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, I-00146 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 Poisson-Nijenhuis 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; Poisson-Nijenhuis manifolds; geometric reduction; R-matrix; modified Yang-Baxter.

pdf (186 kb)   ps (140 kb)   tex (10 kb)

References

1. Takhtajan L.A., Integration of the continuous Heisenberg spin chain through the inverse scattering method, Phys. Lett. A 64 (1977), 235-237.
2. Faddeev L.D., Takhtajan L.A., Hamiltonian methods in the theory of solitons, Springer Series in Soviet Mathematics, Berlin, 1987.
3. Orphanidis S.J., SU(N) Heisenberg spin chain, Phys. Lett. A 75 (1980), 304-306.
4. 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, 61-119.
5. 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, 63-97, hep-th/9211111.
6. Bethe H.A., Zur Theorie der Metalle I. Eigenwerte und Eigenfunktionen der linearen Atomkette, Z. Phys. 71 (1931), 205-226.
7. Magri F., Morosi C., A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds, Quaderno di matematica S19, Dipartimento di matematica, Università di Milano, 1984.
8. Magri F., Morosi C., Ragnisco O., Reduction techniques for infinite-dimensional Hamiltonian systems: some ideas and applications, Comm. Math. Phys. 99 (1985), 115-140.
9. Kosmann-Schwarzbach Y., Magri F., Poisson-Nijenhuis structures, Ann. Inst. H. Poincaré, Phys. Theor. 53 (1990), 35-81.
10. Potts R., Some generalized order-disorder transformations, Proc. Camb. Phil. Soc. 48 (1952), 106-109.
11. Wu F.Y., The Potts model, Rev. Modern Phys. 54 (1982), 235-268.
12. Hartmann A.K., Calculation of partition functions by measuring component distributions, Phys. Rev. Lett. 94 (2005), 050601, 4 pages, cond-mat/0410583.
13. Ragnisco O., Santini P.M., A unified algebraic approach to integral and discrete evolution equations, Inverse Problems 6 (1990), 441-452.
14. Nijenhuis A., Connection-free 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, 171-190.
15. Kreyszig E., Introductory functional analysis with applications, Wiley Classic Library, 1989.
16. Morosi C., The R-matrix theory and the reduction of Poisson manifolds J. Math. Phys. 33 (1992), 941-952.
17. Morosi C., Tondo G., Yang-Baxter equations and intermediate long wave hierarchies, Comm. Math. Phys 122 (1989), 91-103.
18. Morosi C., Tondo G., Some remarks on the bi-Hamiltonian structure of integral and discrete evolution equations, Inverse Problems 6 (1990), 557-566.