A new type of minimizers in lattice energy and its application

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