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Zacatecas Mexico PASSI School . Montauk June (2006).

Introduction to Dynamical Mean Field Theory (DMFT) and its Applications to the Electronic Structure of Correlated Materials. G.Kotliar Physics Department Center for Materials Theory Rutgers University. Zacatecas Mexico PASSI School . Montauk June (2006).

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Zacatecas Mexico PASSI School . Montauk June (2006).

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  1. Introduction to Dynamical Mean Field Theory (DMFT) and its Applications to the Electronic Structure of Correlated Materials G.Kotliar Physics Department Center for Materials Theory Rutgers University. Zacatecas Mexico PASSI School . Montauk June (2006). Collaborators: K. Haule (Rutgers), C. Marianetti (Rutgers ) S. Savrasov (UC Davis)

  2. References • Electronic structure calculations with dynamical mean-field theory: G. Kotliar, S. Savrasov, K. Haule, V. Oudovenko, O. Parcollet, and C. Marianetti, Rev. of Mod. Phys. 78, 000865 (2006).Dynamical Mean Field Theory of Strongly Correlation Fermion Systems and the Limit of Infinite Dimensions:  A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. of Mod. Phys. 68, 13-125 (1996). • Electronic Structure of Strongly Correlated Materials: Insights from Dynamical Mean Field Theory: Gabriel Kotliar and Dieter Vollhardt, Physics Today 57, 53 (2004).

  3. What is a strongly correlated material ?

  4. Electrons in a Solid:the Standard Model Band Theory: electrons as waves. Landau Fermi Liquid Theory. n band index, e.g. s, p, d,,f Rigid bands , optical transitions , thermodynamics, transport……… • Quantitative Tools. Density Functional Theory • Kohn Sham (1964) Static Mean Field Theory. Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965) 2

  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. Kohn Sham reference system Excellent starting point for computation of spectra in perturbation theory in screened Coulomb interaction GW.

  7. - [ - ] = [ - ]-1 = G = W GW approximation (Hedin )

  8. Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965) Self Energy VanShilfgaarde (2005) 3

  9. Strong Correlation Problem:where the standard model fails • Fermi Liquid Theory works but parameters can’t be computed in perturbation theory. • Fermi Liquid Theory does NOT work . Need new concepts to replace of rigid bands ! • Partially filled d and f shells. Competition between kinetic and Coulomb interactions. • Breakdown of the wave picture. Need to incorporate a real space perspective (Mott). • Non perturbative problem. 4

  10. 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 (fully localized or fully itinerant). • Situation realized by applying a control parameters, e.g. pressure. Metal to Insulator Transition. • Some materials have several species of electrons, some localized (f ‘s d’s ) some itinerant (sp, spd) . OSMT. Heavy Fermions. • Introducing carries (electrons or holes) to a Mott insulator. Doping Driven Mott transition.

  11. Why is it worthwhile to study correlated electron materials ?

  12. 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.

  13. Strongly correlated systems • Copper Oxides. High Temperature Superconductivity. • Cobaltates Anomalous thermoelectricity. • Manganites . Colossal magnetoresistance. • Heavy Fermions. Huge quasiparticle masses. • 2d Electron gases. Metal to insulator transitions. • Lanthanides, Transition Metal Oxides, Multiferroics……………….. 5

  14. Basic Questions • How does the electron go from being localized to itinerant. • How do the physical properties evolve. • How to bridge between the microscopic information (atomic positions) and experimental measurements. • New concepts, new techniques

  15. How do we probe SCES experimentally ?

  16. One Particle Spectral Function and Angle Integrated Photoemission e • Probability of removing an electron and transfering energy w=Ei-Ef, and momentum k f(w) A(w, K) M2 • Probability of absorbing an electron and transfering energy w=Ei-Ef, and momentum k (1-f(w)) A(w K ) M2 • Theory. Compute one particle greens function and use spectral function. n n e

  17. Spectral Function Photoemission and correlations e • Probability of removing an electron and transfering energy w=Ei-Ef, and momentum k f(w) A(w, K) M2 • Weak Correlation • Strong Correlation n n Angle integrated spectral function 8

  18. Strong Correlation Problem:where the standard model fails • Fermi Liquid Theory works but parameters can’t be computed in perturbation theory. • Fermi Liquid Theory does NOT work . Need new concepts to replace of rigid bands ! • Partially filled d and f shells. Competition between kinetic and Coulomb interactions. • Breakdown of the wave picture. Need to incorporate a real space perspective (Mott). • Non perturbative problem. 4

  19. How do we approach the problem of strongly correlated electron stystems ?

  20. Two roads for ab-initio calculation of electronic structure of strongly correlated materials Crystal structure +Atomic positions Model Hamiltonian Correlation Functions Total Energies etc.

  21. Strongly correlated systems are usually treated with model Hamiltonians • Tight binding form. Eliminate the “irrelevant” high energy degrees of freedom Add effective Coulomb interaction terms.

  22. One Band Hubbard model • U/t • Doping d or chemical potential • Frustration (t’/t) • T temperature Mott transition as a function of doping, pressure temperature etc.

  23. How do we reduce the many body problem to something tractable ?

  24. DMFT Cavity Construction. Happy marriage of atomic and band physics. Extremize a functional of the local spectra. Local self energy. Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004). G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti Rev. Mod. Phys. 78, 865 (2006) . G. Kotliar and D . Vollhardt Physics 53 Today (2004)

  25. Mean-Field : Classical vs Quantum Classical case Quantum case A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497

  26. Single site DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)] Weissfield

  27. Extension to clusters. Cellular DMFT. C-DMFT. G. Kotliar,S.Y. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) tˆ(K) is the hopping expressed in the superlattice notations. • Other cluster extensions (DCA, nested cluster schemes, PCMDFT ), causality issues, O. Parcollet, G. Biroli and GK cond-matt 0307587 (2003)

  28. What is the structure of the DMFT problem ? • Embedding and truncation

  29. 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)]

  30. How do we generalize this construction realistic systems ?

  31. More general DMFT loop

  32. Dynamical Mean Field Theory. Cavity Construction.A. Georges and G. Kotliar PRB 45, 6479 (1992). A(w) 10

  33. A(w) A. Georges, G. Kotliar (1992) 11

  34. Dynamical Mean Field Theory • Weiss field is a function. Multiple scales in strongly correlated materials. • Exact in the limit of large coordination (Metzner and Vollhardt 89) , kinetic and interaction energy compete on equal footing. • Immediate extension to real materials DFT+DMFT 12

  35. Evolution of the DOS. Theory and experiments 13

  36. DMFT Qualitative Phase diagram of a frustrated Hubbard model at integer filling T/W 14

  37. T=170 T=300 Interaction with Experiments. Photoemission Three peak strucure. V2O3:Anomalous transfer of spectral weight M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995) 15

  38. Photoemission measurements and Theory V2O3 Mo, Denlinger, Kim, Park, Allen, Sekiyama, Yamasaki, Kadono, Suga, Saitoh, Muro, Metcalf, Keller, Held, Eyert, Anisimov, Vollhardt PRL . (2003) . NiSxSe1-xMatsuura Watanabe Kim Doniach Shen Thio Bennett (1998) Poteryaev et.al. (to be published) 16

  39. How do we solve the impurity model ?

  40. Methods of solution : some examples • Iterative perturbation theory. A Georges and G Kotliar PRB 45, 6479 (1992). H Kajueter and G. Kotliar PRL (1996). Interpolative schemes (Oudovenko et.al.) • Exact diag schemes Rozenberg et. al. PRL 72, 2761 (1994)Krauth and Caffarel. PRL 72, 1545 (1994) • Projective method G Moeller et. al. PRL 74 2082 (1995). • NRG R. Bulla PRL 83, 136 (1999)

  41. QMC M. Jarrell, PRL 69 (1992) 168, Rozenberg Zhang Kotliar PRL 69, 1236 (1992) ,A Georges and W Krauth PRL 69, 1240 (1992) M. Rozenberg PRB 55, 4855 (1987). • NCA Prushke et. al. (1993) . SUNCA K. Haule (2003). • Analytic approaches, slave bosons. • Analytic treatment near special points.

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