Ground states and low temperature phases of itinerant electrons interacting with classical fields: a review of rigorous results

We review, from a unified point of view, a general class of models of itinerant electrons interacting with classical fields. Applications to the static Holstein, Kondo, and Hubbard models are discussed. The ground state structure of the classical field is investigated when the electron band is half-filled. Some of the results are also valid when there is a Hubbard interaction between spin up and spin down electrons. It is found that the ground states are either homogeneous or period two Ne´el configurations, depending on the geometry of the lattice and on the magnetic fluxes present in the system. In the specific models, Ne´el configurations correspond to Peierls, magnetic or superconducting instabilities of the homogeneous state. The effect of small thermal and quantum fluctuations of the classical fields are reviewed in the context of the Holstein model. Many of the results described here originate from the work of Elliott Lieb and collaborators.