  We examine the spinless one-dimensional Falicov-Kimball model (FKM) below
half-filling, addressing both the binary alloy and valence transition
interpretations of the model. Using a non-perturbative technique, we derive an
effective Hamiltonian for the occupation of the localized orbitals, providing a
comprehensive description of charge order in the FKM. In particular, we uncover
the contradictory ordering roles of the forward-scattering and backscattering
itinerant electrons: the latter are responsible for the crystalline phases,
while the former produces the phase separation. We find an Ising model
describes the transition between the phase separated state and the crystalline
phases; for weak-coupling we present the critical line equation, finding
excellent agreement with numerical results. We consider several extensions of
the FKM that preserve the classical nature of the localized states. We also
investigate a parallel between the FKM and the Kondo lattice model, suggesting
a close relationship based upon the similar orthogonality catastrophe physics
of the associated single-impurity models.
