Repulsive particles on a two-dimensional lattice

The problem of finding the minimum-energy configurations of particles on a square lattice, subject to short-ranged repulsive interactions, is studied analytically. The study is relevant to charge-ordered states of interacting fermions, as described by the spinless Falicov-Kimball model. A simple model is introduced, corresponding to the limit of a two-body potential which is a very rapidly decreasing convex function of distance. Its ground states are found rigorously for certain density ranges, including half filling. These agree with known properties of neutral ground states of the large-U Falicov-Kimball model, suggesting a characterization of the latter as “most homogeneous” configurations. For lower densities, a family of ground states is found having the novel property that they are aperiodic even when the particle density is a rational number. In some cases, local phase separation suggests an inherent sensitivity to the detailed form of the interaction potential.