THE FALICOV-KIMBALL MODEL - A REVIEW OF EXACT RESULTS AND EXTENSIONS

The Falicov-Kimball model is a simplified version of the the Hubbard m odel in which only one type of electron (e.g. spin down) is allowed to hop. It describes in particular a system of spinless quantum particle s interacting with classical particles (Ising spins). In this review w e present the progress which has been accomplished in the last decade concerning this model, with an emphasis on rigorous results. Our discu ssion includes the one, two, and three dimensional cases. We also show how certain techniques can be applied to other related models such as the static Holstein and Kondo models. Their common feature with the F alicov-Kimball model is that they consist of itinerant spinless electr ons interacting with a classical field, associated with either a discr ete Ising spin, a continuous scalar spin or a vector field. Finally Re discuss a generalized Falicov-Kimball model of spin one-half electron s with on-site Hubbard interaction and interacting also with classical particles, as well as different models where the fermions are replace d by hard-core bosons. For the last class of models the interactions a re truly many body but a limited number of rigorous results can be obt ained using reflection positivity. The main issues discussed in this r eview concern the structure of ground states for the classical particl es, and how they are affected by magnetic fluxes (via orbital coupling ) and quantum statistics. Perturbative as well as non perturbative met hods are used.