Skyrmions and stability of degree ±1 harmonic maps from the plane tothe two-dimensional sphere

Skyrmions are topologically nontrivial patterns in the magnetization of extremely thin ferromagnets. Typically thought of as stabilized by the so-called Dzyaloshinskii-Moriya interaction (DMI), or antisymmetric exchange interaction, arising in such materials, they are of great interest in the physics community due to possible applications in memory devices.

In this talk, I will characterize skyrmions as local minimizers of a two-dimensional limit of the full micromagnetic energy, augmented by DMI and retaining the nonlocal character of the stray field energy. In the regimeof dominating Dirichlet energy, I will provide rigorous predictions for their size and ``wall angles''. The main tool is a quantitative stability result for harmonic maps of degree ± 1 from the plane to the two-dimensional sphere, relating the energy excess of any competitor to the homogeneous H¹-distance to the closest harmonic map.

This is joint work with Anne Bernand-Mantel and Cyrill B. Muratov.