Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 047, 27 pages      arXiv:2006.01417      https://doi.org/10.3842/SIGMA.2020.047

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

Guido Magnano a and Taras Skrypnyk ab
a) Università degli Studi di Torino, via Carlo Alberto 10, 10123, Torino, Italia
b) Bogolyubov Institute for Theoretical Physics, Kyiv, Ukraine

Received December 20, 2019, in final form May 16, 2020; Published online June 02, 2020

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.

Key words: integrable spin chains; quadratic Sklyanin brackets; separation of variables.

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