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arXiv:1112.0839 · Journal of Statistical Mechanics Theory and Experiment

On factorizingF-matrices in \mathcal {Y}(sl_n) and \mathcal {U}_q(\widehat {sl_n}) spin chains

S G McAteer, M Wheeler

Published Apr 30, 2012• 2 citations
arXivHTMLPDF

Abstract

We consider quantum spin chains arising from N -fold tensor products of the fundamental evaluation representations of and . Using the partial F -matrix formalism from the seminal work of Maillet and Sanchez de Santos, we derive a completely factorized expression for the F -matrix of such models and prove its equivalence to the expression obtained by Albert, Boos, Flume and Ruhlig. A new relation between the F -matrices and the Bethe eigenvectors of these spin chains is given.

DOI
10.1088/1742-5468/2012/04/p04016

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Abstract We study the glm∣n {\mathfrak{gl}}_{m\mid n} glm∣n​ gl m ∣ n XXX spin chains defined on tensor products of highest weight glm∣n {\mathfrak{gl}}_{m\mid n} glm∣n​ gl m ∣ n -modules. We show that the on-shell Bethe vectors are eigenvectors of higher transfer matrices and compute the corresponding eigenvalues. Then we take the classical limits and obtain the corresponding results for the glm∣n {\mathfrak{gl}}_{m\mid n} glm∣n​ gl m ∣ n Gaudin models.
Extending the method proposed in [1], we derive QQ-relations (functional relations among Baxter Q-functions) and T-functions (eigenvalues of transfer matrices) for fusion vertex models associated with the twisted quantum affine superalgebras Uq(gl(2r+1|2s)(2)), Uq(gl(2r|2s+1)(2)), Uq(gl(2r|2s)(2)), Uq(osp(2r|2s)(2)) and the untwisted quantum affine orthosymplectic superalgebras Uq(osp(2r+1|2s)(1)) and Uq(osp(2r|2s)(1)) (and their Yangian counterparts, Y(osp(2r+1|2s)) and Y(osp(2r|2s))) as reductions (a kind of folding) of those associated with Uq(gl(M|N)(1)). In particular, we reproduce previously proposed generating functions (difference operators) of the T-functions for the symmetric or anti-symmetric representations, and tableau sum expressions for more general representations for orthosymplectic superalgebras [2], [3], and obtain Wronskian-type expressions (analogues of Weyl-type character formulas) for them. T-functions for spinorial representations are related to reductions of those for asymptotic limits of typical representations of Uq(gl(M|N)(1)).