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arXiv:hep-th/9410193 · Journal of Physics A Mathematical and General

Integrable spin chains associated with slq(n) and slp,q(n)

J. Abad, M Rios

Published Jun 21, 1995• 6 citations
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Abstract

The Hopf structure of the central extension of the U q (sl(n)) algebra is considered. The intertwine matrix induces new integrable chain models. We show the relation of these models and the biparametric spin chain sl p,q (n) models. The cases n=2 are n=3 are discussed, and for n=2 we obtain the model of Dasgupta and Chowdhury (1993). The case n=3 is solved by the nested Bethe ansatz method, and the dependence of the Bethe equations on the second parameter introduced is demonstrated.

DOI
10.1088/0305-4470/28/12/006

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The Hopf structure of the central extension of the U q (sl(n)) algebra is considered. The intertwine matrix induces new integrable chain models. We show the relation of these models and the biparametric spin chain sl p,q (n) models. The cases n=2 are n=3 are discussed, and for n=2 we obtain the model of Dasgupta and Chowdhury (1993). The case n=3 is solved by the nested Bethe ansatz method, and the dependence of the Bethe equations on the second parameter introduced is demonstrated.
We describe a representation for Uq(sl(n)^)U_q(\widehat{sl(n)})Uq​(sl(n)​), when qqq is not a root of unity, based on the fundamental representation of sl(n)sl(n)sl(n). As Uq(sl(n))U_q(sl(n))Uq​(sl(n)) has a Hopf algebra structure with a non-commutative co-product, we look for a intertwine matrix RRR that relates two possible definitions of that co-product. We solve cases for n=2n=2n=2 and n=3n=3n=3, and then we generalize for any nnn. We obtain the hamiltonian associated to such matrix RRR, corresponding to a multi-state chain. As the case for n=2n=2n=2 corresponds to the XXZ model with spin 1/21/21/2, for n>2n>2n>2 we have the generalization of the XXZ model to sl(n)sl(n)sl(n). We show the case for n=3n=3n=3 and its solution by Bethe ansatz.
We have recently constructed a large class of open quantum spin chains which have quantum-algebra symmetry and which are integrable. We show here that these models can be exactly solved using a generalization of the analytical Bethe Ansatz (BA) method. In particular, we determine in this way the spectrum of the transfer matrices of the Uq[su(2)]-invariant spin chains associated with A1(1) and A2(2) in the fundamental representation. The quantum-algebra invariance of these models plays an essential role in obtaining these results. The BA equations for these open chains are “doubled” with respect to the BA equations for the corresponding closed chains.
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