Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University

1. Towards Realistic Electronic Structure Calculations of CorrelatedMaterials Exhibiting a Mott Transition. Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University March Meeting of the American Physical Society Seattle 2001 Collaborators: S. Savrasov, V. Udovenko(Rutgers), R.Chitra(Jussieux) S. Lichtenstein (Nijmeigen) M. Rozenberg (UBA) E Lange (McKinsey)

2. Outline • Introduction to the correlation driven localization delocalization transition ( Mott transition). • Some lessons from very simple models DMFT study of a one band Hubbard with a semicircular density of states. • Extensions to more realistic situations . Case studies in d and f electrons • Outlook for further developments THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

3. The Mott Hubbard problem • Array of H atoms, e is localized a>>aB, extended if a<<aB e_ e_ e_ e_ Momentum space, bands • Real space, atoms High T : local moments Low T: spin orbital order THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

4. The Mott Hubbard problem Evolution of the electronic structure between the atomic limit and the band limit. Basic solid state problem. Solved by band theory when the atoms have a closed shell. Mott’s problem: Open shell situation. The “”in between regime”” is ubiquitous central them in strongly correlated systems. Stimulates the development of new electronic structure methods (LDA+DMFT). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

5. Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

6. A time-honored example: Mott transition in V2O3 under pressure or chemical substitution on V-site THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

7. Kuwamoto Honig and AppellPRB (1980) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

8. Phase Diag: Ni Se2-x SxG. Czek et. al. J. Mag. Mag. Mat. 3, 58 (1976) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

9. DMFT: Model Calculation Weissfield THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

10. Solving the DMFT equations • Wide variety of computational tools (QMC, NRG,ED….) • Analytical Methods THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

11. DMFT: Methods of solution • Iterative perturbation theory. A Georges and G Kotliar PRB 45, 6479 (1992). H Kajueter and G. Kotliar PRL (1996). S. Savrasov et.al (2001). • Projective method G Moeller et. al. PRL 74 2082 (1995). • NRG R. Bulla PRL 83, 136 (1999) • QMC M. Jarrell, PRL 69 (1992) 168, Rozenberg Zhang Kotliar PRL 69, 1236 (1992) ,A Georges and W Karuth PRL 69, 1240 (1992) M. Rozenberg PRB 55, 4855 (1987). • Prushke T. Jarrell M. and Freericks J. Adv. Phys. 44,187 (1995) • A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

12. 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

13. Mott transition in model system • The qualitative features of the Rutgers-ENS results for the Mott transition were challenged in a series of publications: S Kehrein Phys. Rev Lett. 3192 (1998),R. Noack and F. Gebhardt, Phys. Rev. Lett. 82, 1915 (1999), J. Schlipf et. al. Phys. Rev. Lett 82, 4890 (1999). • These works missed subtle aspects of the and non perturbative character of the region near the metal to insulator transition such as the singular behavior of the self energy THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

14. Landau Functional G. Kotliar EPJB (1999) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

15. Functional Approach • The Landau functional offers a direct connection to the atomic energies • Allows us to study states away from the saddle points, • All the qualitative features of the phase diagram, are simple consequences of the non analytic nature of the functional. • Mott transitions and bifurcations of the functional . THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

16. 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

17. Evolution of the Spectral Function with Temperature Anomalous transfer of spectral weight connected to the proximity to an Ising Mott endpoint (Kotliar Lange and Rozenberg PRL 84, 5180 (2000)) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

18. Insights from DMFT: think in term of spectral functions (branch cuts) instead of well defined QP (poles ) Resistivity near the metal insulator endpoint ( Rozenberg et. Al 1995) exceeds the Mott limit THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

19. Anomalous Resistivity and Mott transition Ni Se2-x Sx Miyasaka and Tagaki (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

20. ARPES measurements on NiS2-xSexMatsuura et. Al Phys. Rev B 58 (1998) 3690 . THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

21. Insights from DMFT • Low temperatures several competing phases . Their relative stability depends on chemistry and crystal structure • High temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

22. LDA+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, substract this out by shifting the heavy level (double counting term) The U matrix can be estimated from first principles of viewed as parameters THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

23. LDA+DMFT Self-Consistency loop DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

24. Realistic DMFT loop THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

25. DMFT +LDA : effective action construction (Fukuda, Valiev and Fernando ,Argaman and Makov, Chitra and GK). • Select a set of local orbitals. • Define a frequency dependent, local Greens function by projecting onto the local orbitals. • The exact free energy can be expressed as a functional of the local Greens function and of the density • The functional can be built in perturbation theory in the interaction (well defined diagrammatic rules ) • The functional can also be constructed from the atomic limit. • A useful approximation to the exact functional can be constructed, the DMFT +LDA functional. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

26. 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 full self consistent implementation (2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

27. Delocalization-Localization across the actinide series • f electrons in Th Pr U Np are itinerant . From Am on they are localized. Pu is at the boundary. • Pu has a simple cubic fcc structure,the d phase which is easily stabilized over a wide region in the T,p phase diagram. • The d phase is non magnetic. • Many LDA , GGA studies ( Soderlind et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give an equilibrium volume of the d phaseIs 35% lower than experiment • This is one of the largest discrepancy ever known in DFT based calculations. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

28. Pu: DMFT total energy vs Volume (S. Savrasov ) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

29. Lda vs Exp Spectra THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

30. Pu Spectra DMFT(Savrasov) EXP (Arko et. Al) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

31. Iron and Nickel: band picture at low T, crossover to real space picture at high T THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

32. Photoemission Spectra and Spin Autocorrelation: Fe(Lichtenstein, Katsenelson,GK cond-mat 0102297) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

33. Photoemission and Spin Autocorrelation: Ni THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

34. Iron and Nickel:mgnetic properties (Lichtenstein, Katsenelson,GK cond-mat 0102297) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

35. Ni and Fe: theory vs exp • m( T=.9 Tc)/ mB ordered moment • Fe 1.5 ( theory) 1.55 (expt) • Ni .3 (theory) .35 (expt) meff / mB high T moment Fe 3.09 (theory) 3.12 (expt) Ni 1.50 (theory) 1.62 (expt) Curie Temperature Tc • Fe 1900 ( theory) 1043(expt) • Ni 700 (theory) 631 (expt) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

36. Conclusion • The delocalization delocalization transition is a very relevant problem to the electronic structure of solids. • The character of the localization delocalization in the Hubbard model within DMFT is now fully understood. This has lead to extensions to more realistic models, and a beginning of a first principles approach interpolating between atoms and bands. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

37. Outlook • Need more experience in the estimates of the double counting term and the Coulomb interaction parameters. • Combinations of DMFT and GW. Incorporate effects of long range Coulomb interactions. E-DMFT Model calculation. Mott transtion at T=0 Is first order. R. Chitra and G. Kotliar PRL 84, 3678 (2000). • Extension to multiple site clusters (DCA M. Jarrell et. al., Two impurity DMFT Schiller, Ingersent Georges Kotliar, C-Dmft , E-DMFT …..) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

39. Lda+dmft functional • dynamical Kohn Sham field c Weiss field D DMFTfunctional THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

40. LDA+DMFT functional THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

41. LDA+DMFT loop (in a tight binding basis, e.g. LMTO’s cka U, interaction matrix • 0) Guess r (r), G0ab(i w) • 1) Form Vxc , Solve AIM to get S, and local Greens function of heavy orbitals. • 2) Form LMTO Matrix , overlap matrix and heavy level shift E , form G(k, i w) • 3) Recompute the density and Weiss function G0ab(i w) to go back to 1. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

42. LDA+DMFT • To implement step 3 we use Notice the Weiss field,E and self energies use only heavy block, while H is full. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

43. Sir Nevill Mott THE STATE UNIVERSITY OF NEW JERSEY RUTGERS