
SIGMA 3 (2007), 033, 6 pages nlin.SI/0701006
http://dx.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.
pdf (186 kb)
ps (140 kb)
tex (10 kb)
References
 Takhtajan L.A., Integration of the continuous Heisenberg spin chain through
the inverse scattering method, Phys. Lett. A 64
(1977), 235237.
 Faddeev L.D., Takhtajan L.A., Hamiltonian methods in the theory of solitons, Springer Series in Soviet Mathematics, Berlin, 1987.
 Orphanidis S.J., SU(N) Heisenberg spin chain, Phys. Lett. A 75 (1980), 304306.
 Kulish P.P., Sklyanin E.K., Quantum spectral transform method,
recent developments, in Integrable Quantum Field Theories,
Lecture Notes in Physics, Vol. 151, Editors J. Hietarinta and
C. Montonen, 1982, 61119.
 Sklyanin E.K., Quantum inverse scattering method. Selected topics, in Quantum Groups and Quantum Integrable Systems, Editor M.L. Ge, Nankai Lectures in Mathematical Physics, World Scientific, Singapore, 1992, 6397,
hepth/9211111.
 Bethe H.A., Zur Theorie der Metalle I. Eigenwerte und Eigenfunktionen der linearen Atomkette, Z. Phys. 71 (1931), 205226.
 Magri F., Morosi C., A geometrical characterization of
integrable Hamiltonian systems through the theory of PoissonNijenhuis
manifolds, Quaderno di matematica S19, Dipartimento di matematica,
Università di Milano, 1984.
 Magri F., Morosi C., Ragnisco O., Reduction techniques for infinitedimensional Hamiltonian systems: some ideas and applications,
Comm. Math. Phys. 99 (1985), 115140.
 KosmannSchwarzbach Y., Magri F.,
PoissonNijenhuis structures, Ann. Inst. H. Poincaré, Phys. Theor. 53 (1990), 3581.
 Potts R., Some generalized orderdisorder transformations, Proc. Camb. Phil. Soc. 48 (1952), 106109.
 Wu F.Y., The Potts model, Rev. Modern Phys. 54 (1982), 235268.
 Hartmann A.K., Calculation of partition functions by measuring component distributions, Phys. Rev. Lett. 94 (2005), 050601, 4 pages, condmat/0410583.
 Ragnisco O., Santini P.M., A unified algebraic approach to
integral and discrete evolution equations, Inverse Problems 6 (1990),
441452.
 Nijenhuis A., Connectionfree differential geometry, in Proc. Conf. Differential Geometry and
Applications, (August 28  September 1, 1995, Masaryk
University, Brno, Czech Republic), Editors J. Janyska, I. Kolar and J. Slovak, Masaryk University Brno, 1996, 171190.
 Kreyszig E., Introductory functional analysis with
applications, Wiley Classic Library, 1989.
 Morosi C.,
The Rmatrix theory and the reduction of Poisson manifolds
J. Math. Phys. 33 (1992), 941952.
 Morosi C., Tondo G.,
YangBaxter equations and intermediate long wave hierarchies,
Comm. Math. Phys 122 (1989), 91103.
 Morosi C., Tondo G.,
Some remarks on the biHamiltonian structure of integral and
discrete evolution equations, Inverse Problems 6
(1990), 557566.

