On lattice hexagonal crystallization for non-monotone potentials

Abstract

Let $L=\sqrt{\frac{1}{\Im (z)}}(\mathbb{Z}\oplus z\mathbb{Z})$ where $z\in \mathbb{H}=\{z=x+iyor(x,y)\in \mathbb{C}:y>0\}$ be the two dimensional lattices with unit density. Assuming that $\alpha \ge 1$, we prove that \begin{equation}\aligned\nonumber \min_{L}\sum_{\mathbb{P}\in L, |L|=1}|\mathbb{P}|^2 e^{- πα|\mathbb{P}|^2} \endaligned\end{equation} is achieved at hexagonal lattice. More generally we prove that for $\alpha \ge 1$ \begin{equation}\aligned\nonumber \min_{L}\sum_{\mathbb{P}\in L, |L|=1}(|\mathbb{P}|^2-\frac{b}α) e^{- πα|\mathbb{P}|^2} \endaligned\end{equation} is achieved at hexagonal lattice for $b\le \frac{1}{2\pi }$ and does not exist for $b>\frac{1}{2\pi }$. As a consequence, we provide two classes of non-monotone potentials which lead to hexagonal crystallization among lattices. Our results partially answer some questions raised in \cite{Oreport, Bet2016, Bet2018, Bet2019AMP} and extend the main results in \cite{LW2022} on minima of difference of two theta functions.

33 pages; comments welcome