1. Electronic Structure of Strongly Correlated Materials : a DMFT Perspective Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Boston March 2002

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Introduction to strongly correlated electrons Dynamical Mean Field Theory Model Hamiltonian Studies. Universal aspects insights from DMFT System specific studies: LDA+DMFT Outlook

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strongly Correlated Materials 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.

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strongly Correlated Materials. High Temperature Superconductivity in doped filled Bucky Balls (J. H. Schon et.al Science Express 1064773 (2001)) CHBr 3 C 60 Tc=117K. 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) )

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

6. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Standard Model of Solids  Qualitative predictions: low temperature dependence of thermodynamics and transport.  Optical response, transition between the bands.  Qualitative predictions: filled bands give rise to insulting behavior. Compounds with odd number of electrons are metals.  Quantitative tools: Density Functional Theory with approximations suggested by the Kohn Sham formulation, (LDA GGA) is a successful computational tool for the total energy. Good starting point for perturbative calculation of spectra,eg. GW. Kinetic equations yield transport coefficients.

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

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott : Correlations localize the electron Low densities, electron behaves as a particle,use atomic physics, 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….) 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 Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics.

9. 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 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 b and Hubbard bands.

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Failure of the standard model Mott transition in V 2 O 3 under pressure or chemical substitution on V-site

11. 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)

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

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Failure of the standard model : Anomalous Resistivity :LiV 2 O 4 Takagi et.al. PRL 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 Strong Correlation Problem Large number of compounds (d,f,p….).Qualitative and quantitive failures of the standard model. 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 bands and atoms with open shells.

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

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

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field Theory, cavity construction Phys. Rev. B 45, 6497,1992 A. Georges G. Kotliar

19. 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)

20. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Solving the DMFT equations Wide variety of computational tools (QMC,NRG,ED….)Analytical Methods Extension to ordered states. Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Single site DMFT, functional formulation. Construct a functional of the local Greens function Expressed in terms of Weiss field (semicircularDOS) [G. Kotliar EBJB 99]

22. 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

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic DMFT phase diagram Hubbard model (partial frustration ) M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

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

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

26. 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…

27. 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 and Rozenberg Phys. Rev. Lett. 84, 5180 (2000)

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

29. 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)

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

31. 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

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

33. 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)

34. 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 weight from the coherent to the incoherent part of the spectra. Real and momentum space.

35. 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. Beyond LDA+U approach (Anisimov, Andersen and Zaanen) Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997)

36. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Combining LDA and DMFT The light, SP (or SPD) electrons are extended, well described by LDA The heavy, D (or F) electrons are localized,treat by DMFT. LDA already contains an average interaction of the heavy electrons, subtract this out by shifting the heavy level (double counting term) The U matrix can be estimated from first principles or viewed as parameters

37. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional : effective action construction ( Fukuda, Valiev and Fernando, Chitra and Kotliar ). DFT, consider the exact free energy as a functional of an external potential. Express the free energy as a functional of the density by Legendre transformation.  DFT  (r)] 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  (r),G(R,R)(i  )] A useful approximation to the exact functional can be constructed, the DMFT +LDA functional. Savrasov Kotliar and Abrahams Nature 410, 793 (2001))

38. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT Self-Consistency loop DMFT U E

39. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Case study in f electrons, Mott transition in the actinide series. B. Johanssen 1974 Smith and Kmetko Phase Diagram 1984.

40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Total energy vs Volume (Savrasov Kotliar and Abrahams Nature 410, 793 (2001))

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Small amounts of Ga stabilize the  phase

42.  Pu Problems with density functional treatements of  Pu DFT in the LDA or GGA is a well established tool for the calculation of ground state properties. Many studies (APW Freeman, Koelling 1972, ASA and FP- LMTO, Soderlind et. al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) show an equilibrium volume of the  phase  Is 35% lower than experiment an equilibrium volume of the  phase  Is 35% lower than experiment This is the largest discrepancy ever known in DFT based calculations. LSDA predicts magnetic long range order which is not observed experimentally (Solovyev et.al.) If one treats the f electrons as part of the core LDA overestimates the volume by 30% Weak correlation picture for alpha phase.

43. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu DMFT total energy vs Volume (Savrasov Kotliar and Abrahams Nature 410, 793 (2001)

44. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Lda vs Exp Spectra (Joyce et.al.)

45. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu Spectra DMFT(Savrasov) EXP (Joyce, Arko et.al)

46. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Case study Fe and Ni Archetypical itinerant ferromagnets LSDA predicts correct low T moment Band picture holds at low T Main puzzle: at high temperatures  has a Curie Weiss law with a moment much larger than the ordered moment. Magnetic anisotropy 

47. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Iron and Nickel: crossover to a real space picture at high T (Lichtenstein, Katsnelson and Kotliar Phys Rev. Lett 87, 67205, 2001 )

48. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Iron and Nickel:magnetic properties (Lichtenstein, Katsenelson,GK PRL 01)

49. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ni and Fe: theory vs exp  /   ordered moment Fe 2.5 ( theory) 2.2(expt) Ni.6 (theory).6(expt)  eff    high T moment Fe 3.1 (theory) 3.12 (expt) Ni 1.5 (theory) 1.62 (expt) Curie Temperature T c Fe 1900 ( theory) 1043(expt) Ni 700 (theory) 631 (expt)

50. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Fe and Ni Consistent picture of Fe (more localized) and Ni (more correlated) Satellite in minority band at 6 ev, 30 % reduction of bandwidth, exchange splitting reduction.3 ev Spin wave stiffness controls the effects of spatial flucuations, it is about twice as large in Ni and in Fe Mean field calculations using measured exchange constants(Kudrnovski Drachl PRB 2001) right Tc for Ni but overestimates Fe, RPA corrections reduce Tc of Ni by 10% and Tc of Fe by 50%.

51. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,T/Tc=.8) (Lichtenstein, Katsenelson,Kotliar Phys Rev. Lett 87, 67205, 2001)

52. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission and T/Tc=.8 Spin Autocorrelation: Ni (U=3, J=.9 ev)

53. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Summary Introduction to strongly correlated electrons Dynamical Mean Field Theory Model Hamiltonian Studies. Universal aspects insights from DMFT System specific studies: LDA+DMFT Outlook

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. Cellular DMFT

55. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Challenges Short Range Magnetic Correlations without magnetic order. Single Site DMFT does not capture these effects

56. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook Extensions to take into account longer range correlations and interactions [ Cellular DMFT G. Kotliar S. Savrasov G. Palsson and G. Biroli Phys. Rev. Lett. 87, 186401, 2001] Mott transition magnetic correlations and momentum space differentiation. RVB, multipatch models of transport A. Perali M. Sindel and G. Kotliar Eur. Phys. J. B 24, 487 (2001). Exploration of materials.

57. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Acknowledgements 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 Support: National Science Foundation. Work on Fe and Ni: Office of Naval Research Work on Pu: Departament of Energy and LANL.

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59. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

60. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

61. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT functional  Sum of local 2PI graphs with local U matrix and local G

62. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on LDA+DMFT Static limit of the LDA+DMFT functional, with  =  HF reduces to LDA+U Removes inconsistencies and shortcomings of this approach. DMFT retain correlations effects in the absence of orbital ordering. Only in the orbitally ordered Hartree Fock limit, the Greens function of the heavy electrons is fully coherent Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing.

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

64. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivities: Doped Hubbard Model G. Palsson 1998 Title: Creator: gnuplot Preview: was not saved a preview included in it. Comment: cript printer, but not to other types of printers. IPT NCA

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

66. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA functional Conjugate field, V KS (r)

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

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

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

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

71. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS V2O3 resistivity

72. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT Self-Consistency loop DMFT U E

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

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

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

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

77. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ni and Fe: theory vs exp  ( T=.9 Tc)/   ordered moment Fe 1.5 ( theory) 1.55 (expt) Ni.3 (theory).35 (expt)  eff    high T moment Fe 3.1 (theory) 3.12 (expt) Ni 1.5 (theory) 1.62 (expt) Curie Temperature T c Fe 1900 ( theory) 1043(expt) Ni 700 (theory) 631 (expt)

78. 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)APW methods o ASA and FP-LMTO Soderlind et. Al 1990, Kollar et.al 1997, Boettger 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.

79. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

80. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional The exact functional can be built in perturbation theory in the interaction (well defined diagrammatic rules )The functional can also be constructed expanding around the the atomic limit. No explicit expression exists. DFT is useful because good approximations to the exact density functional  DFT  (r)] exist, e.g. LDA, GGA A useful approximation to the exact functional can be constructed, the DMFT +LDA functional. Savrasov Kotliar and Abrahams Nature 410, 793 (2001))

81. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA functional Conjugate field, V KS (r)