Optimality of the triangular lattice for Lennard-Jones type lattice energies: a computer-assisted method

TL;DR

This paper introduces a computer-assisted method to rigorously prove that the triangular lattice is the optimal configuration for Lennard-Jones type potentials in two dimensions, with improved bounds and broader applicability.

Contribution

A novel computer-assisted approach to establish the global minimality of the triangular lattice for Lennard-Jones potentials, including new bounds and conjectures for general parameters.

Findings

Proved the triangular lattice is globally minimal for the classical (12,6) Lennard-Jones potential.

Derived improved bounds on the inverse density for lattice minimality.

Extended results to other exponents and proposed a new conjecture for universal optimality.

Abstract

It is well-known that any Lennard-Jones type potential energy must have a periodic ground state given by a triangular lattice in dimension 2. In this paper, we describe a computer-assisted method that rigorously shows such global minimality result among $2$-dimensional lattices once the exponents of the potential have been fixed. The method is applied to the widely used classical $(12,6)$ Lennard-Jones potential, which is the main result of this work. Furthermore, a new bound on the inverse density (i.e. the co-volume) for which the triangular lattice is minimal is derived, improving those found in [L. B\'etermin and P. Zhang, \textit{Commun. Contemp. Math.}, 17 (2015), 1450049] and [L. B\'etermin, \textit{SIAM J. Math. Anal.}, 48 (2016), 3236--3269]. The same results are also shown to hold for other exponents as additional examples and a new conjecture implying the global optimality of a triangular lattice for any parameters is stated.