1. Strongly Correlated Electron Systems: a DMFT Perspective Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University

2. REVIEW OF SOLID STATE THEORY. • Chapter 1. The Standard Model of Solids. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

3. The electron in a solid: wave picture Momentum Space (Sommerfeld) Maximum metallic resistivity 200 mohm cm Standard model of solids Periodic potential, waves form bands , k in Brillouin zone Landau: Interactions renormalize away THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

4. Standard Model of Solids RIGID BAND PICTURE. Optical response, transitions between bands. Quantitative tools: DFT, LDA, GGA, total energies,good starting point for spectra, GW,and transport THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

5. Density functional and Kohn Sham reference system Kohn Sham spectra, proved to be an excelent starting point for doing perturbatio theory in screened Coulomb interactions GW. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

6. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

8. Solid State Physics • Chapter 2 . Mott insulators. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

9. 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

10. Mott : Correlations localize the electron 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….) Low densities, electron behaves as a particle,use atomic physics, work in real space. 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

11. Solid State Physics • Chapter 3, strongly correlated electrons. • Status: unfinished. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

12. 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 –GW or LDA+U or Hartree Fock work well. Need approach which interpolates correctly between atoms and bands. Treats QP bands and Hubbard bands. New reference point, to replace the Kohn Sham system. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

13. Failure of the standard model • DMFT is a new reference frame to approach strongly correlated phenomena, and describes naturally , NON RIGID BAND picture, highly resistive states, treats quasiparticle excitations and Hubbard bands on the same footing.. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

14. Correlated Materials do big things • Mott transition.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

15. Pressure Driven Mott transition THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

16. V2O3 resistivity THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

17. Cuprate Superconductors THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

18. Correlated Electron Materials are based on different physical principles outside the “standard model”, exciting perspectives for technological applications (e.g. high Tc). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

19. 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). • Large and ultrafast optical nonlinearities Sr2CuO3 (T Ogasawara et.a Phys. Rev. Lett. 85, 2204 (2000) ) • Huge volume collapses, Ce, Pu. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

20. Large thermoelectric power in a metal with a large number of carriers NaCo2O4 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

21. Large and ultrafast optical nonlinearities Sr2CuO3 (T Ogasawara et.a Phys. Rev. Lett. 85, 2204 (2000) ) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

22. More examples • LiCoO2 • Used in batteries, laptops, cell phones THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

23. Breakdown of standard model • Many Qualitative Failures • Large metallic resistivities exceeding the Mott limit. [Anderson, Emery and Kivelson] • Breakdown of the rigid band picture. • Anomalous transfer of spectral weight in photoemission and optics. [G. Sawatzki] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

26. Breakdown of the standard model :Anomalous transfer of optical weight [D. Van der Marel group ] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

27. Breakdown of the Standard Model • The LDA+GW program fails badly, • Qualitatively incorrect predictions. • Incorrect phase diagrams. • Physical Reason: The one electron spectra, contains both Hubbard Bands and Quasiparticle featurs. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

28. Basic Difficulties • Lack of a small parameter. Kinetic energy is comparable to Coulomb energies. • Relevant degrees of freedom change their form in different energy scales, challenge for traditional RG methods. • WANTED: a simple picture of the physical phenomena, and the physics underlying a given material. • WANTED: a computational tool to replace LDA+GW THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

29. Breakthru: Development of Dynamical Mean Field Theory. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

30. Dynamical Mean Field Theory • Basic idea: reduce the quantum many body problem to a one site or a cluster of sites, in a medium of non interacting electrons obeying a self consistency condition. • Basic idea: instead of using functionals of the density, use more sensitive functionals of the one electron spectral function. [density of states for adding or removing particles in a solid, measured in photoemission] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

33. Pressure Driven Mott transition THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

34. Insight • Phase diagram in the T, U plane of a frustrated ((the magnetic order is supressed)) correlated system at integer filling. • At high temperatures, the phase diagram is generic, insensitive to microscopid details. • At low temperatures, every detail matters. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

35. 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

36. Pressure Driven Mott transition THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

37. Insight, in the strongly correlated region the one particle density of states has a three peak structureLow energy Quasiparticle Peak plus Hubbard bands. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

38. DMFT has bridged the gap between band theory and atomic physics. • Delocalized picture, it should resemble the density of states, (perhaps with some additional shifts and satellites). • Localized picture. Two peaks at the ionization and affinity energy of the atom. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

39. One electron spectra near the Mott transition. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

40. 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

41. 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) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

42. ARPES measurements on NiS2-xSexMatsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998) . THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

43. QP in V2O3 was recently found Mo et.al THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

44. Anomalous metallic resistivities • In the “ in between region “ anomalous resistivities are the rule rather than the exception. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

46. Anomalous Resistivity and Mott transition (Rozenberg et. Al. ) 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

47. More recent work, organics, Limelette et. al. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

48. Anomalous Resistivities when wave picture does not apply. Doped Hubbard model Title: gnuplot Creator: Preview: was not saved a preview included in it. Comment: cript printer, but not to other types of printers. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

49. 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. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

50. Anomalous transfer of spectral weight heavy fermions Rozenberg Kajueter Kotliar (1996) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS