Minimizing Lattice Energy and Hexagonal Crystallization

Abstract

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