Periodic minimizers in 1D local mean field theory

TL;DR

This paper proves the existence and characterizes the structure of minimizers in 1D local mean field models with competing interactions, showing they are either periodic or constant under certain conditions.

Contribution

It extends previous results to mesoscopic free energies, demonstrating the existence and nature of minimizers in 1D systems with competing interactions using reflection positivity.

Findings

Minimizers exist for a class of 1D free-energy models.

All minimizers are either periodic with zero average or of constant sign.

Under convexity conditions, all minimizers are either periodic or constant.

Abstract

Using reflection positivity techniques we prove the existence of minimizers for a class of mesoscopic free-energies representing 1D systems with competing interactions. All minimizers are either periodic, with zero average, or of constant sign. If the local term in the free energy satisfies a convexity condition, then all minimizers are either periodic or constant. Examples of both phenomena are given. This extends our previous work where such results were proved for the ground states of lattice systems with ferromagnetic nearest neighbor interactions and dipolar type antiferromagnetic long range interactions.