Microstructures in a two-dimensional frustrated spin system: Scaling regimes and a discrete-to-continuum limit

TL;DR

This paper investigates pattern formation in a 2D frustrated spin system, deriving a continuum limit and analyzing energy scaling laws, revealing conditions favoring vortex formation.

Contribution

It derives the discrete-to-continuum $ ext{Gamma}$-limit for the $J_1$-$J_3$ spin model at a critical transition, introducing a singularly perturbed multiwell energy functional.

Findings

Vortices can be energetically favorable in certain regimes.

The discrete-to-continuum limit is characterized by a multiwell energy functional.

Scaling laws for minimal energy depend on parameter regimes.

Abstract

We study pattern formation within the $J_1$-$J_3$ - spin model on a two-dimensional square lattice in the case of incompatible (ferromagnetic) boundary conditions on the spin field. We derive the discrete-to-continuum $\Gamma$-limit at the helimagnetic/ferromagnetic transition point, which turns out to be characterized by a singularly perturbed multiwell energy functional on gradient fields. Furthermore, we study the scaling law of the discrete minimal energy. The constructions used in the upper bound include besides rather uniform or complex branching-type patterns also structures with vortices. Our results show in particular that in certain parameter regimes the formation of vortices is energetically favorable.