On the optimality of the rock-salt structure among lattices with charge distributions

Abstract

The goal of this paper is to investigate the optimality of the [Formula: see text]-dimensional rock-salt structure, i.e. the cubic lattice [Formula: see text] of volume [Formula: see text] with an alternation of charges [Formula: see text] at lattice points, among periodic distributions of charges and lattice structures. We assume that the charges are interacting through two types of radially symmetric interaction potentials, according to their signs. We first restrict our study to the class of orthorhombic lattices. We prove that, for our energy model, the [Formula: see text]-dimensional rock-salt structure is always a critical point among periodic structures of fixed density. This holds for a large class of potentials. We then investigate the minimization problem among orthorhombic lattices with an alternation of charges for inverse power laws and Gaussian interaction potentials. High density minimality results and low-density non-optimality results are derived for both types of potentials. Numerically, we investigate several particular cases in dimensions [Formula: see text], [Formula: see text] and [Formula: see text]. The numerics support the conjecture that the rock-salt structure is the global optimum among all lattices and periodic charges, satisfying some natural constraints. For [Formula: see text], we observe a phase transition of the type “triangular-rhombic-square-rectangular” for the minimizer’s shape as the density decreases.