Strongly Correlated Electron Systems: a DMFT Perspective

1. Strongly Correlated Electron Systems: a DMFT Perspective Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Colloquium UBC September (2004)

2. Outline • Introduction to the strong correlation problem. • Essentials of DMFT • The Mott transition problem: some insights from studies of models. • Towards an electronic structure method: applications to materials: Ce, Pu • Outlook 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. 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

6. 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

7. 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 –GW 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. New reference point, to replace the Kohn Sham system. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

8. DFT+GW program has been less succesful in correlated situations. • Strong interactions localize the particles. Atoms with open shells are not easily connected to band theory. • The spectrum in this case, contain Hubbard bands which are NOT simply perturbatively connected to the Kohn Sham orbitals. • Need an alternative reference point for doing perturbation theory! Situation is worse “in between the atomic and the localized limit” • DMFT! THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

9. 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

10. 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

11. Breakdown of standard model • LDA+GW program fails badly. • Large metallic resistivities exceeding the Mott limit. [Anderson, Emery and Kivelson] • Breakdown of the rigid band picture. Need new ways to think about the excitations. • Anomalous transfer of spectral weight in photoemission and optics. [G. Sawatzki] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

12. Strongly correlated systems are usually treated with model Hamiltonians THE STATE UNIVERSITY OF NEW JERSEY RUTGERS In practice other methods (eg constrained LDA are used)

13. Strongly correlated systems are usually treated with model Hamiltonians THE STATE UNIVERSITY OF NEW JERSEY RUTGERS They are hard to derive and hard to solve. In practice other methods (eg. constrained LDA are used)

14. Outline • Introduction to the strong correlation problem and to the Mott transition. • DMFT ideas • Applications to the Mott transition problem: some insights from studies of models. • Towards an electronic structure method: applications to materials: Pu………. • Outlook THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

15. 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

16. Insert transparency from nijmeigen • About infinite dimensions, and about • Greens functions. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

17. DMFT: Effective Action point of view. • Identify observable, A. Construct an exact functional of <A>=a, G [a] which is stationary at the physical value of a. • Example, density in DFT theory. (Fukuda et. al.) • When a is local, it gives an exact mapping onto a local problem, defines a Weiss field. • The method is useful when practical and accurate approximations to the exact functional exist. Example: LDA, GGA, in DFT. • DMFT, build functionals of the LOCAL spectral function. [Density of states for adding or removing and electron] • Exact functionals exist. We also have good approximations! • Extension to an ab initio method. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

18. LDA+DMFT References • V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997). • A Lichtenstein and M. Katsenelson Phys. Rev. B 57, 6884 (1988). • S. Savrasov and G.Kotliar and Abrahams funcional formulation for full self consistent Nature {\bf 410}, 793(2001). • Reviews: Held et.al. , Psi-k Newsletter \#{\bf 56} (April 2003), p. 65 Lichtenstein Katsnelson and and Kotliar cond-mat/0211076: THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

19. How good is the LOCAL approximation? THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

20. C-DMFT: test in one dimension. (Bolech, Kancharla GK cond-mat 2002) Gap vs U, Exact solution Lieb and Wu, Ovshinikov Nc=2 CDMFT vs Nc=1 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

21. N vs mu in one dimension.Compare 2+8 vs exact Bethe Anzats, [M. Capone and M.Civelli] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

22. Outline • Introduction to the strong correlation problem. • Essentials of DMFT • Applications to the Mott transition problem: some insights from studies of models. • Towards an electronic structure method: applications to materials • Outlook THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

25. Resistivity. • Limelette et. al. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

26. How good is the local approximation ? • Single site DMFT study of the Mott transition, based on a study of the Hubbard model on frustrated lattices made several interesting qualitative predictions. • New experiments and reexamination of old ones give credence to that the local picture is quite good. • DMFT is a new reference frame to approach strongly correlated phenomena, and describes naturally , NON RIGID BAND picture, highly resistive states, etc…. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

27. 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 microscopic details. • At low temperatures, details matters. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

28. 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

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

30. 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

31. 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

32. One electron spectra near the Mott transition, three peak structure. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

33. 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

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

35. 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

36. 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

37. 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

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

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

40. 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

41. More recent work, organics, Limelette et. al.(PRL 2003) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

42. 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

43. 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

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

45. Anomalous transfer of optical spectral weight V2O3 :M Rozenberg G. Kotliar and H. Kajuter Phys. Rev. B 54, 8452 (1996). M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

48. Anomalous transfer of optical weight [A. Damascelli D. Van der Marel ] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

49. Anomalous Spectral Weight Transfer: Optics Below energy ApreciableT dependence found. Schlesinger et.al (FeSi) PRL 71 ,1748 , (1993) B Bucher et.al. Ce2Bi4Pt3PRL 72, 522 (1994), Rozenberg et.al. PRB 54, 8452, (1996). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

50. DMFT and the strong correlation anomalies: crossover from momentum space to real space picture • Metals with resistivities which exceed the Mott Ioffe Reggel limit. • Three peak structure of DOS • Transfer of spectral weight which is non local in frequency. • Dramatic failure of DFT based approximations in predicting physical properties. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS