New Separation of Variables for the Classical XXX and XXZ Heisenberg Spin Chains

TL;DR

This paper introduces a novel separation of variables approach for classical XXX and XXZ Heisenberg spin chains with degenerate twist matrices, simplifying the equations and linking some variables to action-angle coordinates.

Contribution

It presents a non-standard separation of variables that interchanges classical functions and constructs a new set of separated variables with simpler equations.

Findings

New separated variables satisfy simpler equations.

In certain cases, variables are identified with action-angle coordinates.

Method applies to spin chains with degenerate twist matrices.

Abstract

We propose a non-standard separation of variables for the classical integrable XXX and XXZ spin chains with degenerate twist matrix. We show that for the case of such twist matrices one can interchange the role of classical separating functions $A(u)$ and $B(u)$ and construct a new full set of separated variables, satisfying simpler equation of separation and simpler Abel equations in comparison with the standard separated variables of Sklyanin. We show that for certain cases of the twist matrices the constructed separated variables can be directly identified with action-angle coordinates.