Strong Coupling Theory for Interacting Lattice Models

Abstract

We develop a strong coupling approach for a general lattice problem. We argue that this strong coupling perspective represents the natural framework for a generalization of the dynamical mean field theory (DMFT). The main result of this analysis is twofold: 1) It provides the tools for a unified treatment of any non-local contribution to the Hamiltonian. Within our scheme, non-local terms such as hopping terms, spin-spin interactions, or non-local Coulomb interactions are treated on equal footing. 2) By performing a detailed strong-coupling analysis of a generalized lattice problem, we establish the basis for possible clean and systematic extensions beyond DMFT. To this end, we study the problem using three different perspectives. First, we develop a generalized expansion around the atomic limit in terms of the coupling constants for the non-local contributions to the Hamiltonian. By analyzing the diagrammatics associated with this expansion, we establish the equations for a generalized dynamical mean-field theory (G-DMFT). Second, we formulate the theory in terms of a generalized strong coupling version of the Baym-Kadanoff functional. Third, following Pairault, Senechal, and Tremblay, we present our scheme in the language of a perturbation theory for canonical fermionic and bosonic fields and we establish the interpretation of various strong coupling quantities within a standard perturbative picture.