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

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.

On minima of difference of theta functions and application to hexagonal crystallization

“…In [23] we have considered the following minimization problem of sum of two theta functions min z∈H θ(α; z) + ρθ(α; z + 1 2 ) (1.4) and showed that the hexagonal-rhombic-square-rectangular transition appears as ρ goes from 0 to +∞. This result can also be generalized to completely monotone functions.…”