Two-parameter families of MPO integrals of motion in Heisenberg spin chains

TL;DR

The paper extends the MPO-based construction of conserved charges in Heisenberg spin chains to two-parameter families with bond dimension 4 for the XXX, XXZ, and XYZ models. Using symmetry constraints and symbolic algebra, it derives explicit MPO tensors $ ext{A}$ and corresponding error terms that yield commuting charges, organizing the results as sphere- and projective-plane parameterizations and showing how local charges arise from expansions of $ ext{M}$. The XXX and XXZ cases yield sphere-parameterized families, while XYZ admits a projective-plane family, plus two one-parameter reductions related to earlier results; the author conjectures that these families capture all local integrals of motion and discusses potential applications to generalized Gibbs ensembles and MPO dynamics. The work provides a practical, algebraic route to constructing MPO charges, with implications for integrability, numerical approaches, and extensions to other one-dimensional models and Floquet-like settings.

Abstract

Recently, Fendley et al. (2025) [arXiv:2511.04674] revealed a new way to demonstrate the integrability of XYZ Heisenberg model by constructing a one-parameter family of integrals of motion in the matrix product operator (MPO) form. In this short note, I report on the discovery of two-parameter families of MPOs that commute with with the Heisenberg spin chain Hamiltonian in the XXX, XXZ, and XYZ cases. I describe a symbolic algebra approach for finding such integrals of motion and speculate about possible applications.