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

Abstract

Let $z=x+iy \in \mathbb {H}:=\{z= x+ i y\in \mathbb {C}: y>0\}$ 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 $L =\sqrt{\frac{1}{{\text {Im}}(z)}}\left( {\mathbb Z}\oplus z{\mathbb Z}\right)$. In this paper we consider the following minimization problem of difference of two theta functions

$$\begin{aligned} \min _{ \mathbb {H} } \left( \theta (\alpha ; z)-\beta \theta (2\alpha ; z)\right) \end{aligned}$$

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 Bétermin (SIAM J Math Anal 48(5):3236–269, 2016), Bétermin (Nonlinearity 31(9):3973–4005, 2018), Bétermin et al. (Models Methods Appl Sci 31(2):293–325, 2021) and Bétermin and Petrache (Anal Math Phys 9(4):2033–2073, 2019) and gives a new proof in the hexagonal crystallization among lattices under Yukawa potential.