Hubbard model

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

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

3. DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)] Weissfield THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

4. A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497 Mean-Field : Classical vs Quantum Quantum case Classical case THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

5. Single site DMFT, functional formulation Local self energy (Muller Hartman 89) • Express in terms of Weiss field (semicircularDOS) • The Mott transition as bifurcation point in functionals oG[G] or F[D], (G. Kotliar EPJB 99) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

8. 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 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

10. Kuwamoto Honig and AppellPRB (1980) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

11. Phase Diag: Ni Se2-x Sx THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

12. 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… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

13. 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) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

14. ARPES measurements on NiS2-xSexMatsuura et. Al Phys. Rev B 58 (1998) 3690 . THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

15. 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 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

16. Anomalous Resistivity and Mott transition Ni Se2-x Sx THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

17. 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. Real and momentum space. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

18. 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 ). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

19. 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) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

20. Anomalous Resistivity:LiV2O4 Takagi et.al. PRL 2000 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

21. Anomalous Resistivities: DopedHubbard Model (Prushke and Jarrell 1993) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

22. Anomalous Resistivities:Doped Hubbard ModelG. Palsson 1998 IPT NCA THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

24. Specific Heat Titanates THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

26. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

27. Standard Model 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 Typical Mott values of the resistivity 200 mOhm-cm Residual instabilites SDW, CDW, SC THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

30. 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) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

32. 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 ). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

33. 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 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

36. LDA functional Conjugate field, VKS(r) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

38. A time-honored example: Mott transition in V2O3 under pressure or chemical substitution on V-site THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

39. 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 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

41. Photoemission LaTiO3 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS