1. Electronic Structure of Strongly Correlated Materials : a DMFT Perspective Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Supported by the NSF -DMR

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to the electronic structure of correlated electrons Dynamical Mean Field Theory Delocalization - Localization Transition. Universal aspects insights from DMFT A case study of system specific properties:  Pu (S. Savrasov) Outlook

3. 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

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

5. 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?

6. 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 ……………………

7. 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

8. 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

9. 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

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott : correlations localize the electron Low densities, electron as a particle, atomic physics, real space One particle excitations: Hubbard bands NiO, CoO MnO…. Atoms: sharp excitation lines corresponding to adding or removing e In solids they broaden by their incoherent motion, Hubbard bands Magnetic and Orbital Ordering at low T Quantitative calculations of Hubbard bands and exchange constants, LDA+ U

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Localization vs Delocalization 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 works well Dynamical Mean Field Theory: Simplest approach to the electronic structure, which interpolates correctly between atoms and bands

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

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

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Failure of the Standard Model: 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

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

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

17. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strong Correlation Problem Large number of f and d electrons based compounds 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

18. 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.

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

20. 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)

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

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

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Reviews of DMFT Prushke T. Jarrell M. and Freericks J. Adv. Phys. 44,187 (1995) A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

24. 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 ).

25. 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)

26. 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

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

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

29. 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

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

31. 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…

32. 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 Mott endpoint (Kotliar Lange and Rozenberg 2000)

33. 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

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

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

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

37. 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

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

39. 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 weigth from the coherent to the incoherent part of the spectra.

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

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Realistic Calculations of the Electronic Structure of Correlated materials Combinining DMFT with state of the art electronic structure methods to construct a first principles framework to describe complex materials Hubbard bands and QP bands The puzzle of elemental plutonium (S. Savrasov)

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Delocalization Localization Transition across the actinide series

43. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Problems with LDA o DFT in the LDA or GGA is a well established tool for the calculation of ground state properties. o Many studies (Freeman, Koelling 1972, ….Beottger et.al 1998, Wills et.al. 1999) give o an equilibrium volume of the  phase  Is 35% lower than experiment o This is the largest discrepancy ever known in DFT based calculations.

44. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu: Complex Phase Diagram (J. Smith LANL)

45. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity J. Smith LANL

46. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu Specific Heat

47. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu: DMFT total energy vs Volume (Savrasov 00)

48. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Lda vs Exp Spectra

49. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu Spectra DMFT(Savrasov) EXP (Arko et. Al)

50. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS S. Savrasov: DMFT Lab

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

52. 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 Realistic Treatments

53. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook Strongly correlated electron exhibit unusual characteristics, complex systems. Two recent examples: large 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.

54. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook  The Strong Correlation Problem:How to deal with a multiplicity of competing low temperature phases and infrared trajectories which diverge in the IR  Strategy: advancing our understanding scale by scale  Generalized cluster methods to capture longer range magnetic correlations  New structures in k space?

55. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT: References Collaborators: V. Anisimov, R. Chitra, V. Dobrosavlevic, D. Fisher, A. Georges, H. Kajueter, W.Krauth, E. Lange, G. Moeller, Y. Motome, G. Palsson, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko, X.Y. Zhang Other work: A. Brandt, W. Nolting, R. Bulla, M. Jarrell, D. Logan, J. Freericks, T. Prushke, W. Metzner, F. Gebhardt, A. Lichtenstein, M. Fleck D. Vollhardt ……………….

56. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Reviews of DMFT Prushke T. Jarrell M. and Freericks J. Adv. Phys. 44,187 (1995) A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

57. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Challenges The photoemission in cuprates has a strong momentum dependence Strong Magnetic Correlations (no orbital degeneracy) Single Site DMFT does not capture these effects

58. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Cuprates: Photoemission – Transfer of Spectral Weight with a) temperature and b) doping

59. 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 ).

60. 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)

61. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Cuprates: Photoemission – Transfer of Spectral Weight with a) temperature and b) doping

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

63. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivities: Doped Hubbard Model (Prushke and Jarrell 1993)

64. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivities: Doped Hubbard Model G. Palsson 1998 IPT NCA

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

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

67. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

68. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS