Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University

1. Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University ISSP-Kashiwa 2001 Tokyo 1st-5th October

2. Evolution of the electronic structure between the atomic limit and the band limit in an open shell situation. The “”in between regime” is ubiquitous central them in strongly correlated systems, gives rise to interesting physics. New insights and new techniques from the solution of the Mott transition problem within dynamical mean field of simple model Hamiltonians Use the ideas and concepts that resulted from this development to give physical insights into real materials. Steps taken to turn the technology developed to solve the toy models into a practical electronic structure method. the Mott phenomena THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

3. Background: DMFT study of the Mott transition in a toy model. Behavior of the compressibility near the Mott transition endpoint. DMFT as an electronic structure method. From Lda to LDA+U to LDA+ DMFT. DMFT results for delta Pu, and some qualitative insights into the “Mott transition across the actinide series” Fe and Ni, a new look at the classic itinerant ferromagnets Outline THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

4. Describe some recent steps taken to make DMFT into an electronic structure tool. model Hamiltonian review see A. Georges talk in this workshop and consult reviews: 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)] Goal of the talk THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

5. Choice of Basis. Realistic self consistency condition Brief Comment on Impurity Solvers Integration with LDA. Effective action formulation. Comparison with LDA and LDA+U Some examples in real materials, transition metals and actinides. Outline: THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

6. S. Lichtenstein (Nijmeigen), E Abrahams (Rutgers) G. Biroli (Rutgers), R. Chitra (Rutgers-Jussieux), V. Udovenko (Rutgers), S. Savrasov (Rutgers-NJIT) G. Palsson, I. Yang (Rutgers) NSF, DOE and ONR Acknowledgements: Collaborators, Colleagues, Support for realistic work…………. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

8. 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. Strategy, look electronic structure problems where this physics is absolutely essential , Fe, Ni, Pu ……………. Good method to study the Mott phenomena THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

9. Definition of the local degrees of freedom Expression of the Weiss field in terms of the local variables (I.e. the self consistency condition) Expression of the lattice self energy in terms of the cluster self energy. Elements of the Dynamical Mean Field Construction and C-DMFT. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

10. Cellular DMFT : Basis selection. Exact spectra is basis independent DMFT results are not. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

11. Lattice action THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

12. Elimination of the medium variables THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

13. Determination of the effective medium. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

14. Connection between cluster and lattice self energy. The estimation of the lattice self energy in terms of the cluster energy has to be done using additional information Ex. Translation invariance • C-DMFT is manifestly causal: causal impurity solvers result in causal self energies and Green functions (GK S. Savrasov G. Palsson and G. Biroli) • Improved estimators for the lattice self energy are available (Biroli and Kotliar) • In simple cases C-DMFT converges faster than other causal cluster schemes. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

15. Convergence of CDMFT, test in a soluble problem (G. Biroli and G. Kotliar) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

16. Realistic DMFT self consistency loop THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

17. Realistic implementation of the self consistency condition • H and S, do not commute • Need to do k sum for each frequency • DMFT implementation of Lambin Vigneron tetrahedron integration V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997). • Transport Coeff (G. Palsson V. Udovenko and G. Kotliar) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

19. DMFT+QMC (A. Lichtenstein, M. Rozenberg) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

20. Multiorbital situation and several atoms per unit cell considerably increase the size of the space H (of heavy electrons). QMC scales as [N(N-1)/2]^3 N dimension of H Fast interpolation schemes (Slave Boson at low frequency, Roth method at high frequency, + 1st mode coupling correction), match at intermediate frequencies. (Savrasov et.al 2001) Solving the impurity THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

21. 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

22. Recent QMC phase diagram of the frustrated Half filled Hubbard model with semicircular DOS ( Joo and Udovenko). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

23. (Uc1)exact = 2.1 (Exact diag, Rozenberg, Kajueter, Kotliar PRB 1996) , (Uc1)IPT =2.4 (Uc2)exact =2.97+_.05(Projective self consistent method, Moeller Si Rozenberg Kotliar Fisher PRL 1995 ) (Uc2)IPT =3.3 (TMIT ) exact =.026+_ .004 (QMC Rozenberg Chitra and Kotliar PRL 1999), (TMIT )IPT =.5 (UMIT )exact =2.38 +- .03 (QMC Rozenberg Chitra and Kotliar PRL 1991), (UMIT )IPT =2.5 For realistic studies errors due to other sources (for example the value of U, are at least of the same order of magnitude). Case study: IPT half filled Hubbard one band THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

24. Interaction driven Mott transition Brinkman Rice . k ~ (Uc –U) Doping driven Mott transition (Gutzwiller, Brinkman Rice, Slave Boson method) . k is non singular Numerical simulations T=0 QMC , . k diverges As 1/ d (Furukawa and Imada) Compressibility near a Mott transition THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

25. At different points in the phase diagram, different behaviors. k vanishes at Uc2 (interaction driven Mott transition) At zero temperature k is non singular, at the doping driven Mott transition Behavior at UMIT TMIT ? The Mott transition as a bifurcation THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

26. The Mott transition as a bifurcation in effective action Zero mode with S=0 and p=0, couples generically Divergent compressibility THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

27. Qualitative phase diagram in the U, T , m plane (Murthy Rozenberg and Kotliar 2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

28. QMC calculationof n vs m (Murthy Rozenberg and Kotliar 2001, 2 band model, U=3.0) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

29. QMC n vs m (Murthy Rozenberg and Kotliar 2001, 2 band, low T THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

30. Compresibility vs T THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

31. Two Roads for calculations of the electronic structure of correlated materials Crystal Structure +atomic positions Model Hamiltonian Correlation functions Total energies etc. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

32. LDA functional Conjugate field, VKS(r) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

33. Minimize LDA functional THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

34. LDA+U functional THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

35. Double counting term (Lichtenstein et.al) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

36. 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 LDA+DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

37. 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)] Spectral Density Functional : effective action construction (Fukuda, Valiev and Fernando , Chitra and GK). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

38. The exact functional can be built in perturbation theory in the interaction (well defined diagrammatic rules )The functional can also be constructed from the atomic limit, but 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. Spectral Density Functional THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

40. Static limit of the LDA+DMFT functional , with F= FHF reduces to LDA+U Removes inconsistencies of this approach, 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. Comments on LDA+DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

41. LDA+DMFTConnection with atomic limit Weiss field THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

44. 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 of a spectral density functional( cond-mat 2001) LDA+DMFT References THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

45. The functional approach offers a direct connection to the atomic energies. One is free to add terms which vanish quadratically at the saddle point. 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 . Functional Approach THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

46. G. Kotliar EPJB (1999) Functional Approach THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

47. Case study in f electrons, Mott transition in the actinide series THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

50. 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. Delocalization-Localization across the actinide series THE STATE UNIVERSITY OF NEW JERSEY RUTGERS