1. Towards Realistic Electronic Structure Calculations of Correlated Materials 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. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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

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

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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).

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

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

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

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phase Diag: Ni Se 2-x S x G. Czek et. al. J. Mag. Mag. Mat. 3, 58 (1976)

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

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

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT: Methods of solution 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)]  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).

12. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic DMFT phase diagram one band Hubbard model (half filling, semicircular DOS, partial frustration) Rozenberg et.al PRL (1995)

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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

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

15. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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.

16. 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…

17. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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))

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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

19. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity and Mott transition Ni Se 2-x S x Miyasaka and Tagaki (2000)

20. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS. ARPES measurements on NiS 2-x Se x Matsuura et. Al Phys. Rev B 58 (1998) 3690

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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

22. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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

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

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

25. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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.

26. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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)

27. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Delocalization-Localization across the actinide series o f electrons in Th Pr U Np are itinerant. From Am on they are localized. Pu is at the boundary. o Pu has a simple cubic fcc structure,the  phase which is easily stabilized over a wide region in the T,p phase diagram. o The  phase is non magnetic. an equilibrium volume of the  phase  Is 35% lower than experiment o 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  phase  Is 35% lower than experiment o This is one of the largest discrepancy ever known in DFT based calculations.

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

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

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

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

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

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

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

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

36. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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.

37. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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 …..)

38. 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)

39. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Lda+dmft functional dynamical Kohn Sham field  Weiss field  DMFTfunctional

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

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT loop (in a tight binding basis, e.g. LMTO’s   U, interaction matrix 0) Guess  (r), G   (i  1) Form Vxc, Solve AIM to get  and local Greens function of heavy orbitals.  Form LMTO Matrix, overlap matrix and heavy level shift E, form G(k, i  3) Recompute the density and Weiss function G   (i  to go back to 1.

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

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