Dynamical Mean Field Theory of the Mott Transition

1. Dynamical Mean Field Theory of the Mott Transition Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Jerusalem Winter School January 2002

2. OUTLINE OF THE COURSE • Motivation . Electronic structure of correlated materials, limiting cases and open problems. The standard model of solids and its failures. • Introduction to the Dynamical Mean Field Theory (DMFT). Cavity construction. Statistical Mechanical Analogies. Lattice Models and Quantum Impurity models. Functional derivation. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

3. Outline • The limit of large lattice coordination. Ordered phases. Correlation functions. • Techniques for solving the Dynamical Mean Field Equations. [ Trieste School June 17-22 2002] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

4. Outline • The Mott transition. Early ideas. Brinkman Rice. Hubbard. Slater. • Analysis of the DMFT equations: existence of a Mott transition. • The Mott transition within DMFT. Overview of some important results of DMFT studies of the Hubbard Model. Electronic Structure of Correlated Materials. Canonical Phase diagram of a fully frustrated Hubbard model. Universal and non universal aspects of the physics of strongly correlated materials. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

5. Outline • Analysis of the DMFT equations. Existence of a Mott transition. Analysis from large U and small U. • The destruction of the metallic phase. Landau analysis. Uc1 . Uc2. • The Mott transition endpoint. • A new look at experiments. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

6. Outline • The electronic structure of real materials. Examples of problems where DMFT gives new insights, and quantitative understanding: itinerant ferromagnetism, Fe, Ni. Volume collapse transitions, actinide physics. Doping driven Mott transition titanites. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

7. Outline • New directions, beyond single site DMFT. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

8. Realistic Theories of Correlated Materials ITP, Santa-Barbara workshop July 29 – December 16 (2002) O.K. Andesen, A. Georges, G. Kotliar, and A. Lichtenstein Contact: kotliar@physics.rutgers.edu Conference: November 25-29, (2002)

9. The promise of Strongly Correlated Materials • Copper Oxides. High Temperature Superconductivity. • Uranium and Cerium Based Compounds. Heavy Fermion Systems. • (LaSr)MnO3 Colossal Magnetoresistence. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

10. The Promise of Strongly Correlated Materials. • High Temperature Superconductivity in doped filled Bucky Balls (B. Battlog et.al Science) • Thermoelectric response in CeFe4 P12 (H. Sato et al. cond-mat 0010017). Large Ultrafast Optical Nonlinearities Sr2CuO3 (T Ogasawara et.al cond-mat 000286) • Theory will play an important role in optimizing their physical properties. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

11. How to think about the electron in a solid? Drude Sommerfeld Bloch, Periodic potential Bands, k in Brillouin zone Maximum metallic resistivity 200 mohm cm THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

12. Standard Model High densities, electron as a wave, band theory, k-space Landau: Interactions Renormalize Away One particle excitations: quasi-particle bands Density Functional Theory in Kohn Sham Formulation, successful computational tool for total energy, and starting point For perturbative calculation of spectra, Si Au, Li, Na …………………… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

13. Hall Coefficient Resistivity Thermopower Specific Heat Susceptibility Standard Model : Metals Predicts low temperature dependence of thermodynamics and transport THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

14. Quantitative Tools : Density Functional Theory with approximations suggested by the Kohn Sham formulation, (LDA GGA) is a successful computational tool for the total energy, and a good starting point for perturbative calculation of spectra, GW, transport.…………………… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

15. Mott : correlations localize the electron • Array of hydrogen atoms is insulating if a>>aB e_ e_ e_ e_ • Superexchange Think in real space , atoms High T : local moments Low T: spin orbital order THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

16. 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….) Rich structure of Magnetic and Orbital Ordering at low T Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

17. Localization vs Delocalization Strong Correlation Problem • A large number of compounds with electrons which are not close to the well understood limits (localized or itinerant). • These systems display anomalous behavior (departure from the standard model of solids). • Neither LDA or LDA+U or Hartree Fock works well • Dynamical Mean Field Theory: Simplest approach to the electronic structure, which interpolates correctly between atoms and bands THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

18. Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

20. Failure of the standard model : AnomalousResistivity:LiV2O4 Takagi et.al. PRL 2000 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

21. Failure of the StandardModel: Anomalous Spectral Weight Transfer Optical Conductivity of FeSi for T=,20,20,250 200 and 250 K from Schlesinger et.al (1993) Neff depends on T THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

22. Strong Correlation Problem • Large number of compounds (d,f,p….). Departure from the standard model. • Hamiltonian is known. Identify the relevant degrees of freedom at a given scale. • 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 that bands and atoms THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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24. 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

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

26. Mean-Field : Classical THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

27. DMFT Impurity cavity construction: [A. Georges, G. Kotliar, PRB, (1992)] Weissfield THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

28. Comments on DMFT • Exact in both atomic and band limits • Weiss field is a function • Multiple energy scales in a correlated electron problem, non linear coupling between them. • Frezes spatial fluctuations but treats quantum fluctuations exactly, local view of the quantum many body problem. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

29. Example: semicircular DOS THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

30. DMFT Impurity cavity construction: [A. Georges, G. Kotliar, PRB, (1992)] Weissfield THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

31. Solving the DMFT equations • Wide variety of computational tools (QMC, NRG,ED….) • Analytical Methods THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

33. Single site DMFT, functional formulation • Express in terms of Weiss field (semicircularDOS) • The Mott transition as bifurcation point in functionals oG[G] or F[D], (G. Kotliar EPJB 99) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

34. DMFT for lattice hamiltonians k independent S k dependent G, Local Approximation Treglia et. al 1980 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

35. How to compute S ? View locally the lattice problem as a (multiorbital) Anderson impurity model The local site is now embedded in a medium characterized by THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

36. How to determine the medium • Use the impurity model to compute S and the impurity local Greens function. Require that impurity local Greens function equal to the lattice local Greens function. Weiss field THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

37. Response functions THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

38. Evaluation of the Free energy. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

39. Solving the DMFT equations • Wide variety of computational tools (QMC, NRG,ED….) • Analytical Methods THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

40. Review of DMFT, technical toolsfor solving DMFT eqs.., applications, references…… • A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

42. Mott transition: Early ideas. Half filling. • Evolution of the one electron spectra [physical quantity measured in photoemission and BIS] as a function of control parameters. ( U/t, pressure, temperature ) • Hubbard, begin in paramagnetic insulator. As U/t is reduced Hubbard bands merge. Gap closure. Mathematical description, closure of equations of motion, starting from atoms (I.e. large U). Incoherent motion, no fermi surface. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

43. Mott transition: early ideas. • Brinkman and Rice. Gutzwiller. Begin in paramagnetic metallic state, as U/t approaches a critical value the effective mass diverges. Luttinger fermi surface. Mathematical description, variational wave function, slave bosons, quantum coherence and double occupancy. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

44. Slave bosons: mean field +fluctuations • Fluctuations of the slave bosons around the saddle point gives rise to Hubbard bands. • Starting from the insulating side, in a paramagnetic state, the gap closes at the same U, where Z vanishes. • No satisfactory treatement of finite temperature properties. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

45. Mott vs Slater • Mott: insulators in the absence of magnetic long range order. e.g. Vanadium Oxide Nickel Oxide. Mott transition in the paramagnetic state . • Slater: insulating behavior as a consequence of antiferromagnetic long range order. Double the unit cell to convert a Mott insulator into a band insulator. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

46. A time-honored example: Mott transition in V2O3 under pressure or chemical substitution on V-site THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

47. Local view of the spectral function Partition function of the Anderson impurity model : gas of kinks [Anderson and Yuval] Metallic state, proliferation of kinks. Insulating state. Kinks are confined. Insulating state Metallic state, THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

48. Local view of the spectral function. • Consistent treatement of quasiparticles and collective modes. • Kinky paths, with may spin fluctuations: low energy resonance [Abrikosov Suhl Resonance] • Confined kinks, straight paths, Hubbard bands. [control the insulator partition function] • Strongly correlated metal has both. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

50. Destruction of the metal The gap is well formed at Uc2, when the metal is destroyed. Hubbard bands are well formed in the metal. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS