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 non-diagonality of the associated linearized 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.