The Commutativity of Integrals of Motion for Quantum Spin Chains and Elliptic Functions Identities

TL;DR

This paper proves the commutativity of certain integrals of motion in quantum spin chains with elliptic interactions and derives new identities for elliptic functions, enhancing understanding of integrable quantum systems.

Contribution

It establishes the commutativity and linear independence of integrals of motion for elliptic quantum spin chains and derives new elliptic function identities.

Findings

Proved commutativity of the first two nontrivial integrals of motion.

Established linear independence for systems with more than 4 spins.

Derived new identities between elliptic Weierstrass functions.

Abstract

We prove the commutativity of the first two nontrivial integrals of motion for quantum spin chains with elliptic form of the exchange interaction. We also show thair linear independence for the numbers of spins larger than 4. As a byproduct, we obtained several identities between elliptic Weierstrass functions of three and four arguments.