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Dynamical Mean Field Approach to Strongly Correlated Electrons

Dynamical Mean Field Approach to Strongly Correlated Electrons. Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University. Field Theory and Statistical Mechanics Rome 10-15 June (2002). Outline. Correlated Electrons and the Mott transition problem.

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Dynamical Mean Field Approach to Strongly Correlated Electrons

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  1. Dynamical Mean Field Approach to Strongly Correlated Electrons Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Field Theory and Statistical Mechanics Rome 10-15 June (2002)

  2. Outline • Correlated Electrons and the Mott transition problem. • Dynamical Mean Field Theory. Cavity construction. Effective action construction. [G Jona-Lasinio, Nuovo Cimento 34, (1964), De Dominicis and Martin, Fukuda ] • Model Hamiltonian Studies of the Mott transition in frustrated systems. Universal aspects. • Application to itinerant ferromagnets: Fe,Ni. • Outlook THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  3. The electron in a solid: wave picture Standard model of solid (Sommerfeld) (Bloch )Periodic potential, waves form bands , k in Brillouin zone . (Landau) Interactions renormalize away. Justification: perturbative RG (Benfatto Gallavotti) Consequences: Maximum metallic resistivity 200 mohm cm THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  4. The electron in a solid: particle picture. • Array of hydrogen atoms is insulating if a>>aB. 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 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  5. Evolution of the spectra from localized to itinerant • Low densities. Electron as particle bound to atom. • High densities. Electrons are waves spread thru the crystal. • Mott transition problem: evolution between the two limits, in the open shell case. • Non perturbative problem. • Key to understanding many interesting solids. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  6. Mott transition in V2O3 under pressure or chemical substitution on V-site THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  7. Failure of the Standard Model: NiSe2-xSx Miyasaka and Takagi (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

  10. Missing in this limit • Short Range Magnetic Correlations without magnetic order. Long wavelength modes. • Trust more in frustrated situations and at high temperatures. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  11. DMFT cavity construction A. Georges G. Kotliar 92 Weissfield THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  12. Mean-Field : Classical vs Quantum Classical case Quantum case A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

  14. Different Extensions • Take larger clusters in the cavity construction, e.g. cellular DMFT.[Kotliar Savrasov Palsson and Biroli], DCA[Jarrell and Krishnamurthy] • Take into account approximately the renormalization of the quartic coupling, e.g. extended DMFT. [Sachdev and Ye, Kajueter Kotliar, Si and Smith] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

  16. C-DMFT functional formulation. Construct a functional of the restriction of the Greens function to the cluster and its supercell translations. Sigma and G are non zero on the selected cluster and its supercell translations and are non zero otherwise. Lattice quantities are inferred or projected out from the local quantities. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

  18. Results: Schematic DMFT phase diagram Hubbard model (partial frustration) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  19. Insights from DMFT • Low temperature Ordered phases . Stability depends on chemistry and crystal structure • High temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT. Role of magnetic frustration. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  20. Kuwamoto Honig and Appell PRB (1980)M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  21. Qualitative phase diagram in the U, T , m plane,full frustration ( GK Murthy and Rozenberg 2002) • Shaded regions :the DMFT equations have a metallic-like and an insulating-like solution). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  22. 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). Foreshadowed by Castellani Di Castro Feinberg Ranninger (1979). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  23. 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… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  24. ARPES measurements on NiS2-xSexMatsuura et. al Phys. Rev B 58 (1998) 3690. Doniach and Watanabe Phys. Rev. B 57, 3829 (1998) . THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  25. Anomalous Spectral Weight Transfer: Optics Below energy THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  26. 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) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

  28. Anomalous Resistivity and Mott transition Ni Se2-x Sx Insights from DMFT: think in term of spectral functions (branch cuts) instead of well defined QP (poles ) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  29. 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. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  30. Realistic Calculationsof 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. • Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997) • Savrasov Kotliar and Abrahams Nature 410, 793 (2001)) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  31. Spectral Density Functional : effective action construction ( Chitra and GK PRB 2001). • DFT, exact free energy as a functional of an external potential. Legendre transform to obtain a functional of the density GDFT[r(r)]. [Hohenberg and Kohn, Lieb, Fukuda] • Introduce local orbitals, caR(r-R)orbitals, and local GF G(R,R)(i w) = • The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing G[r(r),G(R,R)(iw)] • A useful approximation to the exact functional can be constructed. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  32. 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 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

  34. Case study Fe and Ni • Band picture holds at low T. LSDA predicts correct low T moment • At high temperatures c has a Curie Weiss law with a (fluctuating) moment larger than the T=0 ordered moment. • Localization delocalization crossover as a function of T. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

  37. Ni and Fe: theory vs exp • m/ mB ordered moment • Fe 2.5 ( theory) 2.2(expt) • Ni .6 (theory) .6(expt) meff / mB high T moment Fe 3.1 (theory) 3.12 (expt) Ni 1.5 (theory) 1.62 (expt) Curie Temperature Tc • Fe 1900 ( theory) 1043(expt) • Ni 700 (theory) 631 (expt) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

  40. Fe and Ni • Consistent picture of Fe (more localized) and Ni (more itinerant but 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, twice as large in Ni and in Fe • Cluster methods. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  41. Outlook • Many open problems! • Strategy: advancing our understanding scale by scale. • New local physics in plaquettes. • Cluster methods to capture longer range magnetic correlations. New structures in k space. Cellular DMFT • Many applications to real materials. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

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