A baby--Skyrme model with anisotropic DM interaction: Compact skyrmions revisited

We consider a baby--Skyrme model with Dzyaloshinskii--Moriya interaction (DMI) and two types of potential terms. The model has a close connection with the vacuum functional of fermions coupled with $O(3)$ nonlinear $\bm{n}$-fields and with a constant $SU(2)$ gauge background. The energy functional is derived from the heat-kernel expansion for the fermion determinant. The model possesses normal skyrmions with topological charge $Q = 1$. The restricted version of the model also includes both the weak-compacton case (at the boundary, not continuously differentiable) and genuine-compacton case (continuously differentiable). The model consists of only the Skyrme term, and the DMI provides soliton solutions that are known as \textit{skyrmions without any potential}. The BPS equation in the supersymmetric soliton models implies that the impurity coupling is closely related to the DMI. Therefore, the effect of an exponentially localized DMI is also studied in the present model.