  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.
