On lattice hexagonal crystallization for non-monotone potentials

Let $L=\sqrt{\frac{1}{\mathrm{\Im }(z)}}(\mathbb{Z}\oplus z\mathbb{Z})$ where $z\in \mathbb{H}=\{z=x+iy\text{or}(x,y)\in \mathbb{C}:y>0\}$ 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.