On crystallization in the plane for pair potentials with an arbitrary norm

TL;DR

This paper studies two-dimensional crystallization for pair potentials with arbitrary norms, proving minimality of certain lattice patches and exploring phase transitions in energy minimization.

Contribution

It extends crystallization results to arbitrary norms, classifies minimizers, and investigates phase transitions in energy minimization for Lennard-Jones and Epstein zeta potentials.

Findings

Crystallization occurs for any fixed norm with minimizers being triangular or square lattices.

Explicit norms are constructed for which crystallization holds on any given lattice.

Numerical simulations reveal a new phase transition in minimizers with respect to the norm parameter p.

Abstract

We investigate two-dimensional crystallization phenomena, i.e. minimality of a lattice's patch for interaction energies, with pair potentials of type $(x,y)\mapsto V(\|x-y\|)$ where $\|\cdot\|$ is an arbitrary norm on $\mathbb{R}^2$ and $V:\mathbb{R}_+^*\to\mathbb{R}$ is a function. For the Heitmann-Radin sticky disk potential $V=V_{\text{HR}}$, we prove, using Brass' key result from [\textit{Computational Geometry}, 6:195--214, 1996], that crystallization occurs for any fixed norm, with a classification of minimizers and minimal energies according to the kissing number associated to $\|\cdot\|$. The minimizer is proved to be, up to affine transform, a patch of the triangular or the square lattice, which shows how to easily get anisotropy in a crystallization phenomenon. We apply this result to the $p$-norms $\|\cdot\|_p$, $p\geq 1$, which allows us to construct an explicit family of norms for which crystallization holds on any given lattice. We also solve part of a crystallization problem studied in [\textit{Arch. Ration. Mech. Anal.}, 240:987--1053] where points are constrained to be on $\mathbb{Z}^2$. Moreover, we numerically investigate the minimization problem for the energy per point among lattices for the Lennard-Jones potential $V=V_{\text{LJ}}:r\mapsto r^{-12}-2r^{-6}$ as well as the Epstein zeta function associated to a $p$-norm $\|\cdot\|_p$, i.e. when $V=V_s:r\mapsto r^{-s}$, $s>2$. Our simulations show a new and unexpected phase transition for the minimizers with respect to $p$.