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

Abstract

In this work, singularly perturbed energies arising from discrete spin 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 spin 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.