  A fermionic model, built up of q species of localized Fermi particles,
interacting by charge correlations, is isomorphic to a spin-q/2 Ising model.
However, the equivalence is only formal and the two systems exhibit a different
physical behavior. By considering a Bethe lattice with q=1, we have exactly
solved the models. There exists a critical temperature below which there is a
spontaneous breakdown of the particle-hole symmetry for the first model, and of
the spin symmetry for the second. While the spin system is always stable and
exhibits a homogeneous ferromagnetic phase below T_{c}, the fermionic system
for T<T_{c} is unstable against the formation of inhomogeneous phases with
charge separation.
