatures  has a Curie Weiss law with a moment larger than the ordered moment. 

103. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Iron and Nickel: crossover to a real space picture at high T (Lichtenstein, Katsnelson and Kotliar Phys Rev. Lett 87, 67205, 2001 )

104. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,T/Tc=.8) (Lichtenstein, Katsenelson,Kotliar Phys Rev. Lett 87, 67205, 2001)

105. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission and T/Tc=.8 Spin Autocorrelation: Ni (U=3, J=.9 ev)

106. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Iron and Nickel:magnetic properties (Lichtenstein, Katsenelson,GK PRL 01)

107. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ni and Fe: theory vs exp  eff    high T moment Fe 3.1 (theory) 3.12 (expt) Ni 1.5 (theory) 1.62 (expt) Curie Temperature T c Fe 1900 ( theory) 1043(expt) Ni 700 (theory) 631 (expt)

108. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Fe and Ni Consistent picture of Fe (more localized) and Ni (more correlated) Satellite in minority band at 6 ev, 30 % reduction of bandwidth, exchange splitting reduction.3 ev Spin wave stiffness controls the effects of spatial flucuations, it is about twice as large in Ni and in Fe Needed cluster extensions to improve on the critical temperature, nested cluster schemes and causality. Biroli Parcollet and Kotliar cond-mat 0307587.

109. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Case study Cerium. Study the alpha to gamma transition. Test accuracy of the approach, in a well study setting. Differentiate between the Kondo volume collapse picture and the Mott transition picture.

110. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Overview  Various phases : isostructural phase transition (T=298K, P=0.7GPa)  (fcc) phase [ magnetic moment (Curie-Wiess law) ]   (fcc) phase [ loss of magnetic moment (Pauli-para) ] with large volume collapse  v/v  15  (  -phase a  5.16 Å  -phase a  4.8 Å) volumesexp.LDALDA+U  28Å 3 24.7Å 3  34.4Å 3 35.2Å 3   -phase (localized) : High T phase  Curie-Weiss law (localized magnetic moment),  Large lattice constant  Tk around 60-80K   -phase (localized) : High T phase  Curie-Weiss law (localized magnetic moment),  Large lattice constant  Tk around 60-80K   -phase (delocalized:Kondo-physics) : Low T phase  Loss of Magnetism (Fermi liquid Pauli susceptibility) - completely screened magnetic moment  smaller lattice constant  Tk around 1000-2000K   -phase (delocalized:Kondo-physics) : Low T phase  Loss of Magnetism (Fermi liquid Pauli susceptibility) - completely screened magnetic moment  smaller lattice constant  Tk around 1000-2000K

111. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative Ideas. Johanssen, Mott transition of the f electrons as a function of pressure. Ce alpha gamma transition. spd electrons are spectators. Mathematical implementation, “metallic phase” treat spdf electrons by LDA, “insulating phase” put f electron in the core. Allen and Martin. Kondo volume collapse picture. The dominant effect is the spd-f hybridization.

112. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative Ideas “metallic alpha phase” Kondo effect between spd and f takes place. “insulating phase” no Kondo effect (low Kondo temperature). Mathematical implementation, Anderson impurity model in the Kondo limit suplemented with elastic terms. (precursor of DMFT ideas, but without self consistency condition).

113. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission&experiment A. Mc Mahan K Held and R. Scalettar (2002) K. Haule V. Udovenko and GK. (2003)

114. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS X.Zhang M. Rozenberg G. Kotliar (PRL 1993) A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497 Unfortunately photoemission cannot decide between the Kondo collapse picture and the Mott transition picture. Evolution of the spectra as a function of U, half filling full frustration, Hubbard model!!!!

115. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Resolution: Turn to Optics! Qualitative idea. The spd electrons have much larger velocities, so optics will be much more senstive to their behavior. See if they are simple spectators (Mott transition picture ) or wether a Kondo binding unbinding takes pace (Kondo collapse picture).

116. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

117. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Optical conductivity of Ce (expt. Van Der Eb et.al. theory Haule et.al) experiment LDA+DMFT K. Haule V. Udovenko and G Kotliar (2003)

118. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Origin of the features.

119. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion The anomalous temperature dependence and the formation of a pseudogap, suggests that the Kondo collapse picture is closer to the truth for Cerium.  Qualitative agreement with experiments, quantitative discrepancies.

120. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusions. Kappa Organics. Ti2O3. Fe-Ni Ce ………… Localization - Delocalization as a central theme in the electronic structure of solids. Next step: Part III. Self consistent determination of lattice and electrons using DMFT.

121. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion ( Part II). C-DMFT studies of the Mott transition reveal the development of a gap with a d wave symmmetry as the transition is approached. Plaquette as a minimal reference frame. Bond as a reference frame, and important role of the Coulomb interactions to trigger the MIT in Ti2O3. Need cluster treatments of itinerant magnets to obtain accurate critical temperatures. (Nested Cluster Schemes).

122. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

123. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Haule et. al.

124. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

125. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

126. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS La 1-x Sr x O 3 Adding holes to a Mott insulator in three dimensions. For very small doping,(x<.07) interesting spin and orbital order takes place, non universal physics and lattice distortions are important. Small energy scales, larger dopings more robust universal behavior. Canonical Brinkman Rice behavior, good system to test ab-initio DMFT.

127. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS (Tokura et. Al. 1993)A doped Mott insulator:La x Sr 1-x O 3

128. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT calculation U near the Mott transition, Rozenberg et.al 94

129. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Hall Coefficient, electron like.

130. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS La 1-x Sr x TiO 3 photoemission

131. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of spectra with doping U=4

132. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Optical spectral weight

133. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Optical conductivity

134. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Realistic Computation of Optical Properties : La 1-x Sr x TiO 3

135. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The Mott transition in f electron materials:a dynamical mean field theory (DMFT) perspective. Part 3. Gabriel Kotliar Center for Materials Theory Rutgers University

136. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline  The Mott transition in f electron systems. Differences with d electron systems.  Functional approach to DMFT and integration with electronic structure methods.  Case study: alpha-gamma transition in Cerium. Photoemission and Optical Spectroscopy.  Case study: the phases of plutonium, photoemission, total energy and lattice vibrations.  Case study: Americium under pressure.

137. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in f electron systems. Volume collapse in Cerium, Plutonium and Americium. Strong coupling of the electronic structure to the lattice. Presence of light spd electrons near the Fermi level. Heavy +light electrons.

138. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Overview  Various phases : isostructural phase transition (T=298K, P=0.7GPa)  (fcc) phase [ magnetic moment (Curie-Wiess law) ]   (fcc) phase [ loss of magnetic moment (Pauli-para) ] with large volume collapse  v/v  15  (  -phase a  5.16 Å  -phase a  4.8 Å) volumesexp.LDALDA+U  28Å 3 24.7Å 3  34.4Å 3 35.2Å 3   -phase (localized) : High T phase  Curie-Weiss law (localized magnetic moment),  Large lattice constant  Tk around 60-80K   -phase (localized) : High T phase  Curie-Weiss law (localized magnetic moment),  Large lattice constant  Tk around 60-80K   -phase (delocalized:Kondo-physics) : Low T phase  Loss of Magnetism (Fermi liquid Pauli susceptibility) - completely screened magnetic moment  smaller lattice constant  Tk around 1000-2000K   -phase (delocalized:Kondo-physics) : Low T phase  Loss of Magnetism (Fermi liquid Pauli susceptibility) - completely screened magnetic moment  smaller lattice constant  Tk around 1000-2000K

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

140. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Case study in f electrons, Mott transition in the actinide series. B. Johanssen 1974 Smith and Kmetko Phase Diagram 1984.

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

142. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative Ideas. Johanssen, Mott transition of the f electrons as a function of pressure. Ce alpha gamma transition. spd electrons are spectators. Mathematical implementation, “metallic phase” treat spdf electrons by LDA, “insulating phase” put f electron in the core. Allen and Martin. Kondo volume collapse picture. The dominant effect is the spd-f hybridization.

143. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative Ideas “metallic alpha phase” Kondo effect between spd and f takes place. “insulating phase” no Kondo effect (low Kondo temperature). Mathematical implementation, Anderson impurity model in the Kondo limit suplemented with elastic terms. (precursor of DMFT ideas, but without self consistency condition).

144. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic DMFT phase diagram one band Hubbard model. Introduce coupling to the lattice will cause a volume jump across the first order transition. (Majumdar and Krishnamurthy ).

145. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Cerium

146. 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

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

148. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Cerium

149. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Case study in f electrons, Mott transition in the actinide series. B. Johanssen 1974 Smith and Kmetko Phase Diagram 1984.

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

151. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline  Some comments on the role of DMFT in solid state physics problems and the strong correlation problem.  Introduction to DMFT: cavity construction E-DMFT and cluster methods. Introduction to DMFT: functional method.  Interfaces with electronic structure. A truly first principles approach GW-U method. LDA+DMFT.

152. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Two roads for ab-initio calculation of electronic structure of strongly correlated materials Correlation Functions Total Energies etc. Model Hamiltonian Crystal structure +Atomic positions

153. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS A more general perspective on DMFT. DMFT as an exact theory. (Chitra and Kotliar PRB 2001 Savrasov and GK cond- matt 2003) analogy to DFT. DMFT as an approximation (Chitra and Kotliar PRB2002) DMFT as a new starting point for perturbative expansions. ( P. Sun and G.K PRB 2002)

154. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT as an exact theory, analogy with DFT Start with TOE

155. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DFT: effective action construction(Fukuda et.al. ) Chitra and GK

156. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Exchange and correlation energy Exact formal expressions can be given in terms of a coupling constant integration.[Harris-Jones, adiabatic connection] DFT is useful because practical accurate expressions for Exc, exist. LDA, GGA, hybrids,

157. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Kohn Sham reference system

158. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Kohn Sham Greens function is an good point to compute spectra in perturbation theory in screenedCoulomb interaction GW,G0W0 Practical implementations, introduce a finite basis set. Division into valence (active ) degrees of freedom and core.

159. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Functional derivation.

160. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Model Hamiltonian. Exact functional of the local Greens function A +

161. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT for model Hamiltonians. Kohn Sham formulation. Introduce auxiliary field Exact “local self energy”

162. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS About XC functional. One can derive a coupling constant integration formulae (Harris Jones formula) for Generate approximations. The exact formalism generates the local Greens function and  ii is NOT the self energy. However one can use the approach as starting point for computing other quantities.

163. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mapping onto impurity models. The local Greens function A, and the auxilliary quantity  can be computed from the solution of an impurity model that describes the effects of the rest of the lattice on the on a selected central site.

164. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on functional construction Atoms as a reference point. Expansion in t. Locality does not necessarily mean a single point. Extension to clusters. Jii ---  Jii Ji i+  Aii ---  Ai i+   ii ---   i i+  Exact functional  Aii,Ai i+   he lattice self energy and other non local quantities extending beyond the cluster are OUTSIDE the formalism and need to be inferred.

165. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on funct. construction. Construction of approximations in the cluster case requires care to maintain causality. One good approximate construction is the cellular DMFT: a) take a supercell of the desired range,b) c) obtain estimate of the lattice self energy by restoring translational symmetry. Many other cluster approximations (eg. DCA, the use of lattice self energy in self consitency condition, restrictions of BK functional, etc. exist). Causality and classical limit of these methods has recently been clarified [ G Biroli O Parcollet and GK]

166. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Lattice and cluster self energies

167. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT 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 of viewed as parameters

168. 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.

169. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional : effective action construction ( Fukuda, Valiev and Fernando, Chitra and GK ). DFT, consider the exact free energy as a functional of an external potential. Express the free energy as a functional of the density by Legendre transformation.  DFT  (r)] Introduce local orbitals,   R (r-R)orbitals, 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  )]

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

171. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on LDA+DMFT Static limit of the LDA+DMFT functional, with  =  HF reduces to LDA+U Removes inconsistencies of this approach, Only in the orbitally ordered Hartree Fock limit, the Greens function of the heavy electrons is fully coherent Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing.

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

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

174. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Realistic implementation of the self consistency condition H and , do not commute Need to do k sum for each frequency DMFT implementation of Lambin Vigneron tetrahedron integration (Poteryaev et.al 1987)

175. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT References 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 and G.Kotliar, funcional formulation for full self consistent implementation. Savasov Kotliar and Abrahams. Application to delta Pu Nature (2001)

176. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Case study Cerium. Study the alpha to gamma transition. Test accuracy of the approach, in a well study setting. Differentiate between the Kondo volume collapse picture and the Mott transition picture.

177. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Overview  Various phases : isostructural phase transition (T=298K, P=0.7GPa)  (fcc) phase [ magnetic moment (Curie-Wiess law) ]   (fcc) phase [ loss of magnetic moment (Pauli-para) ] with large volume collapse  v/v  15  (  -phase a  5.16 Å  -phase a  4.8 Å) volumesexp.LDALDA+U  28Å 3 24.7Å 3  34.4Å 3 35.2Å 3   -phase (localized) : High T phase  Curie-Weiss law (localized magnetic moment),  Large lattice constant  Tk around 60-80K   -phase (localized) : High T phase  Curie-Weiss law (localized magnetic moment),  Large lattice constant  Tk around 60-80K   -phase (delocalized:Kondo-physics) : Low T phase  Loss of Magnetism (Fermi liquid Pauli susceptibility) - completely screened magnetic moment  smaller lattice constant  Tk around 1000-2000K   -phase (delocalized:Kondo-physics) : Low T phase  Loss of Magnetism (Fermi liquid Pauli susceptibility) - completely screened magnetic moment  smaller lattice constant  Tk around 1000-2000K

178. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative Ideas. Johanssen, Mott transition of the f electrons as a function of pressure. Ce alpha gamma transition. spd electrons are spectators. Mathematical implementation, “metallic phase” treat spdf electrons by LDA, “insulating phase” put f electron in the core. Allen and Martin. Kondo volume collapse picture. The dominant effect is the spd-f hybridization.

179. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative Ideas “metallic alpha phase” Kondo effect between spd and f takes place. “insulating phase” no Kondo effect (low Kondo temperature). Mathematical implementation, Anderson impurity model in the Kondo limit suplemented with elastic terms. (precursor of DMFT ideas, but without self consistency condition).

180. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission&experiment A. Mc Mahan K Held and R. Scalettar (2002) K. Haule V. Udovenko and GK. (2003)

181. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS X.Zhang M. Rozenberg G. Kotliar (PRL 1993) A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497 Unfortunately photoemission cannot decide between the Kondo collapse picture and the Mott transition picture. Evolution of the spectra as a function of U, half filling full frustration, Hubbard model!!!!

182. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Resolution: Turn to Optics! Qualitative idea. The spd electrons have much larger velocities, so optics will be much more senstive to their behavior. See if they are simple spectators (Mott transition picture ) or wether a Kondo binding unbinding takes pace (Kondo collapse picture).

183. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

184. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Optical conductivity of Ce (expt. Van Der Eb et.al. theory Haule et.al) experiment LDA+DMFT K. Haule V. Udovenko and G Kotliar (2003)

185. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Haule et. al.

186. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Origin of the features.

187. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

188. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion The anomalous temperature dependence and the formation of a pseudogap, suggests that the Kondo collapse picture is closer to the truth for Cerium.  Qualitative agreement with experiments, quantitative discrepancies.

189. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT studies of elemental Plutonium

190. 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 L=5, S=5/2, J=5/2, Mtot=Ms=  B gJ =.7  B Crystal fields     GGA+U estimate ML=-3.9 Mtot=1.1 (Savrasov GK 2000) This bit is quenches by the f and spd electrons Neutron Scattering in a field (Lander)

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

192. 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.

193. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission Technique Density of states for removing (adding ) a particle to the sample. Delocalized picture, it should resemble the density of states, (perhaps with some satellites). Localized picture. Two peaks at the ionization and affinity energy of the atom.

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

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

196. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Alpha and delta Pu

197. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Alpha phase is also a correlated metal. It differs from delta in the relative weight of the resonance and the Hubbard band. Consistent with resistivity and specific heat measurements.

198. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonon Spectra Electrons are the glue that hold the atoms together. Vibration spectra (phonons) probe the electronic structure. Phonon spectra reveals instablities, via soft modes. Phonon spectrum of Pu had not been measured. Short distance behavior of the elastic moduli.

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

200. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev. E = E i - E f Q = k i - k f

201. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Expt. Wong et. al.

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

203. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Shear anisotropy. Expt. vs Theory C’=(C11-C12)/2 = 4.78 GPa C’=3.9 GPa C44= 33.59 GPa C44=33.0 GPa C44/C’ ~ 7 Largest shear anisotropy in any element! C44/C’ ~ 8.4

204. 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)

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

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

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

208. 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 Ackland et. al. PRB.65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.

209. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Apply Realistic DMFT to the study of Am metal. S Murthy and GK Fully self consistent LDA+DMFT calculations. Use Hubbard 1 as impurity solver. Ingnore multiple splittings but take large spin orbit coupling Su(6)XSu(8) symmetry.

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

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

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

213. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Recent experimental studies of Am. J.C Griveaux et. al. ITU Non trivial evolution of the electronic structure with pressure. Superconductivity near the Mott transition ?

214. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Griveaux et. al.

215. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Griveaux et. al.

216. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Griveaux et. al.

217. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion Developing DMFT to face the challenges of a first principles theory of correlated materials is a very exciting project. There are many difficulties to surmount, but there is clear evidence that we are making significant progress. The goal is in sight, and we are getting exciting results along the way. Hope you join us!

218. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Two roads for ab-initio calculation of electronic structure of strongly correlated materials Correlation Functions Total Energies etc. Model Hamiltonian Crystal structure +Atomic positions

219. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Start with the TOE

220. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Rewrite the TOE as an electron boson problem.

221. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Build effective action for the local greens functions of the fermion and Bose field r=R+  R unit cell vector  position within the unit cell. Ir>=|R,  Couple sources to

222. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Legendre transfor the sources, eliminating the field  Build exact functional of the correlation functionsW(r R,r’ R) and G (r R,r’ R)

223. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS “Kohn Sham “ decomposition.

224. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS (E)DMFT pproximation to Sum over all LOCAL 2PI graphs (integrations are restricted over the unit cell ) built with W and G Map into impurity model to generate G and W Go beyond this approximation by returning to many body theory and adding the first non local correction.

225. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Test on extended Hubbard model V/U=.25, P Sun and GK

226. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS EDMFT functional.

227. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Returning to many body physics.

228. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Take the solution of the EDMFT equations as an approximation for the TRUE local self energy, and add the leading NON LOCAL corrections to the self energy G_NL W_NL, as a correction. Do it self consistently and as a one shot iteration G0_NL W0_NL and compare the results.