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

Abstract

Let z∈ ℍ:={z∈ℂ:ℑ(z)>0} and Λ:=1/√(ℑ(z))(ℤ⊕ zℤ), 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∈(0,√2). For 0<b<√2, an explicit threshold b₁=0.0767767081⋯∈(0,1/10) separates two distinct types of phase transitions. Type I: 0<b<b₁: it admits a type of ``hexagonal--rhombic--square--rectangular'' phase transition. Type II: b₁≤ b<√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<b₁ and vanishes for b≥ b₁. 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.

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Cite

BibTeX RIS

BibTeX

@misc{deng2026role,
  title = {{The role of density in phase transitions and new perspectives on theta functions}},
  author = {Kaixin Deng and Senping Luo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35437},
  eprint = {2610.35437},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35437},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - The role of density in phase transitions and new perspectives on theta functions
AU  - Kaixin Deng
AU  - Senping Luo
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35437
UR  - https://arxiv.org/abs/2610.35437
AB  - Let z∈ ℍ:={z∈ℂ:ℑ(z)>0} and Λ:=1/√(ℑ(z))(ℤ⊕ zℤ), 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∈(0,√2). For 0<b<√2, an explicit threshold b_1=0.0767767081⋯∈(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≤ b<√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≥ 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.
ER  -

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