XYZ integrability the easy way

TL;DR

This paper presents a simplified method to establish the integrability of the XYZ quantum spin chain by constructing conserved charges via matrix-product operators, extending to impurity interactions and connecting to traditional approaches.

Contribution

It introduces a straightforward derivation of XYZ integrability using matrix-product operators, avoiding complex elliptic function parametrizations and enabling impurity generalizations.

Findings

Constructed conserved charges that commute with the XYZ Hamiltonian.

Extended integrability to impurity interactions and edge Kondo-like problems.

Connected matrix-product operators to eight-vertex transfer matrices.

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.