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Model Hamiltonians and qualitative considerations in the physics of materials. Or what do we want to know? An example from the physics of the Mott transition .

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  1. Model Hamiltonians and qualitative considerations in the physics of materials. Or what do we want to know? An example from the physics of the Mott transition. Merging band structure methods with many body theory, where to improve? A) basis set? B) parameter estimates of your model Hamiltonian C) DMFT impurity solver? D) Improvements of DMFT ? An intro to Cellular DMFT [G. Kotliar S. Savrasov G. Palsson and G. Biroli PRL87, 186401 2001] Outline THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  2. Mott transition in the actinide series. B. Johanssen 1974 Smith and Kmetko Phase Diagram 1984. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  3. 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

  4. Underlying Landau Free energy which is responsible of all the qualitative features of the phase diagram. Of the frustrated Hubbard model in large d [G. Kotliar EPJB 99] Around the finite temperature Mott endpoint, the Free energy has a simple Ising like form as in a liquid gas transition [R. Chitra, G. Kotliar E.Lange M. Rozenberg ] Changing the model (DOS, degeneracy, etc) just changes the coefficients of the Landau theory. Robustness of the finite T results THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  5. Different impurity solvers, different values of the Landau coefficients, as long as they preserve the essential (non) analytic properties of the free energy functional. The functional approach can be generalized to combine DFT and DMFT [R. Chitra G. Kotliar , S. Savrasov and G. Kotliar] Justification for applying simple models to some aspects of the crossover in Ni(SeS)2And V2O3. Robustness of the finite T results and Functional Approach THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  6. Coexistence regions between localized and delocalized spectral functions. Qualitative phase diagram in the U, T , m plane (two band Kotliar Murthy Rozenberg PRL (2002). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  7. QMC calculationof n vs m (Kotliar Murthy Rozenberg PRL 2002, 2 band, U=3.0) k diverges at generic Mott endpoints THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  8. Compressibilty divergence : One band case (Kotliar Murthy and Rozenberg 2001, cond-matt 0110625) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

  11. Two impurity method. [A. Georges and G. Kotliar, A. Schiller PRL75, 113 (1995)] M. Jarrell Dynamical Cluster Approximation [Phys. Rev. B 7475 1998] Continuous version [periodic cluster] M. Katsenelson and A. Lichtenstein PRB 62, 9283 (2000). Extended DMFT [H. Kajueter and G. Kotliar Rutgers Ph.D thesis 2001, Q. Si and J L Smith PRL 77 (1996)3391 ] Coulomb interactions R . Chitra Cellular DMFT [PRL87, 186401 2001] A (non comprehensive )list of extensions of DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  12. DMFT cavity construction Weissfield THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  13. 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 Cellular DMFT, G. Kotliar S. Savrasov G. Palsson and G. Biroli PRL 2001 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  14. Cellular DMFT : Basis selection THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  15. Lattice action THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

  18. 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 PRL 2001) • In simple cases C-DMFT converges faster than other causal cluster schemes. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  19. Improved estimators for the lattice self energy are available (Biroli and Kotliar) Improved estimators THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  20. Real Space Formulation of the DCA approximation of Jarrell et.al. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  21. Affleck Marston model. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  22. Convergence test in the Affleck Marston THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  23. Convergence of the self energy THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  24. A. Perali et.al. cond-mat 2001, two patch model, phenomenological fit of the functional form of the vertex function of C-DMFT to experiments in optimally doped and overdoped cuprates Flexibility in the choice of basis seems important. Recent application to high Tc THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  25. Extended DMFT electron phonon THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  26. Extended DMFT e.ph. Problem THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  27. E-DMFT classical case, soft spins THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  28. E-DMFT classical case Ising limit THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  29. E-DMFT test in the classical case[Bethe Lattice, S. Pankov 2001] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  30. The transition is first order at finite temperatures for d< 4 No finite temperature transition for d less than 2 (like spherical approximation) Improved values of the critical temperature Advantage and Difficulties of E-DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  31. For “first principles work” there are several many body tools waiting to be used, once the one electron aspects of the problem are clarified. E-DMFT or C-DMFT for Ni, and Fe ? Promising problem: Qualitative aspects of the Mott transition within C-DMFT ?? Cuprates? Conclusion THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  32. Realistic Theories of Correlated Materials ITP, Santa-Barbara July 27 – December 13 (2002) O.K. Andesen, A. Georges, G. Kotliar, and A. Lichtenstein http://www.itp.ucsb.edu/activities/future/

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

  34. Recent phase diagram of the frustrated Half filled Hubbard model with semicircular DOS (QMC Joo and Udovenko PRB2001). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  35. (Uc1)exact = 2.1 (Exact diag, Rozenberg, Kajueter, Kotliar 1995) , (Uc1)IPT =2.4 (Uc2)exact =2.95 (Projective self consistent method, Moeller Si Rozenberg Kotliar 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

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

  37. 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 (Poteryaev et.al 1987) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  38. 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

  39. 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 ……………. Good method to study the Mott phenomena THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  40. 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

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

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

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

  44. 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

  45. 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

  46. 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

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

  48. 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

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

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

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