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Advances in Mathematical Physics
Volume 2012, Article ID 193190, 21 pages
Research Article

Multiparameter Statistical Models from 𝑁 2 × 𝑁 2 Braid Matrices: Explicit Eigenvalues of Transfer Matrices T ( 𝑟 ) , Spin Chains, Factorizable Scatterings for All 𝑁

1Laboratoire de Physique Théorique, Université d'Oran Es-Sénia, 31100 Oran, Algeria
2Institut des Sciences et de la Technologie, Centre Universitaire d'Ain Témouchent, 46000 Ain Témouchent, Algeria
3Centre de Physique Théorique, Ecole Polytechnique, 91128 Palaiseau Cedex, France

Received 23 March 2012; Accepted 28 May 2012

Academic Editor: Yao-Zhong Zhang

Copyright © 2012 B. Abdesselam and A. Chakrabarti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


For a class of multiparameter statistical models based on 𝑁 2 × 𝑁 2 braid matrices, the eigenvalues of the transfer matrix 𝐓 ( 𝑟 ) are obtained explicitly for all ( 𝑟 , 𝑁 ) . Our formalism yields them as solutions of sets of linear equations with simple constant coefficients. The role of zero-sum multiplets constituted in terms of roots of unity is pointed out, and their origin is traced to circular permutations of the indices in the tensor products of basis states induced by our class of 𝐓 ( 𝑟 ) matrices. The role of free parameters, increasing as 𝑁 2 with N, is emphasized throughout. Spin chain Hamiltonians are constructed and studied for all N. Inverse Cayley transforms of the Yang-Baxter matrices corresponding to our braid matrices are obtained for all N. They provide potentials for factorizable S-matrices. Main results are summarized, and perspectives are indicated in the concluding remarks.