Energy Concentration in a Two-Dimensional Magnetic Skyrmion Model: Variational Analysis of Lattice and Continuum Theories

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 Γ Γ -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 S² S 2 -valued maps. We then prove a corresponding compactness and Γ Γ -convergence result, thereby bridging the discrete and continuum theories.