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 Physics of Condensed-Matter Systems. Princeton Center for Complex Materials Princeton. July 18-21 (2005).

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Summary 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 a problem.

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Summary Application. Pressure and temperature driven Mott transition. Universal aspects of the Mott transition in transition metal oxides. Three peak structure in the one particle density of states. QP and Hubbard bands. Mott transition is driven by transfer of spectral weight [non rigid band picture ]. Low energy quasiparticle coherence scale. Coherence-incoherent crossover. Place where gap closure occurs differs from the place where coherence disappears. Uc1 vs Uc2.

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Summary Zeroth order picture to confront with experiments in a wide range of materials. Realistic extensions. Interface with band theory. Illustrate with the physics of actinides. Plaquette as a reference frame. Cluster DMFT. Superconductivity as a result of proximity to a Mott insulator singlet state [Anderson’s RVB picture ]

5. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Some References 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). Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)

6. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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………

7. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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.

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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.

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

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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.

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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.

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

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

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strongly correlated systems are usually treated with model Hamiltonians 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)

15. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS One Particle Spectral Function and Angle Integrated Photoemission Probability of removing an electron and transfering energy  =Ei-Ef, and momentum k f(  ) A(  ) M 2 Probability of absorbing an electron and transfering energy  =Ei-Ef, and momentum k (1-f(  )) A(  ) M 2 Theory. Compute one particle greens function and use spectral function. e e

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

17. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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.

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS C. Urano et. al. PRL 85, 1052 (2000) 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.

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

20. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Correlated Materials do big things Huge resistivity changes. Mott transition. V 2 O 3. Copper Oxides..(La 2-x Ba x ) CuO 4 High Temperature Superconductivity. 150 K in the Ca 2 Ba 2 Cu 3 HgO 8. Uranium and Cerium Based Compounds. Heavy Fermion Systems,CeCu 6,m*/m=1000 (La 1-x Sr x )MnO 3 Colossal Magneto- resistance.

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strongly Correlated Materials. Large thermoelectric response in CeFe 4 P 12 (H. Sato et al. cond-mat 0010017). Ando et.al. NaCo 2-x Cu x O 4 Phys. Rev. B 60, 10580 (1999). Gigantic Volume Collapses. Lanthanide and actinides. Large and ultrafast optical nonlinearities Sr 2 CuO 3 (T Ogasawara et.a Phys. Rev. Lett. 85, 2204 (2000) ) ……………….

22. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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.

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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!

24. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT

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

26. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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:

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

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

29. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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:

30. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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)]

31. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Main Omission of this Course Techniques for solving quantitatively the Anderson Impurity Model. G[G0]See Reviews. Qualitative behavior of the solution of the Anderson Impurity Model. Kondo Physics. Extension to describe ordered phases. Superconductivity. Antiferromagnetism. Etc…

32. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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)

33. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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)

34. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS U/t=4. 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) ]

35. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on DMFT. Review of DMFT, technical tools for solving DMFT eqs. A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] CDMFT, instead of studying finite systems with open or periodic boundary conditions, study a system in a medium. Connection with DMRG, infer the density matrix by using a Gaussian anzats, and the periodicity of the system.

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

37. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strongly Correlated Electrons and DMFT. The challenge (besides learning to solve the DMFT equations more accurately or more explicitly) is to identify which strong correlation phenomena can be capture from a local DMFT perspective using sites, linkes, plaquettes, etc as reference systems, and which aspects involve non local and non Gaussian fluctuations.

38. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS V 2 O 3 under pressure or

39. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS NiSe 2-x S x

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

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pressure Driven Mott transition How does the electron go from the localized to the itinerant limit ?

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS T/W Phase diagram of a Hubbard model with partial frustration at integer filling. Thinking about the Mott transition in single site DMFT. High temperature universality M. Rozenberg et. al. Phys. Rev. Lett. 75, 105 (1995)

43. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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.

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

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

46. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative single site DMFT predictions. Spectra of the strongly correlated metallic regime contains both quasiparticle-like and Hubbard band-like features. Mott transition is drive by transfer of spectral weight.

47. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of the Spectral Function with Temperature Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000)

48. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative single site DMFT predictions: Optics Spectra of the strongly correlated metallic regime contains both quasiparticle-like and Hubbard band-like features. Mott transition is drive by transfer of spectral weight. Consequences for optics.

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

50. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous transfer of optical spectral weight, NiSeS. [Miyasaka and Takagi 2000]

51. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Spectral Weight Transfer: Optics Schlesinger et.al (FeSi) PRL 71,1748, (1993) B Bucher et.al. Ce 2 Bi 4 Pt 3 PRL 72, 522 (1994), Rozenberg et.al. PRB 54, 8452, (1996). ApreciableT dependence found. Below energy

52. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity and Mott transition Ni Se 2-x S x Crossover from Fermi liquid to bad metal to semiconductor to paramagnetic insulator.

53. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Single site DMFT and kappa organics

54. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ising critical endpoint! In V 2 O 3 P. Limelette et.al. Science 302, 89 (2003)

55. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Searching for a quasiparticle peak

56. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS. ARPES measurements on NiS 2-x Se x Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998)

57. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

58. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic DMFT phase Implications for transport.

59. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity and Mott transition Ni Se 2-x S x Crossover from Fermi liquid to bad metal to semiconductor to paramagnetic insulator.

60. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

61. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phase Diagram k Organics

62. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Transport in k organics: hysteresis. Limelette et. al.

63. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ising endpoint finally found

64. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of the Spectral Function with Temperature Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000)

65. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS V 2-x Cr x O 3

66. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ising critical endpoint! In VCr 2 O 3 Limelette et.al.

67. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion. An electronic model accounts for all the qualitative features of the finite temperature of a frustrated system at integer occupancy. The electronic degrees of freedom rather than the lattice drives the transition.

68. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion Single site DMFT describes the main features of the experiments at high temperatures using a simple model. Made non trivial predictions. Finite temperature conclusions are robust. At low temperatures clusters will bring refinements of this picture.