1. Strongly Correlated Electron Systems: a DMFT Perspective Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Colloquium UBC September (2004)

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to the strong correlation problem. Essentials of DMFT The Mott transition problem: some insights from studies of models. Towards an electronic structure method: applications to materials: Ce, Pu Outlook

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Momentum Space (Sommerfeld) Standard model of solids Periodic potential, waves form bands, k in Brillouin zone The electron in a solid: wave picture Maximum metallic resistivity 200  ohm cm Landau: Interactions renormalize away

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Standard Model of Solids RIGID BAND PICTURE. Optical response, transitions between bands. Quantitative tools: DFT, LDA, GGA, total energies,good starting point for spectra, GW,and transport

5. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The electron in a solid: particle picture. NiO, MnO, …Array of atoms is insulating if a>>a B. Mott: correlations localize the electron e_ e_ e_ e_ Think in real space, solid collection of atoms High T : local moments, Low T spin-orbital order Superexchange

6. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott : Correlations localize the electron Low densities, electron behaves as a particle,use atomic physics, work in 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….) Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics. H H H+ H H H motion of H+ forms the lower Hubbard band H H H H- H H motion of H_ forms the upper Hubbard band

7. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Localization vs Delocalization Strong Correlation Problem A large number of compounds with electrons in partially filled shells, are not close to the well understood limits (localized or itinerant). Non perturbative problem. These systems display anomalous behavior (departure from the standard model of solids). Neither LDA –GW or LDA+U or Hartree Fock work well. Dynamical Mean Field Theory: Simplest approach to electronic structure, which interpolates correctly between atoms and bands. Treats QP bands and Hubbard bands. New reference point, to replace the Kohn Sham system.

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DFT+GW program has been less succesful in correlated situations. Strong interactions localize the particles. Atoms with open shells are not easily connected to band theory. The spectrum in this case, contain Hubbard bands which are NOT simply perturbatively connected to the Kohn Sham orbitals. Need an alternative reference point for doing perturbation theory! Situation is worse “in between the atomic and the localized limit” DMFT!

9. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Correlated Materials do “big” things Mott transition.Huge resistivity changes V 2 O 3. Copper Oxides..(La 2-x Ba x ) CuO 4 High Temperature Superconductivity. 150 K in the Ca 2 Ba 2 Cu 3 HgO 8. Uranium and Cerium Based Compounds. Heavy Fermion Systems,CeCu 6,m*/m=1000 (La 1-x Sr x )MnO 3 Colossal Magneto- resistance.

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strongly Correlated Materials. Large thermoelectric response in CeFe 4 P 12 (H. Sato et al. cond-mat 0010017). Ando et.al. NaCo 2-x Cu x O 4 Phys. Rev. B 60, 10580 (1999). Large and ultrafast optical nonlinearities Sr 2 CuO 3 (T Ogasawara et.a Phys. Rev. Lett. 85, 2204 (2000) ) Huge volume collapses, Ce, Pu……

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Breakdown of standard model LDA+GW program fails badly. Large metallic resistivities exceeding the Mott limit. [Anderson, Emery and Kivelson] Breakdown of the rigid band picture. Need new ways to think about the excitations. Anomalous transfer of spectral weight in photoemission and optics. [G. Sawatzki]

12. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strongly correlated systems are usually treated with model Hamiltonians In practice other methods (eg constrained LDA are used)

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strongly correlated systems are usually treated with model Hamiltonians They are hard to derive and hard to solve. In practice other methods (eg. constrained LDA are used)

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to the strong correlation problem and to the Mott transition. DMFT ideas Applications to the Mott transition problem: some insights from studies of models. Towards an electronic structure method: applications to materials: Pu………. Outlook

15. 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)

16. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insert transparency from nijmeigen About infinite dimensions, and about Greens functions.

17. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT: Effective Action point of view. Identify observable, A. Construct an exact functional of =a,  [a] which is stationary at the physical value of a. Example, density in DFT theory. (Fukuda et. al.) When a is local, it gives an exact mapping onto a local problem, defines a Weiss field. The method is useful when practical and accurate approximations to the exact functional exist. Example: LDA, GGA, in DFT. DMFT, build functionals of the LOCAL spectral function. [Density of states for adding or removing and electron] Exact functionals exist. We also have good approximations! Extension to an ab initio method.

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT References V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997). A Lichtenstein and M. Katsenelson Phys. Rev. B 57, 6884 (1988). S. Savrasov and G.Kotliar and Abrahams funcional formulation for full self consistent Nature {\bf 410}, 793(2001). Reviews: Held et.al., Psi-k Newsletter \#{\bf 56} (April 2003), p. 65 Lichtenstein Katsnelson and and Kotliar cond-mat/0211076:

19. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS How good is the LOCAL approximation?

20. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS C-DMFT: test in one dimension. (Bolech, Kancharla GK cond-mat 2002) Gap vs U, Exact solution Lieb and Wu, Ovshinikov Nc=2 CDMFT vs Nc=1

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS N vs mu in one dimension. Compare 2+8 vs exact Bethe Anzats, [M. Capone and M.Civelli]

22. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to the strong correlation problem. Essentials of DMFT Applications to the Mott transition problem: some insights from studies of models. Towards an electronic structure method: applications to materials Outlook

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The Mott transition Electronically driven MIT. Forces to face directly the localization delocalization problem. Relevant to many systems, eg V2O3 Techniques applicable to a very broad range or problems.

24. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in V 2 O 3 under pressure or chemical substitution on V-site

25. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Resistivity. Limelette et. al.

26. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS How good is the local approximation ? Single site DMFT study of the Mott transition, based on a study of the Hubbard model on frustrated lattices made several interesting qualitative predictions. New experiments and reexamination of old ones give credence to that the local picture is quite good. DMFT is a new reference frame to approach strongly correlated phenomena, and describes naturally, NON RIGID BAND picture, highly resistive states, etc….

27. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insight Phase diagram in the T, U plane of a frustrated ((the magnetic order is supressed)) correlated system at integer filling. At high temperatures, the phase diagram is generic, insensitive to microscopic details. At low temperatures, details matters.

28. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic DMFT phase diagram one band Hubbard model (half filling, semicircular DOS, partial frustration) Rozenberg et.al PRL (1995)

29. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)

30. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insight, in the strongly correlated region the one particle density of states has a three peak structure low energy quasiparticle peak plus Hubbard bands.

31. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT has bridged the gap between band theory and atomic physics. Delocalized picture, it should resemble the density of states, (perhaps with some additional shifts and satellites). Localized picture. Two peaks at the ionization and affinity energy of the atom.

32. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS One electron spectra near the Mott transition, three peak structure.

33. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS. ARPES measurements on NiS 2-x Se x Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998)

34. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS QP in V2O3 was recently found Mo et.al

35. 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…

36. 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 nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000)

37. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS. ARPES measurements on NiS 2-x Se x Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998)

38. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous metallic resistivities In the “ in between region “ anomalous resistivities are the rule rather than the exception.

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

40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity and Mott transition (Rozenberg et. Al. ) Ni Se 2-x S x Insights from DMFT: think in term of spectral functions (branch cuts) instead of well defined QP (poles )

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS More recent work, organics, Limelette et. al.(PRL 2003)

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivities when wave picture does not apply. Doped Hubbard model Title: Creator: gnuplot Preview: was not saved a preview included in it. Comment: cript printer, but not to other types of printers.

43. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative single site DMFT predictions: Optics Spectra of the strongly correlated metallic regime contains both quasiparticle-like and Hubbard band-like features. Mott transition is drive by transfer of spectral weight. Consequences for optics.

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

45. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous transfer of optical spectral weight V2O3 :M Rozenberg G. Kotliar and H. Kajuter Phys. Rev. B 54, 8452 (1996). M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

46. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous transfer of optical spectral weight, NiSeS. [Miyasaka and Takagi 2000]

47. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous transfer of spectral weight heavy fermions Rozenberg Kajueter Kotliar (1996)

48. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous transfer of optical weight [A. Damascelli D. Van der Marel ]

49. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Spectral Weight Transfer: Optics Schlesinger et.al (FeSi) PRL 71,1748, (1993) B Bucher et.al. Ce 2 Bi 4 Pt 3 PRL 72, 522 (1994), Rozenberg et.al. PRB 54, 8452, (1996). ApreciableT dependence found. Below energy

50. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT and the strong correlation anomalies: crossover from momentum space to real space picture Metals with resistivities which exceed the Mott Ioffe Reggel limit. Three peak structure of DOS Transfer of spectral weight which is non local in frequency. Dramatic failure of DFT based approximations in predicting physical properties.

51. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to the strong correlation problem. Essentials of DMFT Applications to the Mott transition problem: some insights from studies of models. Towards an electronic structure method: applications to materials: Pu, Fe, Ni, Ce, LaSrTiO3, NiO,MnO,CrO2,K3C60,2d and quasi-1d organics, magnetic semiconductors,SrRuO4,V2O3…………. Outlook

52. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Generalized phase diagram T U/W Relax Structure, bands, orbitals

53. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu in the periodic table actinides

54. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Electronic Physics of Pu

55. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DFT studies. Underestimates the volume by 35 % Predicts Pu to be magnetic. Largest quantitative failure of DFT-LDA- GA Fail to predict a stable delta phase.

56. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonon Spectra Electrons are the glue that hold the atoms together. Vibration spectra (phonons) probe the electronic structure. Phonon spectra reveals instablities, via soft modes. Phonon spectrum of Pu had not been measured until recently.

57. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003

58. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Inelastic X ray scattering. Wong et. al. Science 301, 1078 (2003).

59. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu: DMFT total energy vs Volume ( Savrasov Kotliar and Abrahams 2001)

60. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Alpha and delta Pu

61. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to the strong correlation problem. Essentials of DMFT The Mott transition problem: some insights from studies of models. Towards an electronic structure method: applications to materials: Pu Outlook

62. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS What do we want from materials theory? New concepts, qualitative ideas Understanding, explanation of existent experiments, and predictions of new ones. Quantitative capabilities with predictive power. Notoriously difficult to achieve in strongly correlated materials. DMFT is delivering on both fronts.

63. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook Local approach to strongly correlated electrons. Many extensions, make the approach suitable for getting insights and quantitative results in correlated materials.

64. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion The character of the localization delocalization in simple( Hubbard) models within DMFT is now fully understood, nice qualitative insights.  This has lead to extensions to more realistic models, and a beginning of a first principles approach to the electronic structure of correlated materials.

65. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook Systematic improvements, short range correlations, cluster methods, improved mean fields. Improved interfaces with electronic structure. Exploration of complex strongly correlated materials. Correlation effects on surfaces, large molecules, systems out of equilibrium, illumination, finite currents, aeging.

66. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Acknowledgements: Development of DMFT Collaborators: V. Anisimov,G. Biroli, R. Chitra, V. Dobrosavlevic, X. Dai, D. Fisher, A. Georges, H. Kajueter, K. Haujle, W.Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, O. Parcollet, G. Palsson, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko, I. Yang, X.Y. Zhang Support: NSF DMR 0096462 Support: Instrumentation. NSF DMR-0116068 Work on Fe and Ni: ONR4-2650 Work on Pu: DOE DE-FG02-99ER45761 and LANL subcontract No. 03737-001-02

67. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS High Performance Computing Project Department of Physics and Astronomy National Science Foundation - NSF0116068: Acquisition of a Network Cluster of Advanced Workstations for First Principles Electronic Structure Calculations of Complex Materials Department of Physics and Astronomy High Performance Computing http://beowulf.rutgers.edu

68. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS TOP 500 (ICL-UT)

69. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS TOP 500

70. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

71. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Shear anisotropy fcc Pu (GPa) C’=(C11-C12)/2 = 4.78 C44= 33.59 C44/C’ ~ 8 Largest shear anisotropy in any element! LDA Calculations (Bouchet et. al.) C’= -48

72. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dai et. al.

73. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Epsilon Plutonium.

74. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous transfer of spectral weight heavy fermions

75. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous transfer of spectral weight

76. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous transfer of spectral weigth heavy fermions

77. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS V2O3 resistivity

78. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Example: DMFT for lattice model (e.g. single band Hubbard).Muller Hartman 89, Chitra and Kotliar 99. Observable: Local Greens function G ii (  ). Exact functional  [G ii (  )  DMFT Approximation to the functional.

79. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional : effective action construction ( Chitra and GK ). Introduce local orbitals,   R (r-R)orbitals, and local GF G(R,R)(i  ) = The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for  (r) and G and performing a Legendre transformation,  (r),G(R,R)(i  )] Approximate functional using DMFT insights.