Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University

1. Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University

2. The Strong Correlation Problem Two limiting cases of the electronic structure of solids are understood:the high density limit and the limit of well separated atoms. High densities, the is electron be a wave, use band theory, k-space One particle excitations: quasi-particle,quasi-hole bands, collective modes. Density Functional Theory with approximations suggested by the Kohn Sham formulation, (LDA GGA) is a successful computational tool for the total energy, and a good starting point for perturbative calculation of spectra, GW.…………………… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

3. Mott : Correlations localize the electron Low densities, electron behaves as a particle,use atomic physics, real space 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….) Magnetic and Orbital Ordering at low T Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

4. Localization vs Delocalization Strong Correlation Problem • A large number of compounds with electrons which are not close to the well understood limits (localized or itinerant). • These systems display anomalous behavior (departure from the standard model of solids). • Neither LDA or LDA+U or Hartree Fock works well • Dynamical Mean Field Theory: Simplest approach to the electronic structure, which interpolates correctly between atoms and bands THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

5. Outline • Motivation: plutonium puzzles. • Review of Dynamical Mean Field Theory an Extension to realistic systems . DMFT and DFT. • A case study of system specific properties: f .electrons DMFT Results for d Pu. • A case study of system specific properties d electrons in Fe and Ni. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

6. Collborators and References • Reviews of DMFT: A. Georges G. Kotliar W krauth and M . Rozenberg Rev Mod Pnys 68, 13 (1996). DMFT and LDA R. Chitra and G. Kotliar Phys. Rev. B 62,, 12715 (2000). S. Savrasov and G. Kotliar cond-mat cond-mat 0106308. • DMFT study of Plutonium. S. Savrasov, G. Kotiar and E. Abrahams, Nature 410, 793 (2001). S. Savrasov and G. Kotliar • DMFT study of Iron and Nickel. S. Lichtenstein M Katsenelson and G. Kotliar Phys. Rev. Lett 87, (2001). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

7. Outline • Motivation: plutonium puzzles. • Review of Dynamical Mean Field Theory an Extension to realistic systems . DMFT and DFT. • A case study of system specific properties: f .electrons DMFT Results for d Pu. • A case study of system specific properties d electrons in Fe and Ni. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

8. Case study in f electrons, Mott transition in the actinide series. B. Johanssen 1974 Smith and Kmetko Phase Diagram 1984. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

9. Pu: Complex Phase Diagram (J. Smith LANL) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

11. 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

12. 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, using 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

13. Other Methods • LDA+ U (Savrasov and Kotliar Phys. Rev. Lett. 84, 3670, 2000, Bouchet et. al 2000) predicts correct volume of Pu with the constrained LDA estimate of U=4 ev. However, it predicts spurious magnetic long range order and a spectra which is very different from experiments. • Requires U=0 to treat the alpha phase, which has many physical properties in common with the delta phase. • Similar problems with the constrained (4 of the 5f electrons are treated as core ) LDA approach of Erikson and Wills. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

14. Conventional viewpoint • Alpha Pu is a simple metal, it can be described with LDA + correction. In contrast delta Pu is strongly correlated. • Constrained LDA approach (Erickson, Wills, Balatzki, Becker). In Alpha Pu, all the 5f electrons are treated as band like, while in Delta Pu, 4 5f electrons are band-like while one 5f electron is deloclized. • Same situation in LDA + U (Savrasov andGK Bouchet et. al. ) .Delta Pu has U=4,Alpha Pu has U =0. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

15. Problems with the conventional viewpoint of Pu • The specific heat of delta Pu, is only twice as big as that of alpha Pu. • The susceptibility of alpha Pu is in fact larger than that of delta Pu. • The resistivity of alpha Pu is comparable to that of delta Pu. • Only the structural and elastic properties are completely different. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

16. MAGNETIC SUSCEPTIBILITY THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

17. Pu Specific Heat THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

18. Anomalous ResistivityJ. Smith LANL THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

19. Outline • Motivation: plutonium puzzles. • Review of Dynamical Mean Field Theory an Extension to realistic systems . DMFT and DFT. • A case study of system specific properties: f .electrons DMFT Results for d Pu. • A case study of system specific properties d electrons in Fe and Ni. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

20. DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)] Weissfield THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

21. Single site DMFT, functional formulation • Express in terms of Weiss field (semicircularDOS) • The Mott transition as bifurcation point in functionals oG[G] or F[D], (G. Kotliar EPJB 99) Local self energy (Muller Hartman 89) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

23. DMFT • Construction is easily extended to states with broken translational spin and orbital order. • Large number of techniques for solving DMFT equations for a review see • A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

25. 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

26. Localization Delocalization • The Mott transition/crossover is driven by transfer of spectral weight from low to high energy as we approach the localized phase • Control parameters: doping, temperature,pressure… • Intermediate U region is NOT perturbatively accessible. DMFT a new starting point to access this regime. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

27. Qualitative phase diagram in the U, T , m plane (two band Kotliar and Rozenberg (2001)) • Coexistence regions between localized and delocalized spectral functions. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

28. QMC calculationof n vs m (Murthy Rozenberg and Kotliar 2001, 2 band, U=3.0) k diverges at generic Mott endpoints THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

30. Spectral Density Functional : 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, caR(r-R)orbitals, and local GF • 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

31. Spectral Density Functional • The exact functional can be built in perturbation theory in the interaction (well defined diagrammatic rules )The functional can also be constructed expanding around the the atomic limit. No explicit expression exists. • DFT is useful because good approximations to the exact density functional GDFT[r(r)] exist, e.g. LDA, GGA • A useful approximation to the exact functional can be constructed, the DMFT +LDA functional. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

32. LDA+DMFT functnl F Sum of local 2PI graphs with local U matrix and local G THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

34. Comments on LDA+DMFT • Static limit of the LDA+DMFT functional , with F= FHF reduces to LDA+U • Removes inconsistencies and shortcomings of this approach. DMFT retain correlations effects in the absence of orbital ordering. • Only in the orbitally ordered Hartree Fock limit, the Greens function of the heavy electrons is fully coherent • Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

35. Outline • Motivation: plutonium puzzles. • Review of Dynamical Mean Field Theory an Extension to realistic systems . DMFT and DFT. • A case study of system specific properties: f .electrons DMFT Results for d Pu. • A case study of system specific properties d electrons in Fe and Ni. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

36. Pu: DMFT total energy vs Volume Savrasov Kotliar Abrahams to appear in Nature THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

39. PU: ALPHA AND DELTA THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

40. Dynamical Mean Field View of Pu(Savrasov Kotliar and Abrahams, Nature 2001) • Delta and Alpha Pu are both strongly correlated, the DMFT mean field free energy has a double well structure, for the same value of U. One where the f electron is a bit more localized (delta) than in the other (alpha). • Is the natural consequence of the model Hamiltonian phase diagram once the structure is about to vary. • This result resolves one of the basic paradoxes in the physics of Pu. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

41. Minimum of the melting point • Divergence of the compressibility at the Mott transition endpoint. • Rapid variation of the density of the solid as a function of pressure, in the localization delocalization crossover region. • Slow variation of the volume as a function of pressure in the liquid phase THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

42. Minimum in melting curve and divergence of the compressibility at the Mott endpoint THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

43. Cerium: melting T vs p THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

44. Pu: Anomalous thermal expansion (J. Smith LANL) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

45. Double well structure and d Pu Qualitative explanation of negative thermal expansion Sensitivity to impurities which easily raise the energy of the a -like minimum. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

46. Double well structure and d Pu negative thermal expansion Sensitivity to impurities which easily raise the energy of the a -like minimum. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

47. Future directions • Including short range correlations. Less local physics, C-DMFT. • Life without U, including the effects of long range Coulomb interactions, E-DMFT and GW. • Applications are just beginning, many surprises ahead…… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

48. Outline • Motivation: plutonium puzzles. • Review of Dynamical Mean Field Theory an Extension to realistic systems . DMFT and DFT. • A case study of system specific properties: f .electrons DMFT Results for d Pu. • A case study with d electrons in Fe and Ni. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

49. Case study Fe and Ni • Archetypical itinerant ferromagnets • LSDA predicts correct low T moment • Band picture holds at low T • Main puzzle: at high temperatures c has a Curie Weiss law with a moment much larger than the ordered moment. • Magnetic anisotropy THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

50. Iron and Nickel: crossover to a real space picture at high T (Lichtenstein, Katsnelson and GK) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS