Minimizing Lattice Energy and Hexagonal Crystallization

Abstract

Consider the energy per particle on the lattice given by ${\min }_{\hat{I}›}{\sum }_{\mathbb{P}\in \hat{I}›}{\left|\mathbb{P}\right|}^4{e}^{-\ddot{I}€\hat{I}\pm {|\mathbb{P}|}^2}$, where $\hat{I}\pm >0$ and $\hat{I}›$ is a two dimensional lattice. We prove that for $\hat{I}\pm \ge \frac{3}{2}$, among two dimensional lattices with unit density, such energy minimum is attained at $e^{i\frac{\ddot{I}}{€}3}$, corresponding to the hexagonal lattice. Our result partially answers some open questions proposed by Bétermin.

33 pages. Comments are welcome