1. Dynamical Mean Field Approach to Strongly Correlated Electrons Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Field Theory and Statistical Mechanics Rome 10-15 June (2002)

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Correlated Electrons and the Mott transition problem. Dynamical Mean Field Theory. Cavity construction. Effective action construction. [G Jona-Lasinio, Nuovo Cimento 34, (1964), De Dominicis and Martin, Fukuda ] Model Hamiltonian Studies of the Mott transition in frustrated systems. Universal aspects. Application to itinerant ferromagnets: Fe,Ni. Outlook

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Standard model of solid (Sommerfeld) (Bloch )Periodic potential, waves form bands, k in Brillouin zone. (Landau) Interactions renormalize away. Justification: perturbative RG (Benfatto Gallavotti) The electron in a solid: wave picture Consequences: Maximum metallic resistivity 200  ohm cm

4. 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 spin-orbital order

5. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of the spectra from localized to itinerant Low densities. Electron as particle bound to atom. High densities. Electrons are waves spread thru the crystal. Mott transition problem: evolution between the two limits, in the open shell case. Non perturbative problem. Key to understanding many interesting solids.

6. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in V 2 O 3 under pressure or chemical substitution on V-site

7. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Failure of the Standard Model: NiSe 2-x S x Miyasaka and Takagi (2000)

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Hubbard model  U/t  Doping  or chemical potential  Frustration (t’/t)  T temperature Mott transition as a function of doping, pressure temperature etc.

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

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Missing in this limit Short Range Magnetic Correlations without magnetic order. Long wavelength modes. Trust more in frustrated situations and at high temperatures.

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT cavity construction A. Georges G. Kotliar 92 Weiss field

12. 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)

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Solving the DMFT equations Wide variety of computational tools (QMC,ED….)Analytical Methods Extension to ordered states, many models……….. Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Different Extensions Take larger clusters in the cavity construction, e.g. cellular DMFT.[Kotliar Savrasov Palsson and Biroli], DCA[Jarrell and Krishnamurthy] Take into account approximately the renormalization of the quartic coupling, e.g. extended DMFT. [Sachdev and Ye, Kajueter Kotliar, Si and Smith]

15. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Single site DMFT, functional formulation. Construct a functional of the local Greens function Expressed in terms of Weiss field (semicircularDOS) [G. Kotliar EBJB 99]

16. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS C-DMFT functional formulation. Construct a functional of the restriction of the Greens function to the cluster and its supercell translations. Sigma and G are non zero on the selected cluster and its supercell translations and are non zero otherwise. Lattice quantities are inferred or projected out from the local quantities.

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

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Results: Schematic DMFT phase diagram Hubbard model (partial frustration )

19. 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.

20. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Kuwamoto Honig and Appell PRB (1980) M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative phase diagram in the U, T,  plane, full frustration ( GK Murthy and Rozenberg 2002 ) Shaded regions :the DMFT equations have a metallic-like and an insulating-like solution).

22. 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 and Rozenberg Phys. Rev. Lett. 84, 5180 (2000). Foreshadowed by Castellani Di Castro Feinberg Ranninger (1979).

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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…

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

25. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Spectral Weight Transfer: Optics Below energy

26. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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)

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

28. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity and Mott transition Ni Se 2-x S x Insights from DMFT: think in term of spectral functions (branch cuts) instead of well defined QP (poles )

29. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights from DMFT Mott transition as a bifurcation of an effective action Important role of the incoherent part of the spectral function at finite temperature Physics is governed by the transfer of spectral weight from the coherent to the incoherent part of the spectra. Real and momentum space.

30. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Realistic Calculations of the Electronic Structure of Correlated materials Combinining DMFT with state of the art electronic structure methods to construct a first principles framework to describe complex materials. Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997) Savrasov Kotliar and Abrahams Nature 410, 793 (2001))

31. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional : effective action construction ( Chitra and GK PRB 2001 ). DFT, exact free energy as a functional of an external potential. Legendre transform to obtain a functional of the density  DFT  (r)]. [Hohenberg and Kohn, Lieb, Fukuda] Introduce local orbitals,   R (r-R)orbitals, and local GF G(R,R)(i  ) = The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing  (r),G(R,R)(i  )] A useful approximation to the exact functional can be constructed.

32. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Combining LDA and DMFT The light, SP (or SPD) electrons are extended, well described by LDA The heavy, D (or F) electrons are localized,treat by DMFT. LDA already contains an average interaction of the heavy electrons, subtract this out by shifting the heavy level (double counting term) The U matrix can be estimated from first principles or viewed as parameters

33. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT Self-Consistency loop DMFT E dc U E

34. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Case study Fe and Ni Band picture holds at low T. LSDA predicts correct low T moment At high temperatures  has a Curie Weiss law with a (fluctuating) moment larger than the T=0 ordered moment. Localization delocalization crossover as a function of T.

35. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Iron and Nickel: crossover to a real space picture at high T (Lichtenstein, Katsnelson and Kotliar Phys Rev. Lett 87, 67205, 2001 )

36. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Iron and Nickel:magnetic properties (Lichtenstein, Katsnelson,GK PRL 01)

37. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ni and Fe: theory vs exp  /   ordered moment Fe 2.5 ( theory) 2.2(expt) Ni.6 (theory).6(expt)  eff    high T moment Fe 3.1 (theory) 3.12 (expt) Ni 1.5 (theory) 1.62 (expt) Curie Temperature T c Fe 1900 ( theory) 1043(expt) Ni 700 (theory) 631 (expt)

38. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,T/Tc=.8) (Lichtenstein, Katsenelson,Kotliar Phys Rev. Lett 87, 67205, 2001)

39. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission and T/Tc=.8 Spin Autocorrelation: Ni (U=3, J=.9 ev)

40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Fe and Ni Consistent picture of Fe (more localized) and Ni (more itinerant but more correlated) Satellite in minority band at 6 ev, 30 % reduction of bandwidth, exchange splitting reduction.3 ev Spin wave stiffness controls the effects of spatial flucuations, twice as large in Ni and in Fe Cluster methods.

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook  Many open problems!  Strategy: advancing our understanding scale by scale.  New local physics in plaquettes.  Cluster methods to capture longer range magnetic correlations. New structures in k space. Cellular DMFT  Many applications to real materials.

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT functional  Sum of local 2PI graphs with local U matrix and local G

43. 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)