Repulsive particles on a two-dimensional lattice

TL;DR

This paper analytically investigates the minimum-energy configurations of repulsive particles on a 2D lattice, revealing hierarchical phase structures and novel aperiodic arrangements, with implications for charge order in fermionic systems.

Contribution

It characterizes the ground states of repulsive lattice particles, linking them to Falicov-Kimball model configurations and discovering new aperiodic phases at various densities.

Findings

Minimum-energy states match Falicov-Kimball ionic configurations.

Hierarchical phase sequence described by Farey tree.

Discovery of aperiodic configurations at lower densities.

Abstract

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.