Generalizations of dynamical mean field theory by the lace expansion

We use the so-called lace expansion of mathematical random walk theory to derive several generalizations of dynamical mean-field theory (DMFT) for strongly correlated electron systems (SCES). DMFT is based on a self-consistence condition, by which a lattice Hamiltonian for SCES is mapped onto the corresponding impurity problem. It is a local theory with a density of states of the impurity problem following from the local band Green function. So in the Fourier-transformed version the self-energy is independent of momenta. From a mathematical point of view, the self-consistence condition follows from a physical self-avoiding loop. We discuss non-local as well as local corrections and examine them numerically for the Anderson lattice in the large U limit. All calculations are performed at finite spatial dimensions.