Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University

1. Introduction to Strongly Correlated Electron Materials, Dynamical Mean Field Theory (DMFT) and its extensions. Application to the Mott Transition. Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University • Strongly Correlated Electrons: diverse examples and unifying themes. Cargese August 8-20 (2005).

2. Plan of the course. Lecture I. • Motivation: Strongly Correlated Electron Systems require a new starting point or (non-Gaussian) reference system for their description. • DMFT provides such a reference frame, mapping the full many body problem on the lattice to a much simpler system, a quantum impurity model in a self consistent medium. DMFT a first stab at correlated electron materials. Pedagogical derivations of mean field theories, Weiss theory, density functional theory, DMFT. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

3. Plan of the course. Lecture II. • Motivation: The temperature and pressure driven Mott transition. The most basic competition: kinetic energy vs Coulomb, itineracy vs localization. • Single site DMFT in action. Some results on the frustrated Hubbard model. Comparison with earlier theories. Comparison with experiments. • General lessons, and system specific extensions. [LDA+DMFT]. Mott transition across the actinide series. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

4. Plan of the course:Lecture III. • The plaquette as a reference frame. • Cluster DMFT studies of the doped Mott insulator and the problem of high temperature superconductivity. • Connection with the d-wave RVB approach and with some experiments. • Correlated superconductivity in Am ? THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

5. References http://www.physics.rutgers.edu/~kotliar/publications.html • Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, (1996). • Reviews: G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti. Submitted to RMP (2005). • The Mott transition. G. Kotliar in 50 years of Condensed Matter Physics. P. Ong and R. Bahtt editors. Princeton University Press . • Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

6. Weakly correlated electrons:band theory. • Fermi Liquid Theory. Simple conceptual picture of the ground state, excitation spectra, transport properties of many systems (simple metals, semiconductors,….). • In a certain low energy regime, adiabatic Continuity to a Reference Systen of Free Fermions with renormalized parameters. Rigid bands , optical transitions , thermodynamics, transport……… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

7. Standard Model of Solids • Qualitative predictions: low temperature dependence of thermodynamics and transport. • Optical response, transition between the bands. • Filled bands give rise to insulting behavior. Compounds with odd number of electrons are metals. • Kinetic Boltzman equations for QP. scattering off phonons or disorder, ee. int etc. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

8. Quantitative Tools of Electronic Structure. Kohn Sham reference system Static mean field theory. Derived from a functional which gives the total energy. Excellent starting point for computation of spectra in perturbation theory in screened Coulomb interaction GW. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

9. - [ - ] = [ - ]-1 = G = W GW approximation (Hedin )

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

11. LDA+GW: semiconducting gaps THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

12. DFT +GW approaches do not always work. …. Solid State Physics Chapter 2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

13. 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 Anderson superexchange. spin-orbital order ,RVB. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

14. 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

15. 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

16. Two paths for calculation of electronic structure of strongly correlated materials Crystal structure +Atomic positions Model Hamiltonian Correlation Functions Total Energies etc. DMFT ideas can be used in both cases. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

17. Model Hamiltonians: Hubbard model • U/t • Doping d or chemical potential • Frustration (t’/t) • T temperature THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

18. Strongly correlated systems are usually treated with model Hamiltonians THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conceptually one wants to restrict the number of degrees of freedom by eliminating high energy degrees of freedom. In practice other methods (eg constrained LDA , GW, etc. are used)

19. 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 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

20. Photoemission and the Theory of Electronic Structure Local Spectral Function Limiting case itinerant electrons Limiting case localized electrons Hubbard bands THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

21. Strong Correlation effects appear in 3d- 4f (and sometimes 5f) systems. Because their wave functions are more localized. Many compounds. Also p electron in organic materials with large volumes can be strongly correlated. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

22. Breakdown of the Standard Model. Strong Correlation Anomalies cannot be understood within the Breakdown of standard model of solids. Metallic “resistivities beyond the Mott limit. C. Urano et. al. PRL 85, 1052 (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

23. Optical conductivity of a periodic Anderson model with two electrons per site,U=3, V=.25,t=1,T=.001,.005,.001,.02,.03, weak disorder G =.05 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

24. Failure of the StandardModel: Anomalous Spectral Weight Transfer as a function of T. Optical Conductivity Schlesinger et.al (1993) Neff depends on T THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

25. Non local transfer of spectral weight in photoemission. For a review Damascelli et. al. RMP. Figure from Norman et. al. cond-mat/0507031 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

26. Correlated Materials do big things • Huge resistivity changes. Mott transition. 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

27. 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). • Gigantic Volume Collapses. Lanthanide and actinides. • Large and ultrafast optical nonlinearities Sr2CuO3 (T Ogasawara et.a Phys. Rev. Lett. 85, 2204 (2000) ) • ………………. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

28. 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 (say DFT-GW) in predicting physical properties. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

29. Basic competition between kinetic energy and Coulomb interactions. • One needs a tool that treats quasiparticle bands and Hubbard bands on the same footing to contain the band and atomic limit. • The approach should allow to incorporate material specific information. • When the neither the band or the atomic description applies, a new reference point for thinking about correlated electrons is needed. • DMFT! THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

30. Philosophy of the Approach • Study correlated materials by using a small number of correlated sites in a medium as a reference frame. [one site, a link, a plaquette etc…] • Goal is to understand what aspects of the experimental data can be understood from this very simple framework. After this is done, we can see whether if what is left out, longer wavelenght non Gaussian physics is important. How to incorporate that, is one of the greatest theoretical challenges in the field. A lot of the physics of correlated materials can be understood within DMFT! THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

31. DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

32. Limit of large lattice coordination Metzner Vollhardt, 89 Neglect k dependence of self energy Muller-Hartmann 89 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

33. DMFT mapping (Georges Kotliar 1992) Notice that if the self energy is local it is the self energy of an Anderson impurity model. Determine the bath of the impurity model from: THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

34. Single site DMFT cavity construction: A. Georges, G. Kotliar, PRB, (1992)] Weissfield Semicircular density of states. Behte lattice. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

35. DMFT mapping (Georges Kotliar 1992) Notice that if the self energy is local it is the self energy of an Anderson impurity model. Determine the bath of the impurity model from: THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

36. Technical Intermission : Construction on Mean Field Theories. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

37. Effective Action point of view. • Identify observable, A. Construct a free energy functional of <A>=a, G [a] which is stationary at the physical value of a. • Example, density in DFT theory. (Fukuda et. al.). • DMFT (R. Chitra and G.K (2000) (2001). • H=H0+l H1. G [a,J0]=F0[J0 ]–a J0 _ + Ghxc [a] • Functional of two variables, a ,J0. • H0 +A J0 Reference system to think about H. • J0 [a] Is the functional of a with the property <A>0 =a < >0 computed with H0+A J0 • Many choices for H0 and for A THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

38. Constructing mean field functionals • Extremize a to get G[J0]= exta G [a,J0]. Functional of Weiss field J0 only. • While the focus is on observable a, for example spin, the formalism leads to approximations to correlation functions. Orenstein-Zernicke form. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

39. Mean-Field Classical Systems. Cavity construction and functional approach. A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

40. DFT: effective action construction THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

41. DFT: Kohn Sham formulation = THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

42. 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

43. 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

44. Ex: Baym Kadanoff functional,a= G, H0 = free electrons J0[a]=S Viewing it as a functional of J0, Self Energy functional(Potthoff) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

45. DMFT as an approximation to the Baym Kadanoff functional THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

46. CDMFT and NCS as truncations of the Baym Kadanoff functional THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

47. Medium of free electrons : impurity model. Solve for the medium using Self Consistency G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

48. 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) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

49. Testing CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) ) with two sites in the Hubbard model in one dimension V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][[M.Capone M.Civelli V Kancharla C.Castellani and GK PR B 69,195105 (2004) ] U/t=4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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