Two-parameter families of matrix product operator integrals of motion in Heisenberg spin chains

TL;DR

This paper discovers two-parameter families of matrix product operator integrals of motion in Heisenberg spin chains, extending previous one-parameter solutions and exploring their limits and applications.

Contribution

It introduces new two-parameter MPO families of integrals of motion for various anisotropic Heisenberg models, expanding the understanding of their integrability.

Findings

Found two-parameter MPO integrals of motion for multiple Heisenberg models.

Connected solutions through limiting procedures.

Derived Floquet charges for certain protocols.

Abstract

Recently, Fendley et al. (2025) [arXiv:2511.04674] revealed a new simple 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 with bond dimension 4. In this work, I report on the discovery of two-parameter families of MPOs that commute with Heisenberg spin chain Hamiltonian in case of various anisotropies (XXX, XXZ, XX, XY and XYZ). These solutions are connected by taking appropriate limits. For all cases except XYZ, I also write down Floquet charges of two-step Floquet protocols corresponding to the Trotterization. I describe a symbolic algebra approach for finding such integrals of motion and speculate about possible generalizations and applications.