The role of density in phase transitions and new perspectives on theta functions

Abstract

Let $z\in \mathbb{H}:=\{z\in\mathbb{C}:\Im(z)>0\}$ and $Λ:=\frac{1}{\sqrt{\Im(z)}}\big({\mathbb Z}\oplus z{\mathbb Z}\big)$, and set $ θ(α;z):=θ(α;Λ)=\sum_{(m,n)\in\mathbb{Z}^2}e^{-πα\frac{|m+nz|^2}{\Im(z)}}. $ We study the minimization of the functional $θ(α;z)-bθ(2α;z)$ and establish a complete phase diagram for all $α>0$ and $b\in(0,\sqrt{2})$. For $0<b<\sqrt{2}$, an explicit threshold $b_1=0.0767767081\cdots\in(0,1/10)$ separates two distinct types of phase transitions. Type I: $0<b<b_1$: it admits a type of ``hexagonal--rhombic--square--rectangular'' phase transition. Type II: $b_1\le b<\sqrt{2}$: it admits a type of ``hexagonal--square--rectangular'' phase transition, without passing through the rhombic lattices. These phase transitions contrast with Montgomery's theorem \cite{Mon1988}, where the hexagonal lattice minimizes the single theta function at all scales, and the recent Bétermin-Faulhuber-Steinerberger theorem \cite{BFS2026} for translated Gaussian lattice sums, where the hexagonal lattice prevails in a min-max structure. In our setting, the density parameter $α$ governs the sequence of optimal lattice shapes, while the competition parameter $b$ controls the emergence of the rhombic phase, which exists for $b<b_1$ and vanishes for $b\ge b_1$. As an application, we resolve Bétermin's conjecture \cite{Betermin2021AHP}. Bétermin predicted a Type I phase transition for the functional $θ(α;z)-(1/10)θ(2α;z)$ with $b=1/10$. We instead rigorously establish that no rhombic lattice is a global minimizer at any density and prove that the system undergoes a Type II phase transition, thereby settling the conjecture in corrected form.