A new type of minimizers in lattice energy and its application

TL;DR

This paper introduces a novel class of minimizers in lattice energy problems, demonstrating the existence of hexagonal to skinny-rhombic minimizers in a specific complex analysis setting.

Contribution

It characterizes a new minimization problem involving lattice sums and proves the existence of previously unreported minimizers with hexagonal to rhombic shapes.

Findings

Existence of hexagonal to rhombic minimizers.

New characterization of lattice energy minimization.

Application to complex lattice sum analysis.

Abstract

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.