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arXiv:1705.08953 · Journal of Statistical Mechanics Theory and Experiment

Boundary algebraic Bethe Ansatz for a nineteen vertex model with U q [osp(2|2) (2) ] symmetry

R S Vieira, A Lima-Santos

Published Sep 1, 2017• 1 citation
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Abstract

Abstract The boundary algebraic Bethe Ansatz for a supersymmetric nineteen vertex-model constructed from a three-dimensional representation of the twisted quantum affine Lie superalgebra U q [ o s p ( 2 | 2 ) ( 2 ) ] ≃ U q [ C ( 2 ) ( 2 ) ] is presented. The eigenvalues and eigenvectors of Sklyanin’s transfer matrix, with diagonal reflection K -matrices, are calculated and the corresponding Bethe Ansatz equations are obtained. A numerical analysis for a small-length chain is also presented.

DOI
10.1088/1742-5468/aa85ca

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Abstract The boundary algebraic Bethe Ansatz for a supersymmetric nineteen vertex-model constructed from a three-dimensional representation of the twisted quantum affine Lie superalgebra U q [ o s p ( 2 | 2 ) ( 2 ) ] ≃ U q [ C ( 2 ) ( 2 ) ] is presented. The eigenvalues and eigenvectors of Sklyanin’s transfer matrix, with diagonal reflection K -matrices, are calculated and the corresponding Bethe Ansatz equations are obtained. A numerical analysis for a small-length chain is also presented.
On Bethe Ansatz for a Supersymmetric Vertex Model with Uq[osp(2∣2)(2)]\mathcal{U}_{\rm q}[{\rm osp}(2|2)^{(2)}]Uq​[osp(2∣2)(2)]
The Algebraic Bethe ansatz for a supersymmetric nineteen vertex-model constructed from a three-dimensional representation of the twisted quantum affine Lie superalgebra Uq[osp(2∣2)(2)]\mathcal{U}_{q}[\mathrm{osp}(2|2)^{(2)}]Uq​[osp(2∣2)(2)] is presented in detail. The eigenvalues and eigenvectors of the row-to-row transfer matrix are calculated and the corresponding Bethe Ansatz equations are obtained and analyzed numerically.
It is shown that the transfer matrix of the inhomogeneous nineteen-vertex model with certain diagonal twisted boundary conditions possesses a simple eigenvalue. This is achieved through the identification of a simple and completely explicit solution of its Bethe equations. The corresponding eigenvector is computed by means of the algebraic Bethe ansatz and both a simple component and its square norm are expressed in terms of the Izergin–Korepin determinant. In the homogeneous limit, the vector coincides with a supersymmetry singlet of the twisted spin-one XXZ chain. It is shown that in a natural polynomial normalisation scheme its square norm and the simple component coincide with generating functions for weighted enumeration of alternating sign matrices.