On the formation of microstructure and the occurrence of vortices in a singularly perturbed energy related to helimagnetism: a scaling law result

TL;DR

This paper derives a scaling law for minimal energies in a singularly perturbed model related to helimagnetism, revealing conditions under which vortices necessarily form due to boundary incompatibilities and energy contributions.

Contribution

It introduces a novel analysis of energies with non-gradient admissible fields linked to vortices, extending previous results by incorporating curl conditions and a modified ball-construction technique.

Findings

Scaling law for minimal energy with respect to boundary, regularization, and interatomic parameters.

Vortices necessarily form in certain parameter regimes.

Extension of analysis to non-gradient fields with topological singularities.

Abstract

In this work, singularly perturbed energies arising from discrete $J_1$-$J_3$-models are studied. The energies under consideration consist of a non-convex bulk term and a higher-order regularizing term and are subject to incompatible boundary conditions. In contrast to existing results in the literature, in this work, admissible fields are not necessarily gradient fields, instead their curl is linked to topological singularities, so-called vortices, in the discrete $J_1$-$J_3$-model. The main result of this work is a scaling law for the minimal energy with respect to three parameters: one measuring the incompatibility of the boundary conditions, the second measuring the strength of the regularizing term, and the third being related to the interatomic distance in the discrete model. The shown result implies in particular that in certain parameter regimes, minimizers necessarily develop vortices. A key tool in the analysis is a careful modification of the celebrated ball-construction technique that, due to a lack of rigidity, considers simultaneously both the bulk energy and the regularizing term.