From phase separation to long-range order in a system of interacting electrons

We study a system composed of fermions (electrons), hopping on a square lattice, and of immobile particles (ions), that is described by the spinless Falicov–Kimball Hamiltonian augmented by a next-nearest-neighbor attractive interaction between the ions (a nearest-neighbor repulsive interaction between the ions can be included and does not alter the results). A part of the grand-canonical phase diagram of this system is constructed rigorously, when the coupling between the electrons and ions is much stronger than the hopping intensity of electrons. The obtained diagram implies that, at least for a few rational densities of particles, by increasing the hopping intensity the system can be driven from a state of phase separation to a state with a long-range order. This kind of transitions occurs also, when the hopping fermions are replaced by hopping hard-core bosons.