1. Diagrammatic Theory of Strongly Correlated Electron Systems

2. Outline Introduction  Metal-insulator transition  Intersite interactions in DMFT Extended DMFT  Pseudogap – Incoherent metal  Luttinger’s theorem?  Thermodynamics  Transport Anderson impurity model at finite U  Motivation  Definition of the SUNCA approximation  Results of SUNCA Summary

3. Outline Introduction  Metal-insulator transition  Intersite interactions in DMFT Extended DMFT  Pseudogap – Incoherent metal  Luttinger’s theorem?  Thermodynamics  Transport Anderson impurity model at finite U  Motivation  Definition of the SUNCA approximation  Results of SUNCA Summary

4. Use of HTc  Magnetic levitation (Japan 1999, 343 m.p.h)  Magnetic resonance imaging  Fault current limiters of 6.4MVA, response time ms  E-bombs (strong EM pulse)  5000-horsepower motor made with sc wire (July 2001) (July 2001)  Electric generators, 99% efficiency  Energy storage 3MW

5. Use of HTc  Underground cable in Copenhagen (for 150000 citizens,30 meters long, May 2001)   Researching the possibility to build petaflop computers  Market $200 billion by the year 2010

6. Materials undergoing MIT  High temperature superconductors (2D systems, transition with doping)  Other 3d transition metal oxides (Nickel,Vanadium,Titanium,…) 2D and 3D, transition with doping or pressure  Many f-electron systems Hubbard model – generic model for materials undergoing MIT generic model for materials undergoing MIT E= -2t 2 /U E= 0

7. Dynamical mean-field theory & MIT mapping fermionic bath Zhang, Rozenberg and Kotliar 1992 U

8. Doping Mott insulator – DMFT perspective  Metallic system always Fermi liquid  Im    Fermi surface unchanged (volume and shape)  Narrow quasiparticle peak of width Z  F   at the Fermi level  Effective mass (m*/m  1/Z) diverges at the transition  High-temperature (T>> Z  F ) almost free spin Georges, Kotliar, Krauth and Rozenberg 1996 LHB UHB quasip. peak 

9. Outline Introduction  Metal-insulator transition  Intersite interactions in DMFT Extended DMFT  Pseudogap – Incoherent metal  Luttinger’s theorem?  Thermodynamics  Transport Anderson impurity model at finite U  Motivation  Definition of the SUNCA approximation  Results of SUNCA Summary

10. Nonlocal interaction in DMFT?  Local quantum fluctuations (between states ) completely taken into account within DMFT  Nonlocal quantum fluctuations are mostly lost in DMFT (nonlocal RKKY inter.) (residual ground-state entropy of par. Mott insulator is ln2  2 N deg. states) (residual ground-state entropy of par. Mott insulator is ln2  2 N deg. states) Why? Metzner Vollhardt 89 mean-field description of the exchange term is exact within DMFT J disappears completely in the paramagnetic phase !

11. For simplicity, take the infinite U limit  t-J model: How does intersite exchange J change Mott transition? Hubbard model

12. Outline Introduction  Metal-insulator transition  Intersite interactions in DMFT Extended DMFT  Pseudogap – Incoherent metal  Luttinger’s theorem?  Thermodynamics  Transport Anderson impurity model at finite U  Motivation  Definition of the SUNCA approximation  Results of SUNCA Summary

13. Extended DMFT J and t equally important: fermionic bath mapping bosonic bath fluctuating magnetic field Si & Smith 96, Kajuter & Kotliar 96 Source of the inelasting scattering

14. Still local and conserving theory Local quantities can be calculated from the corresponding impurity problem  Long range fluctuations frozen  Strong inelasting scattering due to local magnetic fluctuations Fermion bubble is zero in the paramagnetic state

15. Outline Introduction  Metal-insulator transition  Intersite interactions in DMFT Extended DMFT  Pseudogap – Incoherent metal  Luttinger’s theorem?  Thermodynamics  Transport Anderson impurity model at finite U  Motivation  Definition of the SUNCA approximation  Results of SUNCA Summary

16. Pseudogap – Incoherent metal Im Pseudogap due to strong inelasting scattering from local magnetic fluctuations Not due to finite ranged fluctuating antiferromagnetic (superconducting) domains highly incoherent response

17. Local spectral function

18. Outline Introduction  Metal-insulator transition  Intersite interactions in DMFT Extended DMFT  Pseudogap – Incoherent metal  Luttinger’s theorem?  Thermodynamics  Transport Anderson impurity model at finite U  Motivation  Definition of the SUNCA approximation  Results of SUNCA Summary

19. Luttinger’s theorem?  Re  zt

20. A(k,  )  =0.02 kxkx kyky k A(k,0) A(k,  ) White lines corresponds to noninteracting system

21. A(k,  )  =0.04 kxkx kyky k A(k,0) A(k,  ) White lines corresponds to noninteracting system

22. A(k,  )  =0.06 kxkx kyky k A(k,0) A(k,  ) White lines corresponds to noninteracting system

23. A(k,  )  =0.08 kxkx kyky k A(k,0) A(k,  ) White lines corresponds to noninteracting system

24. A(k,  )  =0.10 kxkx kyky k A(k,0) A(k,  ) White lines corresponds to noninteracting system

25. A(k,  )  =0.12 kxkx kyky k A(k,0) A(k,  ) White lines corresponds to noninteracting system

26. A(k,  )  =0.14 kxkx kyky k A(k,0) A(k,  ) White lines corresponds to noninteracting system

27. A(k,  )  =0.16 kxkx kyky k A(k,0) A(k,  ) White lines corresponds to noninteracting system

28. A(k,  )  =0.18 kxkx kyky k A(k,0) A(k,  ) White lines corresponds to noninteracting system

29. A(k,  )  =0.20 kxkx kyky k A(k,0) A(k,  ) White lines corresponds to noninteracting system

30. A(k,  )  =0.22 kxkx kyky k A(k,0) A(k,  ) White lines corresponds to noninteracting system

31. A(k,  )  =0.24 kxkx kyky k A(k,0) A(k,  ) White lines corresponds to noninteracting system

32. Outline Introduction  Metal-insulator transition  Intersite interactions in DMFT Extended DMFT  Pseudogap – Incoherent metal  Luttinger’s theorem?  Thermodynamics  Transport Anderson impurity model at finite U  Motivation  Definition of the SUNCA approximation  Results of SUNCA Summary

33. Entropy EMDT+NCA ED 20 sites ED: Jaklič & Prelovšek, 1995 Experiment: LSCO (T/t  0.07) Cooper & Loram

34.  &  EMDT+NCA ED 20 sites

35. Outline Introduction  Metal-insulator transition  Intersite interactions in DMFT Extended DMFT  Pseudogap – Incoherent metal  Luttinger’s theorem?  Thermodynamics  Transport Anderson impurity model at finite U  Motivation  Definition of the SUNCA approximation  Results of SUNCA Summary

36. Hall coefficient T~1000K LSCO: Nishikawa, Takeda & Sato (1994)

37. Outline Introduction  Metal-insulator transition  Intersite interactions in DMFT Extended DMFT  Pseudogap – Incoherent metal  Luttinger’s theorem?  Thermodynamics  Transport Anderson impurity model at finite U  Motivation  Definition of the SUNCA approximation  Results of SUNCA Summary

38. Outline Introduction  Metal-insulator transition  Intersite interactions in DMFT Extended DMFT  Pseudogap – Incoherent metal  Luttinger’s theorem?  Thermodynamics  Transport Anderson impurity model at finite U  Motivation  Definition of the SUNCA approximation  Results of SUNCA Summary

39. Motivation  Numerical renormalization group (NRG)  Quantum Monte Carlo simulation (QMC)  Exact diagonalization (ED)  Iterated perturbation theory (IPT)  Resummations of perturbation theory (NCA, CTMA) A need to solve the DMFT impurity problemA need to solve the DMFT impurity problem for real materials with orbital degeneracy Quantum dots in mesoscopic structuresQuantum dots in mesoscopic structures Several methods available to solve AIM: Either slow or less flexible

40. Auxiliary particle technique

41. NCA  Simple fast and flexible method  Works for T>0.2 T K  Works only in the case of U=   Naive extension very badly fails  T K several orders of magnitude too small

42. Schrieffer-Wolff transformation

43. Outline Introduction  Metal-insulator transition  Intersite interactions in DMFT Extended DMFT  Pseudogap – Incoherent metal  Luttinger’s theorem?  Thermodynamics  Transport Anderson impurity model at finite U  Motivation  Definition of the SUNCA approximation  Results of SUNCA Summary

44. Luttinger-Ward functional for SUNCA

45. Self-energies and Green’s function

46. Outline Introduction  Metal-insulator transition  Intersite interactions in DMFT Extended DMFT  Pseudogap – Incoherent metal  Luttinger’s theorem?  Thermodynamics  Transport Anderson impurity model at finite U  Motivation  Definition of the SUNCA approximation  Results of SUNCA Summary

47. Comparison of various approximations

48. Scaling of T K

49. Comparison with NRG

50. Outline Introduction  Metal-insulator transition  Intersite interactions in DMFT Extended DMFT  Pseudogap – Incoherent metal  Luttinger’s theorem?  Thermodynamics  Transport Anderson impurity model at finite U  Motivation  Definition of the SUNCA approximation  Results of SUNCA Summary

51. Summary EDMFT Purely local magnetic fluctuations can  induce pseudogap  suppress large entropy at low doping  induce strongly growing R H with decreasing T and  Luttinger’s theorem is not applicable in the incoherent regime (  <0.20) Fermi liquid is recovered only when  *>J SUNCA Infinite series of skeleton diagrams is needed to recover correct low energy scale of the AIM at finite Coulomb interaction U

52. Extended Dynamical Mean Field

53. Metal-insulator transition el-el correlations not important:  band insulator: the lowest conduction band is fullthe lowest conduction band is full (possible only for even number of electrons) (possible only for even number of electrons) gap due to the periodic potential – few eVgap due to the periodic potential – few eV  simple metal Conduction band partially occupiedConduction band partially occupied  semiconductor el-el correlations important:  Mott insulator despite the odd number of electrons  Cannot be explained within the independent-electron picture (many body effect)  Several competing mechanisms and several energy scales zt F*F*F*F* Zhang, Rozenberg and Kotliar 1992 U

54. Doping Mott insulator – DMFT perspective  Metallic system always Fermi liquid  Im    Fermi surface unchanged (volume and shape)  Narrow quasiparticle peak of width Z  F   at the Fermi level  Effective mass (m*/m  1/Z) diverges at the transition  High-temperature (T>> Z  F ) almost free spin Georges, Kotliar, Krauth and Rozenberg 1996 LHB UHB quasip. peak 

55. Still local and conserving theory Local quantities can be calculated from the corresponding impurity problem  Long range fluctuations frozen  Strong inelasting scattering due to local magnetic fluctuations

56. Diagrammatic auxiliary particle impurity solver NCA impurity solver This bubble is zero in the paramagnetic state

57.  Independent electron picture not adequate  Yields both bandlike and localized behaviour  Favor local magnetic moments  Lead to a conventional band spectrum