A new type of minimizers in lattice energy and its application

Abstract

Let $z\in \mathbb{H}:=\{z=x+iy\in \mathbb{C}:y>0\}$ and $\mathcal{K}(\alpha ;z):={\sum }_{(m,n)\in {\mathbb{Z}}^2}\frac{{|mz+n|}^2}{\Im (z)}{e}^{-\pi \alpha \frac{{|mz+n|}^2}{\Im (z)}}.$ In this paper, we characterize the following minimization problem$:$ ${\min }_{\mathbb{H}}(\mathcal{K}(\alpha ;z)-b\mathcal{K}(2\alpha ;z)).$ We prove that there exist hexagonal to skinny-rhombic minimizers, which is a novel finding in the literature.

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