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This article is cited in 9 scientific papers (total in 9 papers)
Continuous and Discrete (Classical) Heisenberg Spin Chain Revised
Orlando Ragniscoab, Federico Zulloab a Istituto Nazionale di Fisica Nucleare Sezione di Roma Tre, Via Vasca Navale 84, I-00146 Roma, Italy
b Dipartimento di Fisica, Universittà di Roma Tre
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 $N\times N$ matrix whose eigenvalues are the $N^{\rm th}$ 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 $N\times N$ generalization of the CHSC through the reduction technique for Poisson–Nijenhuis manifolds, and exhibit some explicit, and hopefully interesting, examples for $3\times 3$
and $4\times 4$ 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.
Keywords:
integrable systems; Heisenberg chain; Poisson–Nijenhuis manifolds; geometric reduction; $R$-matrix; modified Yang–Baxter.
Received: December 29, 2006; Published online February 26, 2007
Citation:
Orlando Ragnisco, Federico Zullo, “Continuous and Discrete (Classical) Heisenberg Spin Chain Revised”, SIGMA, 3 (2007), 033, 6 pp.
Linking options:
https://www.mathnet.ru/eng/sigma159 https://www.mathnet.ru/eng/sigma/v3/p33
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