1. Dynamical Mean Field Theory of the Mott Transition Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Jerusalem Winter School January 2002

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS OUTLINE OF THE COURSE Motivation. Electronic structure of correlated materials, limiting cases and open problems. The standard model of solids and its failures. Introduction to the Dynamical Mean Field Theory (DMFT). Cavity construction. Statistical Mechanical Analogies. Lattice Models and Quantum Impurity models. Functional derivation.

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline The limit of large lattice coordination. Ordered phases. Correlation functions. Techniques for solving the Dynamical Mean Field Equations. [ Trieste School June 17- 22 2002]

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline The Mott transition. Early ideas. Brinkman Rice. Hubbard. Slater. Analysis of the DMFT equations: existence of a Mott transition. The Mott transition within DMFT. Overview of some important results of DMFT studies of the Hubbard Model. Electronic Structure of Correlated Materials. Canonical Phase diagram of a fully frustrated Hubbard model. Universal and non universal aspects of the physics of strongly correlated materials.

5. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Analysis of the DMFT equations. Existence of a Mott transition. Analysis from large U and small U. The destruction of the metallic phase. Landau analysis. Uc1. Uc2. The Mott transition endpoint. A new look at experiments.

6. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline The electronic structure of real materials. Examples of problems where DMFT gives new insights, and quantitative understanding: itinerant ferromagnetism, Fe, Ni. Volume collapse transitions, actinide physics. Doping driven Mott transition titanites.

7. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline New directions, beyond single site DMFT.

8. Realistic Theories of Correlated Materials ITP, Santa-Barbara workshop July 29 – December 16 (2002) O.K. Andesen, A. Georges, G. Kotliar, and A. Lichtenstein Contact: kotliar@physics.rutgers.edukotliar@physics.rutgers.edu Conference: November 25-29, (2002)

9. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The promise of Strongly Correlated Materials Copper Oxides. High Temperature Superconductivity. Uranium and Cerium Based Compounds. Heavy Fermion Systems. (LaSr)MnO3 Colossal Magnetoresistence.

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The Promise of Strongly Correlated Materials. High Temperature Superconductivity in doped filled Bucky Balls (B. Battlog et.al Science) Thermoelectric response in CeFe 4 P 12 (H. Sato et al. cond-mat 0010017). Large Ultrafast Optical Nonlinearities Sr 2 CuO 3 (T Ogasawara et.al cond-mat 000286) Theory will play an important role in optimizing their physical properties.

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Drude Sommerfeld Bloch, Periodic potential Bands, k in Brillouin zone How to think about the electron in a solid? Maximum metallic resistivity 200  ohm cm

12. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Standard Model High densities, electron as a wave, band theory, k- space Landau: Interactions Renormalize Away One particle excitations: quasi-particle bands Density Functional Theory in Kohn Sham Formulation, successful computational tool for total energy, and starting point For perturbative calculation of spectra, Si Au, Li, Na ……………………

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Standard Model : Metals Hall Coefficient Resistivity Thermopower Specific Heat Susceptibility Predicts low temperature dependence of thermodynamics and transport

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Quantitative Tools : Density Functional Theory with approximations suggested by the Kohn Sham formulation, (LDA GGA) is a successful computational tool for the total energy, and a good starting point for perturbative calculation of spectra, GW, transport.……………………

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

16. 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….) Rich structure of Magnetic and Orbital Ordering at low T Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics.

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

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

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

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

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Failure of the Standard Model: Anomalous Spectral Weight Transfer Optical Conductivity of FeSi for T=,20,20,250 200 and 250 K from Schlesinger et.al (1993) Neff depends on T

22. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strong Correlation Problem Large number of compounds (d,f,p….). Departure from the standard model. Hamiltonian is known. Identify the relevant degrees of freedom at a given scale. Treat the itinerant and localized aspect of the electron The Mott transition, head on confrontation with this issue Dynamical Mean Field Theory simplest approach interpolating between that bands and atoms

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

24. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Hubbard model  U/t  Doping d or chemical potential  Frustration (t’/t)  T temperature Mott transition as a function of doping, pressure temperature etc.

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

26. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mean-Field : Classical

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

28. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on DMFT Exact in both atomic and band limits Weiss field is a function Multiple energy scales in a correlated electron problem, non linear coupling between them. Frezes spatial fluctuations but treats quantum fluctuations exactly, local view of the quantum many body problem.

29. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Example: semicircular DOS

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

31. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Solving the DMFT equations Wide variety of computational tools (QMC, NRG,ED….) Analytical Methods

32. 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)

33. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Single site DMFT, functional formulation Express in terms of Weiss field (semicircularDOS) The Mott transition as bifurcation point in functionals o  G  or F[  ], (G. Kotliar EPJB 99)

34. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT for lattice hamiltonians k independent  k dependent G, Local Approximation Treglia et. al 1980

35. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS How to compute  View locally the lattice problem as a (multiorbital) Anderson impurity model The local site is now embedded in a medium characterized by

36. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS How to determine the medium Use the impurity model to compute  and the impurity local Greens function. Require that impurity local Greens function equal to the lattice local Greens function. Weiss field

37. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Response functions

38. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evaluation of the Free energy.

39. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Solving the DMFT equations Wide variety of computational tools (QMC, NRG,ED….) Analytical Methods

40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Review of DMFT, technical tools for solving DMFT eqs.., applications, references…… A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT: Methods of Solution

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition: Early ideas. Half filling. Evolution of the one electron spectra [physical quantity measured in photoemission and BIS] as a function of control parameters. ( U/t, pressure, temperature ) Hubbard, begin in paramagnetic insulator. As U/t is reduced Hubbard bands merge. Gap closure. Mathematical description, closure of equations of motion, starting from atoms (I.e. large U). Incoherent motion, no fermi surface.

43. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition: early ideas. Brinkman and Rice. Gutzwiller. Begin in paramagnetic metallic state, as U/t approaches a critical value the effective mass diverges. Luttinger fermi surface. Mathematical description, variational wave function, slave bosons, quantum coherence and double occupancy.

44. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Slave bosons: mean field +fluctuations Fluctuations of the slave bosons around the saddle point gives rise to Hubbard bands. Starting from the insulating side, in a paramagnetic state, the gap closes at the same U, where Z vanishes. No satisfactory treatement of finite temperature properties.

45. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott vs Slater Mott: insulators in the absence of magnetic long range order. e.g. Vanadium Oxide Nickel Oxide. Mott transition in the paramagnetic state. Slater: insulating behavior as a consequence of antiferromagnetic long range order. Double the unit cell to convert a Mott insulator into a band insulator.

46. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS A time-honored example: Mott transition in V 2 O 3 under pressure or chemical substitution on V-site

47. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Local view of the spectral function Partition function of the Anderson impurity model : gas of kinks [Anderson and Yuval] Insulating state Metallic state, Metallic state, proliferation of kinks. Insulating state. Kinks are confined.

48. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Local view of the spectral function. Consistent treatement of quasiparticles and collective modes. Kinky paths, with may spin fluctuations: low energy resonance [Abrikosov Suhl Resonance] Confined kinks, straight paths, Hubbard bands. [control the insulator partition function] Strongly correlated metal has both.

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

50. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Destruction of the metal The gap is well formed at Uc2, when the metal is destroyed. Hubbard bands are well formed in the metal.

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

52. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Destruction of the insulator Continue the insulating solution below Uc2. Coexistence of two solutions between Uc1 and Uc2 Mott Hubbard gap vanishes linearly at Uc1.

53. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Recent calculation of the phase diagram of the frustrated Half filled Hubbard model with semicircular DOS (QMC Joo and Udovenko PRB2001).

54. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Case study: IPT half filled Hubbard one band (Uc1) exact = 2.1 (Exact diag, Rozenberg, Kajueter, Kotliar 1995), (Uc1) IPT =2.4 (Uc2) exact =2.95 (Projective self consistent method, Moeller Si Rozenberg Kotliar PRL 1995 ) (Uc 2 ) IPT =3.3 (T MIT ) exact =.026+_.004 (QMC Rozenberg Chitra and Kotliar PRL 1999), (T MIT ) IPT =.5 (U MIT ) exact =2.38 +-.03 (QMC Rozenberg Chitra and Kotliar PRL 1991), (U MIT ) IPT =2.5 For realistic studies errors due to other sources (for example the value of U, are at least of the same order of magnitude).

55. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic DMFT phase diagram Hubbard model (partial frustration) Rozenberg et.al. PRL (1995)

56. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Kuwamoto Honig and Appell PRB (1980)

57. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phase Diag: Ni Se 2-x S x

58. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights from DMFT  Low temperatures several competing phases. Their relative stability depends on chemistry and crystal structure  High temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT

59. 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…

60. 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 2000)

61. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS. ARPES measurements on NiS 2-x Se x Matsuura et. Al Phys. Rev B 58 (1998) 3690

62. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights from DMFT: think in term of spectral functions (branch cuts) instead of well defined QP (poles ) Resistivity near the metal insulator endpoint ( Rozenberg et. Al 1995) exceeds the Mott limit

63. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity and Mott transition Ni Se 2-x S x

64. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous resisitivity near Mott transition.

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

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

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

68. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous transfer of spectral weight in heavy fermions [Rozenberg etal]

69. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights from DMFT Mott transition as a bifurcation of an effective action Important role of the incoherent part of the spectral function at finite temperature Physics is governed by the transfer of spectral weight from the coherent to the incoherent part of the spectra. [Non local in frequency] Real and momentum space. –

70. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

71. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity:LiV 2 O 4 Takagi et.al. PRL 2000

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

73. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

74. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Standard Model Typical Mott values of the resistivity 200  Ohm- cm Residual instabilites SDW, CDW, SC Odd # electrons -> metal Even # electrons -> insulator  Theoretical foundation: Sommerfeld, Bloch and Landau  Computational tools DFT in LDA  Transport Properties, Boltzman equation, low temperature dependence of transport coefficients

75. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Failure of the “Standard Model”: Cuprates Anomalous Resistivity

76. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT  Formulation as an electronic structure method (Chitra and Kotliar)  Density vs Local Spectral Function  Extensions to treat strong spatial inhomogeneities. Anderson Localization (Dobrosavlevic and Kotliar),Surfaces (Nolting),Stripes (Fleck Lichtenstein and Oles)  Practical Implementation (Anisimov and Kotliar, Savrasov, Katsenelson and Lichtenstein)

77. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT  Spin Orbital Ordered States  Longer range interactions Coulomb, interactions, Random Exchange (Sachdev and Ye, Parcollet and Georges, Kajueter and Kotliar, Si and Smith, Chitra and Kotliar,)  Short range magnetic correlations. Cluster Schemes. (Ingersent and Schiller, Georges and Kotliar, cluster expansion in real space, momentum space cluster DCA Jarrell et.al., C-DMFT Kotliar et. al ).

78. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strongly Correlated Electrons  Competing Interaction  Low T, Several Phases Close in Energy  Complex Phase Diagrams  Extreme Sensitivity to Changes in External Parameters  Need for Quantitative Methods

79. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Landau Functional G. Kotliar EPJB (1999)

80. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA functional Conjugate field, V KS (r)

81. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Minimize LDA functional Kohn Sham eigenvalues, auxiliary quantities.

82. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ising character of the transfer of spectral weight Ising –like dependence of the photo-emission intensity and the optical spectral weight near the Mott transition endpoint

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

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