Low-temperature phases of itinerant fermions interacting with classical phonons: The static Holstein model

We consider models of independent itinerant fermions interacting with classical continuous or discrete variables (spins), the static Holstein model being a special case. We prove for all values of the fermion-spin coupling and a special value of the fermion chemical potential and classical magnetic field, at which the average fermion density is one-half and the average total spin is zero, that there are two degenerate ground states of period two with antiferromagnetic order for the spins and fermions. The existence of two corresponding low-temperature phases is proven for large coupling and dimension two or more by using a Peierls argument. This generalizes results of Kennedy and Lieb for the Falicov-Kimball model.