Dynamical Mean Field Theory DMFT and electronic structure calculations. Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University. ICTP Trieste August 2003. Outline.

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Dynamical Mean Field Theory DMFT and electronic structure calculations

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Outline • Pedagogical Introduction to DMFT for correlated electron systems, parallel with DFT GW .[R. Chitra , P Sun, S. Savrasov and GK] • How good is the local approximation ? a) a brief look at some recent experiments. b) compare with exact results in one dimension. c) look at corrections. What new effects do cluster corrections bring on top of single site DMFT ? Some answers on a model of kappa organics (O. Parcollet G. Biroli and GK) • Some system specific calculations for materials near the Mott transition: La1-x Srx TiO3 , , Pu……… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Weakly correlated electrons:band theory. • Simple conceptual picture of the ground state, excitation spectra, transport properties of many systems (simple metals, semiconductors,….). • A methods for performing quantitative calculations. (Density functional theory, in various approximations and GW). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Start with TOE THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Exchange and correlation energy • Exact formal expressions can be given in terms of a coupling constant integration.[Harris-Jones, adiabatic connection] • DFT is useful because practical accurate expressions for Exc, exist. • LDA, GGA, hybrids, THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Kohn Sham Greens function is an good point to compute spectra in perturbation theory in screenedCoulomb interaction GW,G0W0 Practical implementations, introduce a finite basis set. Division into valence (active ) degrees of freedom and core. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DFT+GW program has been less succesful in correlated situations. • Strong interactions localize the particles. Atoms with open shells are not easily connected to band theory. • The spectrum in this case, contain Hubbard bands which are NOT simply perturbatively connected to the Kohn Sham orbitals. • Need an alternative reference point for doing perturbation theory! • Need to treat bands and atomic excitations on the same footing. • DMFT! THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Strongly correlated systems are usually treated with model Hamiltonians THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conceptually one wants to restrict the number of degrees of freedom by eliminating high energy degrees of freedom. In practice other methods (eg constrained LDA are used)

About XC functional. • One can derive a coupling constant integration formulae (Harris Jones formula) for • Generate approximations. • The exact formalism generates the local Greens function and S ii is NOT the self energy. However one can use the approach as starting point for computing other quantities. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Comments on functional construction • Atoms as a reference point. Expansion in t. • Locality does not necessarily mean a single point. Extension to clusters. • Jii --- Jii Ji i+d • Aii --- Ai i+d • S ii --- S i i+d • Exact functional G[Aii ,Ai i+d] • The lattice self energy and other non local quantities extending beyond the cluster are OUTSIDE the formalism and need to be inferred. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Comments on funct. construction. • Construction of approximations in the cluster case requires care to maintain causality. • One good approximate construction is the cellular DMFT: a) take a supercell of the desired range,b) • c) obtain estimate of the lattice self energy by restoring translational symmetry. • Many other cluster approximations (eg. DCA, the use of lattice self energy in self consitency condition, restrictions of BK functional, etc. exist). Causality and classical limit of these methods has recently been clarified [ G Biroli O Parcollet and GK] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Mapping onto impurity models. • The local Greens function A, and the auxilliary quantity S, can be computed from the solution of an impurity model that describes the effects of the rest of the lattice on the on a selected central site. • One can arrive at the same concept via the cavity construction. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Solving the DMFT equations • Wide variety of computational tools (QMC,ED….)Analytical Methods • Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

(E)DMFT pproximation to Sum over all LOCAL 2PI graphs (integrations are restricted over the unit cell ) built with W and G Map into impurity model to generate G and W Go beyond this approximation by returning to many body theory and adding the first non local correction. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Take the solution of the EDMFT equations as an approximation for the TRUE local self energy, and add the leading NON LOCAL corrections to the self energy G_NL W_NL, as a correction. • Do it self consistently and as a one shot iteration G0_NL W0_NL and compare the results. 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, 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. Savrasov Kotliar and Abrahams . Application to delta Pu Nature (2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

How good is the local approximation ? • Single site DMFT study of the Mott transition, based on a study of the Hubbard model on frustrated lattices made several interesting qualitative predictions. • New experiments and reexamination of old ones give credence to that the local picture is quite good. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

NiSe2-xSx THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Qualitative single site DMFT predictions. • Spectra of the strongly correlated metallic regime contains both quasiparticle-like and Hubbard band-like features. • Mott transition is drive by transfer of spectral weight. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Schematic DMFT phase diagram Hubbard model (partial frustration) [observation of temperature dependent transfer of spectral weight in optics] M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Insights from DMFT • The Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phase • Control parameters: doping, temperature,pressure… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Evolution of the Spectral Function with Temperature Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Schematic DMFT phase diagram Hubbard model (partial frustration). Evidence for QP peak in V2O3 from optics. M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS