Froth-like minimizers of a non local free energy functional with competing interactions

TL;DR

This paper analyzes a one-dimensional non-local free energy functional with competing ferromagnetic and antiferromagnetic interactions, revealing froth-like spin profiles with regular alternations and quantifying their properties.

Contribution

It introduces a novel combination of coarse graining and reflection positivity methods to study the emergence of froth-like minimizers in non-local free energy models.

Findings

Confirmation of froth-like minimizers with regular oscillation scale

Quantitative estimates of the profiles' distance from periodic reference profiles

Development of a combined coarse graining and reflection positivity approach

Abstract

We investigate the ground and low energy states of a one dimensional non local free energy functional describing at a mean field level a spin system with both ferromagnetic and antiferromagnetic interactions. In particular, the antiferromagnetic interaction is assumed to have a range much larger than the ferromagnetic one. The competition between these two effects is expected to lead to the spontaneous emergence of a regular alternation of long intervals on which the spin profile is magnetized either up or down, with an oscillation scale intermediate between the range of the ferromagnetic and that of the antiferromagnetic interaction. In this sense, the optimal or quasi-optimal profiles are "froth-like": if seen on the scale of the antiferromagnetic potential they look neutral, but if seen at the microscope they actually consist of big bubbles of two different phases alternating among each other. In this paper we prove the validity of this picture, we compute the oscillation scale of the quasi-optimal profiles and we quantify their distance in norm from a reference periodic profile. The proof consists of two main steps: we first coarse grain the system on a scale intermediate between the range of the ferromagnetic potential and the expected optimal oscillation scale; in this way we reduce the original functional to an effective "sharp interface" one. Next, we study the latter by reflection positivity methods, which require as a key ingredient the exact locality of the short range term. Our proof has the conceptual interest of combining coarse graining with reflection positivity methods, an idea that is presumably useful in much more general contexts than the one studied here.