Abstract

I will present an overview of the current results on existence and asymptotic properties of magnetic skyrmions defined as topologically nontrivial maps of degree +1 from the plane to a sphere which minimize a micromagnetic energy containing the exchange, perpendicular magnetic anisotropy  and interfacial Dzyaloshinskii-Moriya interaction (DMI) terms. In ultrathin films, the stray field energy simply renormalizes the anisotropy constant at leading order, but in finite samples it also produces additional non-trivial contributions at the sample edges, promoting nontrivial spin textures. Starting with the whole space problem, I will first discuss the existence of single skyrmions as global energy minimizers at sufficiently small DMI strength. Then, using the quantitative rigidity of the harmonic maps I will present the asymptotic characterization of single skyrmion profiles both in infinite and finite samples. Lastly, I will touch upon the question of existence of multi-skyrmion solutions as minimizers with higher topological degree and present recent existence results obtained jointly with T. Simon and V. Slastikov.