A new type of minimizers in lattice energy and its application

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