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Applications of DMFT to correlated electrons.

Applications of DMFT to correlated electrons. G. Kotliar Physics Department and Center for Materials Theory Rutgers University. Outline. Introduction to the strong correlation problem. Essentials of DMFT Applications to the Mott transition problem: some insights from studies of models.

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Applications of DMFT to correlated electrons.

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  1. Applications of DMFT to correlated electrons. G. Kotliar Physics Department and Center for Materials Theory Rutgers University

  2. 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 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  3. The electron in a solid: wave picture Momentum Space , bands, k in Brillouin zone is good quantum number. Maximum metallic resistivity 200 mohm cm Landau Fermi liquid theory interactions renormalize away at low energy, simple band picture in effective field holds. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  4. Standard Model of Solids • Qualitative predictions: low temperature dependence of thermodynamics and transport. • Optical response, transitions between 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. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  5. Success story : Density Functional Linear Response Tremendous progress in ab initio modelling of lattice dynamics & electron-phonon interactions has been achieved (Review: Baroni et.al, Rev. Mod. Phys, 73, 515, 2001) (Savrasov, PRB 1996)

  6. The electron in a solid: particle picture. • NiO, MnO, …Array of 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

  7. 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. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  8. 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 bands and Hubbard bands. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  9. Strong correlation anomalies • Metals with resistivities which exceed the Mott Ioffe Reggel limit. • Transfer of spectral weight which is non local in frequency. • Dramatic failure of DFT based approximations in predicting physical properties. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  10. Correlated Materials do big things • Huge resistivity changes V2O3. • Copper Oxides. .(La2-x Bax) CuO4 High Temperature Superconductivity.150 K in the Ca2Ba2Cu3HgO8 . • Uranium and Cerium Based Compounds. Heavy Fermion Systems,CeCu6,m*/m=1000 • (La1-xSrx)MnO3 Colossal Magneto-resistance. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  11. Strongly Correlated Materials. • Large thermoelectric response in CeFe4 P12 (H. Sato et al. cond-mat 0010017). Ando et.al. NaCo2-xCuxO4 Phys. Rev. B 60, 10580 (1999). • Huge volume collapses, Ce, Pu…… • Large and ultrafast optical nonlinearities Sr2CuO3 (T Ogasawara et.a Phys. Rev. Lett. 85, 2204 (2000) ) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

  14. Universal and non universal features. Top to bottom approach to correlated materials. • Some aspects at high temperatures, depend weakly on the material (and on the model). • Low temperature phase diagram, is very sensitive to details, in experiment (and in the theory). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  15. 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

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

  17. Phase Diagrams :V2O3, Ni Se2-x SxMc Whan et. Al 1971,. Czek et. al. J. Mag. Mag. Mat. 3, 58 (1976), THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  18. Outline • Introduction to the strong correlation problem and to the Mott transition. • Summary of the essential concepts 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, LaSrTiO3, NiO,…………. • Outlook THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  19. 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

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

  21. DMFT: Effective Action point of view.R. Chitra and G. Kotliar Phys Rev. B.(2000). • Identify observable, A. Construct an exact functional of <A>=a, G [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. • It is useful to introduce a Lagrange multiplier l conjugate to a, G [a, l ]. • It gives as a byproduct a additional lattice information. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  22. Example: DMFT for lattice model (e.g. single band Hubbard). • Observable: Local Greens function Gii (w). • Exact functional G [Gii (w) ]. • DMFT Approximation to the functional. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  23. Extensions of DMFT. • Renormalizing the quartic term in the local impurity action. EDMFT. • Taking several sites (clusters) as local entity. CDMFT • Combining DMFT with other methods. LDA+DMFT, GWU. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  24. 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, LaSrTiO3, NiO…………. • Outlook THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

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

  28. 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) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  29. 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

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

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

  32. 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

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

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

  35. Anomalous Spectral Weight Transfer: Optics Below energy ApreciableT dependence found. Schlesinger et.al (FeSi) PRL 71 ,1748 , (1993) B Bucher et.al. Ce2Bi4Pt3PRL 72, 522 (1994), Rozenberg et.al. PRB 54, 8452, (1996). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  36. 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

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

  38. 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

  39. Qualitative phase diagram in the U, T , m plane (two band Kotliar Murthy Rozenberg PRL. Phys. Rev. Lett. 89, 046401 (2002) • Coexistence regions between localized and delocalized spectral functions. • k diverges at generic Mott endpoints THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  41. Ultrasound study of Fournier et. al. (2002) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  42. Minimum of the melting point • Divergence of the compressibility at the Mott transition endpoint. • Rapid variation of the density of the solid as a function of pressure, in the localization delocalization crossover region. • Slow variation of the volume as a function of pressure in the liquid phase • Elastic anomalies, more pronounced with orbital degeneracy. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  43. Minimum in melting curve and divergence of the compressibility at the Mott endpoint THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  44. Cerium THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

  47. Conclusions: generic aspects • Three peak structure, quasiparticles and Hubbard bands. • Non local transfer of spectral weight. • Large resistivities. • Finite temperature divergence of the compressibility. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  48. 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 pictures are needed as synthesized in DMFT. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  49. 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 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  50. Interface DMFT with electronic structure. • Derive model Hamiltonians, solve by DMFT (or cluster extensions). Total energy? • Full many body aproach, treat light electrons by GW or screened HF, heavy electrons by DMFT . Combining EDMFT with GW. Ping Sun and Phys. Rev. B 66, 085120 (2002). G.  Kotliar and S. Savrasov in New Theoretical Approaches to Strongly Correlated Systems, A. M. Tsvelik Ed. 2001 Kluwer Academic Publishers. 259-301 . cond-mat/0208241 • Treat correlated electrons with DMFT and light electrons with DFT (LDA, GGA +DMFT) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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