1. Strongly Correlated Electron Systems: a DMFT Perspective Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Statistical Mechanics Conference Rutgers December 2003

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to strongly correlated electrons. Introduction to Dynamical Mean Field Theory (DMFT) The Mott transition problem: some theoretical insights from DMFT studies of simple models. Some recent Experiments. Towards a DMFT based electronic structure method. Some highlight of recent results.

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to strongly correlated electrons. Essentials of Dynamical Mean Field Theory (DMFT) The Mott transition problem: some theoretical insights from DMFT studies of simple models. Some recent Experiments. Towards a DMFT based electronic structure method. Some highlight of recent results.

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Momentum Space (Sommerfeld) Standard model of solids Periodic potential, waves form bands, k in Brillouin zone. The electron in a solid: wave picture Maximum metallic resistivity 200  ohm cm Landau: Interactions renormalize away

5. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The electron in a solid: particle picture. NiO, MnO, …solid as a collection of atoms. e_ e_ e_ e_ High T : local moments, Low T spin-orbital order Superexchange One particle excitations: Hubbard bands. Excited atomic states adding or removing electrons, broaden into bands. 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

6. 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 (large metallic resistivities, optical responses that cannot be interpreted in terms of rigid bands, ……..). None of the standard electronic structure tools (LDA –GW 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.

7. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Correlated Materials do “big” things Mott transition.Huge resistivity changes V 2 O 3. Copper Oxides..(La 2-x Ba x ) CuO 4 High Temperature Superconductivity. 150 K in the Ca 2 Ba 2 Cu 3 HgO 8. Uranium and Cerium Based Compounds. Heavy Fermion Systems,CeCu 6,m*/m=1000 (La 1-x Sr x )MnO 3 Colossal Magneto-resistance. Huge Volume Collapses in Lanthanides (Ce, Pr…) and Actinides ( Pu, Am…) …………………………….. Unusual behavior, large resistivity, non-rigid bands, failures of the standard model.

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The Mott transition Pressure driven MIT. Forces to face directly a central issue of the strongly correlated electron systems. Localization delocalization problem. Relevant to many materials, eg V2O3,organics Techniques applicable to a very broad range or problems.

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

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to the strong correlation problem and to the Mott transition. DMFT ideas Applications to the Mott transition problem: some insights from studies of models. Towards an electronic structure method: applications to materials: Pu………. Outlook

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Limit of large lattice coordination Metzner Vollhardt, 89 Muller-Hartmann 89

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

13. 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)

14. 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)

15. 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.

16. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS How good is the LOCAL approximation?

17. 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

18. 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)]

19. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to the strong correlation problem. Essentials of DMFT Applications to the Mott transition problem: some insights from studies of models. A look at recent experiments. Towards an electronic structure method: applications to materials Outlook

20. 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)

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT results. Hubbard model on a frustrated lattice. Phase diagram in the T, U plane of a frustrated ((the magnetic order is supressed)) correlated system at integer filling. Purely electronic model is a good starting point. At high temperatures, the phase diagram is generic, insensitive to microscopic details. At low temperatures, details matters.

22. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insight, in the strongly correlated region the one particle density of states [Density of states for adding and removing one particle. Measureable in photoemission and inverse photoemission]has a three peak structure low energy quasiparticle peak plus Hubbard bands.

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Density of states for adding and removing one particle. Measureable in photoemission and inverse photoemission. Delocalized picture, it should resemble the density of states, (perhaps with some additional shifts and satellites). Localized picture. Two peaks at the ionization and affinity energy of the atom.

24. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS One electron spectra near the Mott transition, three peak structure.

25. 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)

26. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS QP in V2O3 was recently found Mo et.al

27. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous metallic resistivities In the “ in between region “ anomalous resistivities are the rule rather than the exception.

28. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity and Mott transition (Rozenberg et.al. ) Ni Se 2-x S x Insights from DMFT: think in term of spectral functions (branch cuts) instead of well defined QP (poles )

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

30. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights from DMFT  The Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phase  Control parameters: doping, temperature,pressure…

31. 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)

32. 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)

33. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Recent study of V2O3 under pressure Limlette et.al. Science 2003. Ising behavior at the endpoint finally found!

34. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ising critical endpoint! In V 2 O 3 P. Limelette et.al.

35. 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.

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

37. 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)

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

39. 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

40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT studies of the Mott transition Single site DMFT study of the Mott transition, based on a study of the Hubbard model on frustrated lattices made several interesting qualitative predictions. Confirmed by CDMFT study on 2 by2 cluster with intersting modifications. [O. Parcollet G. Biroli and G. Kotliar cond-matt 0308577] New experiments and of reexamination of old ones give credence to that the local picture is quite good. DMFT is a new reference frame to approach strongly correlated phenomena, and describes naturally, NON RIGID BAND picture, highly resistive states, etc….

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to strongly correlated electrons. Introduction to Dynamical Mean Field Theory (DMFT) The Mott transition problem: some theoretical insights from DMFT studies of simple models. Some recent Experiments. Towards a DMFT based electronic structure method. Some highlight of recent results.

42. 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 cond-matt 0308053. (2003.) Reviews: Held et.al., Psi-k Newsletter 56 (April 2003), p. 65 A. Lichtenstein M. Katsnelson and G. Kotliar cond- mat/0211076:

43. 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

44. 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)

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

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

47. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Electronic Physics of Pu

48. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DFT studies. Underestimates the volume by 35 % Predicts Pu to be magnetic. Largest quantitative failure of DFT-LDA- GA Fail to predict a stable delta phase.

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

50. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Alpha and delta Pu photoemission and dos (S. Savrasov and G. Kotliar)

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

52. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Inelastic X ray scattering. Wong et. al. Science 301, 1078 (2003).

53. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to strongly correlated electrons. Introduction to Dynamical Mean Field Theory (DMFT) The Mott transition problem: some theoretical insights from DMFT studies of simple models. Some recent Experiments. Towards a DMFT based electronic structure method. Some highlight of recent results.

54. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

55. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of the Spectral FunctionU/D=2, U/D=2.25 (Parcollet Biroli and Kotliar 2003.) Uc=2.35+-.05, Tc/D=1/44

56. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conjecture Formation of hot regions is a more general phenomena due to the proximity to the Mott point.

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

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

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

60. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

61. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

62. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to the strong correlation problem. Essentials of DMFT The Mott transition problem: some insights from studies of models. Towards an electronic structure method: applications to materials: Pu Outlook

63. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS What do we want from materials theory? New concepts, qualitative ideas Understanding, explanation of existent experiments, and predictions of new ones. Quantitative capabilities with predictive power. Notoriously difficult to achieve in strongly correlated materials. DMFT is delivering on both fronts.

64. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook Local approach to strongly correlated electrons. Many extensions, make the approach suitable for getting insights and quantitative results in correlated materials.

65. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion The character of the localization delocalization in simple( Hubbard) models within DMFT is now fully understood, nice qualitative insights.  This has lead to extensions to more realistic models, and a beginning of a first principles approach to the electronic structure of correlated materials.

66. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook Systematic improvements, short range correlations, cluster methods, improved mean fields. Improved interfaces with electronic structure. Exploration of complex strongly correlated materials. Correlation effects on surfaces, large molecules, systems out of equilibrium, illumination, finite currents, aeging.

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

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

69. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dai et. al.

70. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS V2O3 resistivity

71. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission&experiment

72. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Example: DMFT for lattice model (e.g. single band Hubbard).Muller Hartman 89, Chitra and Kotliar 99. Observable: Local Greens function G ii (  ). Exact functional  [G ii (  )  DMFT Approximation to the functional.

73. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional : effective action construction ( Chitra and GK ). 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  )] Approximate functional using DMFT insights.

74. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455

75. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in V 2 O 3 under pressure or chemical substitution on V-site

76. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Single site DMFT, functional formulation Local self energy (Muller Hartman 89)

77. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT functional constructions. Construction of approximations in the cluster case requires care to maintain causality. 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 consistency condition, restrictions of BK functional, etc. exist). Causality and classical limit of these methods has recently been clarified [ G Biroli O Parcollet and GK]

78. 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)]

79. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Failure of the Standard Model: NiSe 2-x S x Miyasaka and Takagi (2000)

80. 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)

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

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

83. 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

84. 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)

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

86. 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.