Ground-state properties of multicomponent Falicov-Kimball-like models I

We consider a classical lattice gas that consists of more than one "species" of particles (like a spin-3/2 Ising model or the atomic limit of the extended Hubbard model), whose ground-state phase diagram is macroscopically degenerate. This gas is coupled component-wise and in the Falicov-Kimball-like manner to a multicomponent free-fermion gas. We show rigorously that a component-wise coupling of the classical subsystem to the quantum one orders the classical subsystem so that the macroscopic degeneracy is removed.