THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Excitation spectra

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

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Summary 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).

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

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

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

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

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA functional Conjugate field, V KS (r)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Minimize LDA functional

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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.  DFT  (r)] Introduce local orbitals,   R (r-R)orbitals, and local GF G(R,R)(i  ) = The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for  (r) and G and performing a Legendre transformation,  (r),G(R,R)(i  )]

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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 from the atomic limit, but no explicit expression exists. DFT is useful because good approximations to the exact density functional  DFT  (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 LDA+DMFT functional  Sum of local 2PI graphs with local U matrix and local G

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on LDA+DMFT Static limit of the LDA+DMFT functional, with  =  HF 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.

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Realistic DMFT loop

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT References V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, (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)

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

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

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Lda vs Exp Spectra

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

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion 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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook 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)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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 A (non comprehensive )list of extensions of DMFT Two impurity method. [A. Georges and G. Kotliar, A. Schiller PRL75, 113 (1995)] M. Jarrell Dynamical Cluster Approximation [Phys. Rev. B ] 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, ]

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT cavity construction Weiss field

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Elements of the Dynamical Mean Field Construction and Cellular DMFT, G. Kotliar S. Savrasov G. Palsson and G. Biroli PRL 2001 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.

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Lattice action

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

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

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

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Affleck Marston model.

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

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Recent application to high Tc 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.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Extended DMFT electron phonon

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

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

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

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Advantage and Difficulties of E-DMFT 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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion 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?

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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 Realistic implementation of the self consistency condition H and , 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 Good method to study the 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 …………….

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Functional Approach 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.

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

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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  phase of Pu, when full potential LMTO (Soderlind and Wills). This is usually taken as an indication that  Pu is a weakly correlated system

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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 and Kotliar, Bouchet et. Al. ) Delta Pu has U=4, Alpha Pu has U =0. 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 simple elements

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Connection with atomic limit Weiss field