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.
Categories: math.AP