XYZ integrability the easy way

TL;DR

XYZ spin chain integrability has traditionally been shown via the eight-vertex model and elliptic functions through the Yang-Baxter framework. This work builds an explicit matrix-product operator that generates an extensive hierarchy of charges commuting with the XYZ Hamiltonian for both periodic and open boundaries, including boundary fields and impurities. It also connects the MPO to products of eight-_vertex transfer matrices and derives an R-matrix that satisfies the Yang-Baxter equation, thereby tying the new construction to traditional integrability. The results provide a simpler, self-contained route to integrability, reveal a direct link between the strong zero mode and conserved charges, and enable integrable impurity physics such as a lattice Kondo model in a gapped bulk, with potential implications for Bethe-ansatz based analyses in non-U(1)-invariant systems.

Abstract

Sutherland showed that the XYZ quantum spin-chain Hamiltonian commutes with the eight-vertex model transfer matrix, so that Baxter's subsequent tour de force proves the integrability of both. The proof requires parametrising the Boltzmann weights using elliptic theta functions and showing they satisfy the Yang-Baxter equation. We here give a simpler derivation of the integrability of the XYZ chain by explicitly constructing an extensive sequence of conserved charges from a matrix-product operator. We show that they commute with the XYZ Hamiltonian with periodic boundary conditions or an arbitrary boundary magnetic field. A straightforward generalisation yields impurity interactions that preserve the integrability. Placing such an impurity at the edge gives an integrable generalisation of the Kondo problem with a gapped bulk. We make contact with the traditional approach by relating our matrix-product operator to products of the eight-vertex model transfer matrix.