Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University

1. Spectral Density Functional: a first principles approach to the electronic structure of correlated solids Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University 2001 JRCAT-CERC Workshop on Phase Control on Correlated Electron Systems

2. Outline • Motivation. Some universal aspects of simple DMFT the Mott transition endpoint in frustrated systems. • Non universal physics requires detailed material modeling. Combining DMFT and band structure a new functional for electronic structure calculations (S. Savrasov and GK) • Results: d electrons Fe and Ni. (Lichtenstein, Katsenelson and GK, PRL in press) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

3. Outline • Results: f electrons delta Pu ( S. Savrasov G. K and E. Abrahams,Nature (2001)) • Conclusions: further extensions the approach. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

4. Importance of Mott phenomena 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. Some unorthodox examples Fe, Ni, Pu. Solution of this problem and advances in electronic structure theory (LDA +DMFT) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

7. Mott transition in layered organic conductors Ito et al. (1986) Kanoda (1987) Lefebvre et al. (2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

8. Theoretical Approach to the Mott endpoint. • DMFT.Mean field approach to quantum many body systems, constructing equivalent impurity models embedded in a bath to be determined self consistently . Use exact numerical techniques (QMC, ED ) as well as semianalytical (IPT) approaches to solve this simplified problem. • Study simple model Hamiltonians (such as the one band model on simple lattices) • Understand the results physically in terms of a Landau theory :certain high temperature aspects are independent of the details of the model and the approximations used. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

9. DMFT Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] Weissfield THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

10. DMFT Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] Weissfield THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

13. 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

14. Insights into the Mott phenomena • 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

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

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

17. Ising character of Mott endpoint • Singular part of the Weiss field is proportional to h a Max{ (p-pc) (T- Tc)}1/d d=3 in mean field and 5 in 3d • h couples to all physical quantities which then exhibit a kink at the Mott endpoint. Resistivity, double occupancy,photoemission intensity, integrated optical spectral weight, etc. • Divergence of the the compressibility ,in particle hole asymmetric situations, e.g. Furukawa and Imada THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

18. Compressibility THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

19. Mott transition endpoint Rapid variation has been observed in optical measurements in vanadium oxide and nises mixtures Experimental questions: width of the critical region. Ising exponents or classical exponents, validity of mean field theory Building of coherence in other strongly correlated electron systems. condensation of doubly occupied sites and onset of coherence . THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

20. Insights from DMFT: think in term of spectral functions , the density is not changing! Resistivity near the metal insulator endpoint ( Rozenberg et.al 1995) exceeds the Mott limit THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

23. Two Roads for first principles calculations of correlated materials using DMFT. Correlation functions etc.. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

25. 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

26. DMFT +LDA : effective action construction (Fukuda, Valiev and Fernando , Chitra and GK). • DFT, consider the exact free energy as a functional of an external potential. Express the free energy as a functional of the density by Legendre transformation. GDFT[r(r)] • Introduce local orbitals, andf local Greens function by projecting onto the local orbitals.G(R,R)(i w) = • The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for r(r) and G and performing a Legendre transformation. G[r(r),G(R,R)(iw)] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

27. LDA+DMFT • 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. • DFT is useful because e good approximations to the exact density functional GDFT[r(r)] exist, e.g. LDA…. • A useful approximation to the exact functional can be constructed, the DMFT +LDA functional. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

28. LDA+DMFT functional Sum of local 2PI graphs with local U matrix and local G Double counting correction THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

29. Spectral density functionalConnection with atomic limit THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

32. LDA functional Double counting correction THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

33. 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, funcional formulation for full self consistent implementation (2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

35. Photoemission Spectra and Spin Autocorrelation: Fe(U=2, J=.9ev) (Lichtenstein, Katsenelson,GK prl in press) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

36. Photoemission and Spin Autocorrelation: Ni (U=3, J=.9 ev) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

38. 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

39. Fe and Ni • Spin wave stiffness controls the effects of spatial flucuations, it is about twice as large in Ni and in Fe • Classical calculations using measured exchange constants (Kudrnovski Drachl PRB 2001) Weiss mean field theory gives right Tc for Ni but overestimates Fe , RPA corrections reduce Tc of Ni by 10% only but reduce Tc of Fe by nearly factor of 2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

40. 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

41. Small amounts of Ga stabilize the d phase THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

42. Problems with LDA • DFT in the LDA or GGA is a well established tool for the calculation of ground state properties. • Many studies (Freeman, Koelling 1972)APW methods • ASA and FP-LMTO 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 the largest discrepancy ever known in DFT based calculations. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

43. Problems with LDA • LSDA predicts magnetic long range order which is not observed experimentally (Solovyev et.al.) • If one treats the f electrons as part of the core LDA overestimates the volume by 30% • LDA predicts correctly the volume of the a phase of Pu, when full potential LMTO (Soderlind and Wills). This is usually taken as an indication that a Pu is a weakly correlated system THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

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

47. Conclusion • The character of the localization delocalization in simple( Hubbard) models within DMFT is now fully understood. (Rutgers –ENS), nice qualitative insights. 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

48. Conclusions • Systematic improvements, short range correlations. • Take a cluster of sites, include the effect of the rest in a G0 (renormalization of the quadratic part of the effective action). What to take for G0: • DCA (M. Jarrell et.al) , CDMFT ( Savrasov and GK ) • include the effects of the electrons to renormalize the quartic part of the action (spin spin , charge charge correlations) E. DMFT (Kajueter and GK, Si et.al) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

49. Conclusions Extensions of DMFT implemented on model systems, carry over to more realistic framework. Better determination of Tcs. First principles approach: determination of the Hubbard parameters, and the double counting corrections long range coulomb interactions E-DMFT Improvement in the treatement of multiplet effects in the impurity solvers. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS