$su(2)$ symmetry of XX spin chains

TL;DR

This work shows that an inhomogeneous open XX spin chain, upon a suitable uniform transverse-field adjustment, exhibits an exact $\;su(2)\;$ symmetry generated by inhomogeneous nonlocal operators $S^x$ and $S^y$ built from a fermionic zero mode. Using the Jordan–Wigner mapping to a free-fermion problem, the authors construct the zero-mode operators and prove the $su(2)$ algebra closes with a constant Casimir $\vec{S}^{\,2}=\tfrac{3}{4}\mathbb{I}$, implying a twofold degeneracy of the spectrum. They provide explicit realizations for two discrete-orthogonal-polynomial chains—the homogeneous chain and the Krawtchouk chain—giving closed-form expressions for the single-particle eigenfunctions $\phi_n(\omega_k)$ and the coefficients $a_n=\phi_n(0)$ that generate the inhomogeneous charges. The results connect to semilocal charges and coproduct-like structures, and suggest avenues for extending the symmetry to XXZ models and to higher-rank algebras, with potential implications for quench dynamics and protected edge phenomena.

Abstract

We show that, after suitably adjusting a uniform transverse magnetic field, the generic inhomogeneous open XX spin chain has a two-fold degeneracy, and an exact $su(2)$ symmetry whose "inhomogeneous" nonlocal generators depend on coefficients that can be explicitly computed for models associated with discrete orthogonal polynomials.