Some symmetry remarks on the ground-state configurations of the two-dimensional Bosonic Falicov-Kimball model

We show that the ground-state configuration of the nuclei in a system composed by itinerant bosons in two dimensions described by the Falicov-Kimball Hamiltonian is composed by symmetric intervals centered on 45° lines on the Cartesian plane. This adds up to the previous result about this problem which asserted symmetry of intervals centered on the coordinate axes. The combination of these facts strongly suggest the convexity of the nuclear distributions.