Minimizing Lattice Energy and Hexagonal Crystallization

TL;DR

This paper proves that for certain parameters, the hexagonal lattice minimizes a specific lattice energy among two-dimensional lattices with unit density, addressing open questions in lattice energy minimization.

Contribution

It establishes that the hexagonal lattice minimizes the given energy for lpha  /2 among two-dimensional lattices with unit density, partially answering open questions.

Findings

Hexagonal lattice minimizes the energy for lpha  /2.

The result applies to two-dimensional lattices with unit density.

Addresses open questions by Be9tre9n.

Abstract

Consider the energy per particle on the lattice given by $\min_{ \Lambda }\sum_{ \mathbb{P}\in \Lambda} \left|\mathbb{P}\right|^4 e^{-\pi \alpha \left|\mathbb{P}\right|^2 }$, where $\alpha >0$ and $\Lambda$ is a two dimensional lattice. We prove that for $\alpha\geq\frac{3}{2}$, among two dimensional lattices with unit density, such energy minimum is attained at $e^{i\frac{\pi}{3}}$, corresponding to the hexagonal lattice. Our result partially answers some open questions proposed by B\'etermin.