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

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 [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 [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 $p$.