Optimal and non-optimal lattices for non-completely monotone interaction potentials

TL;DR

This paper explores the energy minimization of d-dimensional Bravais lattices with various pairwise potentials, extending beyond completely monotone functions, and identifies conditions for optimality and non-optimality, including numerical and theoretical results.

Contribution

It introduces criteria for lattice minimality based on inverse Laplace transforms, constructs non-completely monotone potentials with specific minimizers, and links lattice energy minimization to the kissing problem.

Findings

Triangular lattice minimizes energy for certain non-completely monotone potentials.

Square lattice can have lower energy than triangular for specific one-well potentials.

Numerical evidence supports optimality of particular lattices across exponents.

Abstract

We investigate the minimization of the energy per point $E\_f$ among $d$-dimensional Bravais lattices, depending on the choice of pairwise potential equal to a radially symmetric function $f(|x|^2)$. We formulate criteria for minimality and non-minimality of some lattices for $E\_f$ at fixed scale based on the sign of the inverse Laplace transform of $f$ when $f$ is a superposition of exponentials, beyond the class of completely monotone functions. We also construct a family of non-completely monotone functions having the triangular lattice as the unique minimizer of $E\_f$ at any scale. For Lennard-Jones type potentials, we reduce the minimization problem among all Bravais lattices to a minimization over the smaller space of unit-density lattices and we establish a link to the maximum kissing problem. New numerical evidence for the optimality of particular lattices for all the exponents are also given. We finally design one-well potentials $f$ such that the square lattice has lower energy $E\_f$ than the triangular one. Many open questions are also presented.