Maximal Theta Functions -- Universal Optimality of the Hexagonal Lattice for Madelung-Like Lattice Energies

TL;DR

This paper proves that the hexagonal lattice uniquely maximizes certain generalized lattice theta functions, establishing its universal optimality among various two-dimensional lattice configurations.

Contribution

It introduces two new families of lattice theta functions and demonstrates the hexagonal lattice's unique maximization property among these functions, extending Montgomery's results.

Findings

Hexagonal lattice uniquely maximizes the new theta functions.

Universal optimality of the hexagonal lattice among charged and shifted lattices.

New generalizations of classical theta functions are introduced.

Abstract

We present two families of lattice theta functions accompanying the family of lattice theta functions studied by Montgomery in [H.~Montgomery. Minimal theta functions. \textit{Glasgow Mathematical Journal}, 30(1):75--85, 1988]. The studied theta functions are generalizations of the Jacobi theta-2 and theta-4 functions. Contrary to Montgomery's result, we show that, among lattices, the hexagonal lattice is the unique maximizer of both families of theta functions. As an immediate consequence, we obtain a new universal optimality result for the hexagonal lattice among two-dimensional alternating charged lattices and lattices shifted by the center of their unit cell.