
SIGMA 3 (2007), 033, 6 pages nlin.SI/0701006
https://doi.org/10.3842/SIGMA.2007.033
Contribution to the Proceedings of the Coimbra Workshop on
Geometric Aspects of Integrable Systems
Continuous and Discrete (Classical) Heisenberg Spin Chain Revised
Orlando Ragnisco and Federico Zullo
Dipartimento di Fisica, Università di Roma Tre and
Istituto Nazionale di Fisica Nucleare Sezione di Roma Tre, Via
Vasca Navale 84, I00146 Roma, Italy
Received December 29, 2006; Published online February 26, 2007
Abstract
Most of the work done in the past on the integrability
structure of the Classical Heisenberg Spin Chain (CHSC) has been
devoted to studying the su(2) case, both at the continuous and
at the discrete level. In this paper we address the problem of
constructing integrable generalized ''Spin Chains'' models, where
the relevant field variable is represented by a N × N
matrix whose eigenvalues are the N^{th} roots of unity. To
the best of our knowledge, such an extension has never been
systematically pursued. In this paper, at first we obtain the
continuous N × N generalization of the CHSC through the
reduction technique for PoissonNijenhuis manifolds, and exhibit
some explicit, and hopefully interesting, examples for 3 × 3
and 4 × 4 matrices; then, we discuss the much more
difficult discrete case, where a few partial new results are
derived and a conjecture is made for the general case.
Key words:
integrable systems; Heisenberg chain; PoissonNijenhuis manifolds; geometric reduction; Rmatrix; modified YangBaxter.
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