Diagrammatic Theory of Strongly Correlated Electron Systems

1. Diagrammatic Theory of Strongly Correlated Electron Systems

2. 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 Outline

3. 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 Outline

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) • 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 E= -2t2/U E= 0

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

8. Doping Mott insulator – DMFT perspective • Metallic system always Fermi liquid ImS(w)w2 • Fermi surface unchanged (volume and shape) • Narrow quasiparticle peak of width ZeFd at the Fermi level • Effective mass (m*/m1/Z) diverges at the transition • High-temperature (T>> ZeF) almost free spin LHB UHB quasip. peak d Georges, Kotliar, Krauth and Rozenberg 1996

9. 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 Outline

10. mean-field description of the exchange term is exact within DMFT 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  2N deg. states) Why? Metzner Vollhardt 89 J disappears completely in the paramagnetic phase!

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

12. 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 Outline

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

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

15. 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 Outline

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

17. Local spectral function

18. 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 Outline

19. Luttinger’s theorem? (m-ReS(0))/zt

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

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

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

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

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

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

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

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

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

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

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

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

32. 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 Outline

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

34.  &  EMDT+NCA ED 20 sites

35. 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 Outline

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

37. 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 Outline

38. 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 Outline

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

40. Auxiliary particle technique

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

42. 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 Outline

43. Luttinger-Ward functional for SUNCA

44. 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 Outline

45. Scaling of TK

46. Comparison with NRG

47. 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 Outline

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

49. Extended Dynamical Mean Field

50. Metal-insulator transition • el-el correlations not important: • band insulator: • the lowest conduction band is full • (possible only for even number of electrons) • gap due to the periodic potential – few eV • simple metal • Conduction band partially occupied • semiconductor zt • 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 U eF* Zhang, Rozenberg and Kotliar 1992