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Lattice Energy Minimization for the Difference of Theta and Epstein Zeta Functions

Jingxuan Sun,Zhen Song,Wenming Zou

Published 2025 in Studies in applied mathematics (Cambridge)

ABSTRACT

Let z∈H≔{z=x+iy∈C:y>0}$z\in \mathbb {H}\coloneqq \lbrace z=x+iy\in \mathbb {C}:y>0\rbrace$ , and L≔Z⊕zZ$L\coloneqq \mathbb {Z} \oplus z \mathbb {Z}$ be the lattice in R2$\mathbb {R}^2$ . Let θ(α;z)≔∑P∈L∖{0},|L|=1e−απ∥P∥2$\theta (\alpha;z)\coloneqq \sum _{\mathbb {P}\in L\backslash \lbrace 0\rbrace,\,|L|=1}\text{e}^{-\alpha \pi \Vert \mathbb {P}\Vert ^2}$ be the Theta function, and ζ(s;z)≔∑P∈L∖{0},|L|=11∥P∥2s$\zeta (s;z)\coloneqq \sum _{\mathbb {P}\in L\backslash \lbrace 0\rbrace,\,|L|=1}\frac{1}{\Vert \mathbb {P}\Vert ^{2s}}$ be the Epstein zeta function. Motivated by the widely used Buckingham potential V(r)=a1e−απr−a21r6$ V(r)=a_1e^{-\alpha\pi r}-a_2\frac{1}{r^6}$ in physics, in this paper, we explore the lattice minimization problem of minz∈H(ζ(6;z)−bθ(α;z))$\min_{z\in\mathbb{H}}(\zeta(6;z)-b\theta(\alpha;z))$ for any b∈R$b\in\mathbb{R}$ . A key finding of this work is that the coefficient α$\alpha$ significantly influences the optimal lattice configuration, leading to three distinct phase transition patterns as b$ b$ varies from −∞$ -\infty$ to +∞$ +\infty$ : For α=1$ \alpha = 1$ , the optimal lattice undergoes a hexagonal →$\rightarrow$ wide rhombic →$\rightarrow$ square →$\rightarrow$ rectangular transition. For α>α∗=2.39⋯$ \alpha > \alpha ^* = 2.39\cdots$ , the transition follows hexagonal →$\rightarrow$ narrow rhombic. For 1

PUBLICATION RECORD

  • Publication year

    2025

  • Venue

    Studies in applied mathematics (Cambridge)

  • Publication date

    2025-08-01

  • Fields of study

    Not labeled

  • Identifiers
    DOI 10.1111/sapm.70092
  • External record

    Open on Semantic Scholar

  • Source metadata

    Semantic Scholar

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