Uniqueness and radial symmetry of magnetic skyrmions

Abstract

This paper establishes radial symmetry and uniqueness of magnetic skyrmions as global minimizers of the simplest local micromagnetic energy. The strategy is to show that any radial critical point with strictly decreasing out-of-plane component is the unique minimizer in the class of general degree-one maps with the prescribed limit at infinity. To this end, we identify a class of null Lagrangians that is sufficiently flexible to contain a calibration, giving a uniform lower bound for the energy that is uniquely saturated on the critical point. Our arguments provide a blueprint for proofs of symmetry and uniqueness of minimizers in vectorial variational problems under topological constraints.