Conserving approximations in direct perturbation theory: new semianalytical impurity solvers and their application to general lattice problems

For the treatment of interacting electrons in crystal lattices, approximations based on the picture of effective sites, coupled in a self-consistent fashion, have proven very useful. Particularly in the presence of strong local correlations, a local approach to the problem, combining a powerful method for the short-ranged interactions with the lattice propagation part of the dynamics, determines the quality of results to a large extent. For a considerable time the noncrossing approximation (NCA) in direct perturbation theory, an approach originally developed by Keiter for the Anderson impurity model, was a standard for the description of the local dynamics of interacting electrons. In the last couple of years exact methods like the numerical renormalization group (NRG), as pioneered by Wilson, have surpassed this approximation as regarding the description of the low-energy regime. We present an improved approximation level of direct perturbation theory for finite Coulomb repulsion U, the crossing approximation 1 (CA1), and discuss its connections with other generalizations of NCA. CA1 incorporates all processes up to fourth order in the hybridization strength V in a self-consistent skeleton expansion, retaining the full energy dependence of the vertex functions. We reconstruct the local approach to the lattice problem from the point of view of cumulant perturbation theory in a very general way and discuss the proper use of impurity solvers for this purpose. Their reliability can be tested in applications to, for example, the Hubbard model and the Anderson-lattice model. We point out shortcomings of existing impurity solvers and improvements gained with CA1 in this context.