On minima of difference of theta functions and application to hexagonal crystallization

Abstract

Let $z=x+iy\in \mathbb{H}:=\{z=x+iy\in \mathbb{C}:y>0\}$ and $\theta (\alpha ;z)={\sum }_{(m,n)\in {\mathbb{Z}}^2}{e}^{-\alpha \frac{\pi }{y}|mz+n{|}^2}$ be the theta function associated with the lattice $L=\mathbb{Z}\oplus z\mathbb{Z}$. In this paper we consider the following minimization problem of difference of two theta functions \begin{equation}\aligned\nonumber \min_{ \mathbb{H} } \Big(θ(α; z)-βθ(2α; z)\Big) \endaligned\end{equation} where $\alpha \ge 1$ and $\beta \in (-\infty ,+\infty )$. We prove that there is a critical value ${\beta }_c=\sqrt{2}$ (independent of $\alpha$) such that if $\beta \le {\beta }_c$, the minimizer is $\frac{1}{2}+i\frac{\sqrt{3}}{2}$ (up to translation and rotation) which corresponds to the hexagonal lattice, and if $\beta >{\beta }_c$, the minimizer does not exist. Our result partially answers some questions raised in \cite{Bet2016, Bet2018, Bet2020, Bet2019AMP} and gives a new proof in the crystallization of hexagonal lattice under Yukawa potential.

29 pages; comments welcome