Local Energy Optimality of Periodic Sets

TL;DR

This paper investigates the local energy optimality of periodic point sets in Euclidean space for Gaussian core models, providing criteria for criticality and demonstrating local optimality of certain sets in high dimensions.

Contribution

It introduces a rigorous framework for analyzing local energy optimality of periodic sets and characterizes criticality in terms of spherical designs, with new results on the optimality of specific families.

Findings

Characterizes $f_c$-critical periodic sets using spherical 2-designs.

Derives Hessian expressions to certify local optimality.

Shows $ extsf{D}^+_n$ sets are locally $f_c$-optimal for large $c$ when $n \\geq 9$.

Abstract

We study the local optimality of periodic point sets in $\mathbb{R}^n$ for energy minimization in the Gaussian core model, that is, for radial pair potential functions $f_c(r)=e^{-c r}$ with $c>0$. By considering suitable parameter spaces for $m$-periodic sets, we can locally rigorously analyze the energy of point sets, within the family of periodic sets having the same point density. We derive a characterization of periodic point sets being $f_c$-critical for all $c$ in terms of weighted spherical $2$-designs contained in the set. Especially for $2$-periodic sets like the family $\mathsf{D}^+_n$ we obtain expressions for the hessian of the energy function, allowing to certify $f_c$-optimality in certain cases. For odd integers $n\geq 9$ we can hereby in particular show that $\mathsf{D}^+_n$ is locally $f_c$-optimal among periodic sets for all sufficiently large~$c$.