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The spinless Falicov-Kimball model on a two-dimensional square lattice is studied using the method of restricted phase diagrams constructed in the grand canonical ensemble. The results are compared with the one-dimensional model. Although the two-dimensional phase diagrams are more complex, with several distinct families of ion configurations occurring as ground states, there are surprising similarities with the one dimensional case. Within each family of configurations, the ground states form a devil's staircase structure and the configurations are constructed according to a composition rule identical to that in one dimension. It is also found that, as in one dimension, segregation occurs in the non-neutral model for large ion-electron interaction strength. Some features of the phase diagrams are understood by examining the effective two body ion interaction.

The problem of finding the minimum-energy configuration of particles on a lattice, subject to a generic short-ranged repulsive interaction, is studied analytically. The study is relevant to charge ordered states of interacting fermions, as described by the spinless Falicov-Kimball model. For a range of particle density including the half-filled case, it is shown that the minimum-energy states coincide with the large-U neutral ground state ionic configurations of the Falicov-Kimball model, thus providing a characterization of the latter as "most homogeneous" ionic arrangements. These obey hierarchical rules, leading to a sequence of phases described by the Farey tree. For lower densities, a new family of minimum-energy configurations is found, having the novel property that they are aperiodic even when the particle density is a rational number. In some cases there occurs local phase separation, resulting in an inherent sensitivity of the ground state to the detailed form of the interaction potential.

It is known that one-dimensional lattice problems with a discrete, finite set of states per site 'generically' have periodic ground states (GSs). We consider slightly less generic cases, in which the Hamiltonian is constrained by either spin (S) or spatial (1) inversion symmetry (or both). We show that such constraints give rise to the possibility of disordered GSs over a finite fraction of the coupling-parameter space-that is, without invoking any nongeneric 'fine tuning' of coupling constants, beyond that arising from symmetry. We find that such disordered GSs can arise for many values of the number of states (k) at each site, and the range r of the interaction. The Ising (k = 2) case is the least prone to disorder: I symmetry allows for disordered GSs (without fine tuning) only for r 2 5, while S symmetry 'never' gives rise to disordered GSs.

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