BPS soliton-impurity models and supersymmetry

We find supersymmetric extensions of the half-BPS soliton-impurity models in (1+1) dimensions which preserve half of the $\mathcal{N}=1$ supersymmetry. This is related to the fact that in the bosonic sector (i.e., the half-BPS soliton-impurity model), only one soliton (for example, the kink) is a BPS configuration which solves the pertinent Bogomolnyi equation and saturates the topological energy bound. On the other hand, the topological charge conjugate state (the antikink) is not a BPS solution. This means that it obeys the full Euler-Lagrange equation and does not saturate the topological energy bound. The supersymmetric approach also allows us to construct half-BPS soliton-impurity models in (2+1) dimensions. Concretely, in the case of the $CP^1$ model, its BPS impurity generalisation preserves one-quarter of the $\mathcal{N}=2$ SUSY, while for the Abelian Higgs model at critical coupling both impurity generalisations preserving one-quarter (the case of a new, so-called Higgs impurity) as well as one-half of the $\mathcal{N}=2$ SUSY (the case of the previously known magnetic impurity) are possible. We also discuss a possible relation between the BPS $CP^1$-impurity model and the Dzyaloshinskii-Moriya interaction energy.