Minimizing Lattice Energy and Hexagonal Crystallization

Consider the energy per particle on the lattice given by ${\min }_{\mathrm{\Lambda }}{\sum }_{\mathbb{P}\in \mathrm{\Lambda }}{\mid \mathbb{P}\mid }^4{e}^{-\pi \alpha {\mid \mathbb{P}\mid }^2}$, where $\alpha >0$ and $\mathrm{\Lambda }$ is a two dimensional lattice. We prove that for $\alpha \ge \frac{3}{2}$, 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 Bétermin.