Continuous and Discrete (Classical) Heisenberg Spin Chain Revised
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 C...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2007 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2007
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/147831 |
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| Zitieren: | Continuous and Discrete (Classical) Heisenberg Spin Chain Revised / O. Ragnisco, F. Zullo // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 18 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862728502466314240 |
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| author | Ragnisco, O. Zullo, F. |
| author_facet | Ragnisco, O. Zullo, F. |
| citation_txt | Continuous and Discrete (Classical) Heisenberg Spin Chain Revised / O. Ragnisco, F. Zullo // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 18 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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 Nth 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.
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| first_indexed | 2025-12-07T19:09:28Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147831 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T19:09:28Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ragnisco, O. Zullo, F. 2019-02-16T09:00:04Z 2019-02-16T09:00:04Z 2007 Continuous and Discrete (Classical) Heisenberg Spin Chain Revised / O. Ragnisco, F. Zullo // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 18 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37K05; 37K10 https://nasplib.isofts.kiev.ua/handle/123456789/147831 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 Nth 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. This paper is a contribution to the Proceedings of the Workshop on Geometric Aspects of Integrable Systems (July 17–19, 2006, University of Coimbra, Portugal). O. Ragnisco would like to thank the organizers of the Geomis workshop, and in particular Joana Nunes da Costa, for the admirable work they have done in preparing and directing the meeting, and for their kind and warm hospitality in Coimbra. Also, O.R. acknowledges illuminating discussions with his long-time colleague and friend Franco Magri. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Continuous and Discrete (Classical) Heisenberg Spin Chain Revised Article published earlier |
| spellingShingle | Continuous and Discrete (Classical) Heisenberg Spin Chain Revised Ragnisco, O. Zullo, F. |
| title | Continuous and Discrete (Classical) Heisenberg Spin Chain Revised |
| title_full | Continuous and Discrete (Classical) Heisenberg Spin Chain Revised |
| title_fullStr | Continuous and Discrete (Classical) Heisenberg Spin Chain Revised |
| title_full_unstemmed | Continuous and Discrete (Classical) Heisenberg Spin Chain Revised |
| title_short | Continuous and Discrete (Classical) Heisenberg Spin Chain Revised |
| title_sort | continuous and discrete (classical) heisenberg spin chain revised |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147831 |
| work_keys_str_mv | AT ragniscoo continuousanddiscreteclassicalheisenbergspinchainrevised AT zullof continuousanddiscreteclassicalheisenbergspinchainrevised |