Energy concentration in a two-dimensional magnetic skyrmion model: variational analysis of lattice and continuum theories

Abstract

We investigate the formation of singularities in a baby Skyrme type energy model, which describes magnetic solitons in two-dimensional ferromagnetic systems. In presence of a diverging anisotropy term, which enforces a preferred background state of the magnetization, we establish a weak compactness of its topological charge density, which converges to an atomic measure with quantized weights. We characterize the $\Gamma$-limit of the energies as the total variation of this measure. In the case of lattice type energies, we first need to carefully define a notion of discrete topological charge for $\mathbb{S}^2$-valued maps. We then prove a corresponding compactness and $\Gamma$-convergence result, thereby bridging the discrete and continuum theories.