On universally optimal lattice phase transitions and energy minimizers of completely monotone potentials

TL;DR

This paper investigates energy minimization for lattice configurations with completely monotone potentials, revealing universal optimality conditions and phase transitions relevant to physical, biological, and number-theoretic systems.

Contribution

It proves that minima of combined monotone functions on lattices lie on a boundary curve and establishes the square lattice as universally optimal in certain regimes.

Findings

Minima are located on a boundary curve of the fundamental region.

Square lattice is universally optimal within a specific interval.

Results apply to physical systems and number theory, revealing phase transitions.

Abstract

We consider the minimizing problem for energy functionals with two types of competing particles and completely monotone potential on a lattice. We prove that the minima of sum of two completely monotone functions among lattices is located exactly on a special curve which is part of the boundary of the fundamental region. We also establish a universal result for square lattice being the optimal in certain interval, which is surprising. Our result establishes the hexagonal-rhombic-square-rectangular transition lattice shapes in many physical and biological system (such as Bose-Einstein condensates and two-component Ginzburg-Landau systems). It turns out, our results also apply to locating the minimizers of sum of two Eisenstein series, which is new in number theory.