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## Excitation spectra

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**Excitation spectra**THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Gabriel Kotliar**Rutgers University Trieste 2002 Comments on realistic calculations using DMFGT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**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**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**Basis set, bands , DOS**THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Computation of U’s**THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**U is a basis dependent concept.**Dynamical mean field theory is a basis dependent technique. Comments 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 Model Hamiltonian Correlation functions Total energies etc. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**LDA functional**Conjugate field, VKS(r) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Minimize LDA functional**THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**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**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**LDA+DMFT functional**F Sum of local 2PI graphs with local U matrix and local G THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**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**LDA+DMFT Self-Consistency loop**E U DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Realistic DMFT loop**THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**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**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**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**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**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**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**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**DMFT cavity construction**Weissfield THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**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**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 for the lattice self energy are**available (Biroli and Kotliar) Improved estimators 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**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**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**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**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**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/