Statistical mechanics of the lattice sphere packing problem

TL;DR

This paper introduces an efficient Monte Carlo approach to study lattice sphere packings in high dimensions, discovering densest packings and revealing phase transition behaviors in the statistical mechanics framework.

Contribution

It presents a novel Monte Carlo method for high-dimensional lattice sphere packings and provides new insights into phase transitions in the problem.

Findings

Discovered densest lattice packings in dimensions 9-20.

Evidence of a first-order crystallization transition in high dimensions.

Possible indication of a glass transition at higher dimensions.

Abstract

We present an efficient Monte Carlo method for the lattice sphere packing problem in d dimensions. We use this method to numerically discover de novo the densest lattice sphere packing in dimensions 9 through 20. Our method goes beyond previous methods not only in exploring higher dimensions but also in shedding light on the statistical mechanics underlying the problem in question. We observe evidence of a phase transition in the thermodynamic limit $d\to\infty$. In the dimensions explored in the present work, the results are consistent with a first-order crystallization transition, but leave open the possibility that a glass transition is manifested in higher dimensions.