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Excitation spectra

Excitation spectra. Gabriel Kotliar Rutgers University Trieste 2002. Comments on realistic calculations using DMFGT. Spectral Evolution at T=0 half filling full frustration. X.Zhang M. Rozenberg G. Kotliar (PRL 1993) Joo and Udovenko (20010). Basis set LMTO (Savrasov)

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Excitation spectra

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  1. Excitation spectra THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  2. Gabriel Kotliar Rutgers University Trieste 2002 Comments on realistic calculations using DMFGT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  3. Spectral Evolution at T=0 half filling full frustration X.Zhang M. Rozenberg G. Kotliar (PRL 1993) Joo and Udovenko (20010) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  4. Basis set LMTO (Savrasov) Materials Information and Design Lab. (Savrasov’s MINDLAB) Computations of U (Anisimov) Derivation of model hamiltonian Solution via DMFT: mapping onto degenerate Anderson model in a self consistent bath. Solution of the multiorbital anderson model Using QMC (Rozenber and Lichtenstein). Summary THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  5. Basis set, bands , DOS THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  6. Computation of U’s THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  7. U is a basis dependent concept. Dynamical mean field theory is a basis dependent technique. Comments THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  8. Unitary transformation K dependent! THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  9. 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

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

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

  12. 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

  13. 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

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

  15. 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

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

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

  18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  19. 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 (1998). S. Savrasov and G.Kotliar, funcional formulation for full self consistent implementation Nature (2001) LDA+DMFT References THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  20. Look for situations which Are in between atomic and band behavior. Many Many Many Compounds Oxides. BUT ALSO SOME ELEMENTS! Applications THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

  22. Pu: DMFT total energy vs Volume(S. Savrasov 2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

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

  26. The character of the localization delocalization in simple( Hubbard) models within DMFT is now fully understood, nice qualitative insights. This has lead to extensions to more realistic models, and a beginning of a first principles approach interpolating between atoms and band, encouraging results for many systems Conclusion THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  27. 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: Cluster DMFT, periodic clusters (Lichtenstein and Katsnelson)DCA (M. Jarrell et.al) , CDMFT ( 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) Outlook THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  28. C-DMFT: test in one dimension. (Bolech, Kancharla and Gk2002) Gap vs U, Exact solution Lieb and Wu, Ovshinikov PRL 20,1445 (1968) Nc=2 CDMFT vs Nc=1 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  29. 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 GK Savrasov Palsson and Biroli [PRL87, 186401 2001] A (non comprehensive )list of extensions of DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

  31. 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

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

  33. Lattice action THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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

  36. 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

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

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

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

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

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

  42. 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

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

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

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

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

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

  48. 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

  49. 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

  50. Realistic Theories of Correlated Materials ITP, Santa-Barbara July 27 – December 13 (2002) Conference November15-19 (2002) O.K. Andesen, A. Georges, G. Kotliar, and A. Lichtenstein http://www.itp.ucsb.edu/activities/future/

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