Existence of higher degree minimizers in the magnetic skyrmion problem

TL;DR

This paper proves the existence of higher degree skyrmion minimizers in a ferromagnetic model, showing they concentrate on point-like configurations under certain conditions.

Contribution

It introduces a new variational approach to establish the existence of higher degree topological minimizers in magnetic skyrmion models.

Findings

Existence of higher degree minimizers in the model.

Minimizers concentrate on point-like skyrmionic configurations.

The domain size or shape influences the existence of these minimizers.

Abstract

We demonstrate existence of topologically nontrivial energy minimizing maps of a given positive degree from bounded domains in the plane to $\mathbb S^2$ in a variational model describing magnetizations in ultrathin ferromagnetic films with Dzyaloshinskii-Moriya interaction. Our strategy is to insert tiny truncated Belavin-Polyakov profiles in carefully chosen locations of lower degree objects such that the total energy increase lies strictly below the expected Dirichlet energy contribution, ruling out loss of degree in the limits of minimizing sequences. The argument requires that the domain be either sufficiently large or sufficiently slender to accommodate a prescribed degree. We also show that these higher degree minimizers concentrate on point-like skyrmionic configurations in a suitable parameter regime.