Minimal Soft Lattice Theta Functions

TL;DR

This paper investigates the minimality properties of new 'soft' theta functions related to lattices, showing that certain minimality results extend from point measures to radially symmetric measures, with applications to various physical lattice structures.

Contribution

It extends minimality results of theta functions from point measures to radially symmetric measures and introduces a method based on a generalized Jacobi transformation formula.

Findings

Radially symmetric measures preserve minimality properties.

Center of honeycomb lattice minimizes the soft theta function.

Applications to physical lattices like triangular and cubic lattices.

Abstract

We study the minimality properties of a new type of "soft" theta functions. For a lattice $L\subset \mathbb{R}^d$, a $L$-periodic distribution of mass $\mu_L$ and an other mass $\nu_z$ centred at $z\in \mathbb{R}^d$, we define, for all scaling parameter $\alpha>0$, the translated lattice theta function $\theta_{\mu_L+\nu_z}(\alpha)$ as the Gaussian interaction energy between $\nu_z$ and $\mu_L$. We show that any strict local or global minimality result that is true in the point case $\mu=\nu=\delta_0$ also holds for $L\mapsto \theta_{\mu_L+\nu_0}(\alpha)$ and $z\mapsto \theta_{\mu_L+\nu_z}(\alpha)$ when the measures are radially symmetric with respect to the points of $L\cup \{z\}$ and sufficiently rescaled around them (i.e. at a low scale). The minimality at all scales is also proved when the radially symmetric measures are generated by a completely monotone kernel. The method is based on a generalized Jacobi transformation formula, some standard integral representations for lattice energies and an approximation argument. Furthermore, for the honeycomb lattice $\mathsf{H}$, the center of any primitive honeycomb is shown to minimize $z\mapsto \theta_{\mu_{\mathsf{H}}+\nu_z}(\alpha)$ and many applications are stated for other particular physically relevant lattices including the triangular, square, cubic, orthorhombic, body-centred-cubic and face-centred-cubic lattices.