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

TL;DR

This paper demonstrates the stability of the Bloch wall in an anisotropic XY model by applying the Bogomolnyi decomposition in elliptic coordinates, showing it minimizes energy via a BPS bound.

Contribution

It introduces a novel application of the Bogomolnyi-Prasad-Sommerfield construction to analyze stability in a one-dimensional continuum spin model.

Findings

The Bloch wall saturates the Bogomolnyi bound, indicating it is an energy-minimizing solution.

The method simplifies stability analysis despite the non-diagonality of the linearized operator.

The approach extends BPS techniques to one-dimensional models beyond their traditional scope.

Abstract

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.