Stability of the Bloch wall via the Bogomolnyi decomposition in elliptic coordinates

We consider the one-dimensional anisotropic XY model in the continuum limit. Stability analysis of its Bloch wall solution is hindered by the nondiagonality of the associated linearised operator and the hessian of energy. We circumvent this difficulty by showing that the energy admits a Bogomolnyi bound in elliptic coordinates and that the Bloch wall saturates it – that is, the Bloch wall renders the energy minimum. Our analysis provides a simple but nontrivial application of the BPS (Bogomolnyi - Prasad - Sommerfield) construction in one dimension, where its use is often believed to be limited to reproducing results obtainable by other means.