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Journal of Physics A Mathematical and General

Off-shell Bethe ansatz for the XXZ vertex model and solution of the trigonometric Knizhnik-Zamolodchikov equations

H. Babujian

Published Dec 7, 1994• 10 citations
Publisher

Abstract

We prove that the wavevector of the off-shell Bethe ansatz equation for the inhomogeneous XXZ vertex model in the quasiclassical limit gives the solution of the trigonometric Knizhnik-Zamolodchikov equation. We also observe that this solution is the quasiclassical limit of the solution of the quantum Knizhnik-Zamolodchikov equation corresponding to U q (sll 2 ).

DOI
10.1088/0305-4470/27/23/018

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We prove that the wavevector of the off-shell Bethe ansatz equation for the inhomogeneous XXZ vertex model in the quasiclassical limit gives the solution of the trigonometric Knizhnik-Zamolodchikov equation. We also observe that this solution is the quasiclassical limit of the solution of the quantum Knizhnik-Zamolodchikov equation corresponding to U q (sll 2 ).
The semiclassical limit of the algebraic Bethe Ansatz for the Izergin-Korepin 19-vertex model is used to solve the theory of Gaudin models associated with the twisted A2(2)A_{2}^{(2)}A2(2)​ R-matrix. We find the spectra and eigenvectors of the N−1N-1N−1 independents Gaudin Hamiltonians. We also use the off-shell Bethe Ansatz method to show how the off-shell Gaudin equation solves the associated trigonometric system of Knizhnik-Zamolodchikov equations.
We present a comprehensive treatment of the non-periodic trigonometric sℓ(2) Gaudin model with triangular boundary, with an emphasis on specific freedom found in the local realization of the generators, as well as in the creation operators used in the algebraic Bethe ansatz. First, we give Bethe vectors of the non-periodic trigonometric sℓ(2) Gaudin model both through a recurrence relation and in a closed form. Next, the off-shell action of the generating function of the trigonometric Gaudin Hamiltonians with general boundary terms on an arbitrary Bethe vector is shown, together with the corresponding proof based on mathematical induction. The action of the Gaudin Hamiltonians is given explicitly. Furthermore, by careful choice of the arbitrary functions appearing in our more general formulation, we additionally obtain: i) the solutions to the Knizhnik-Zamolodchikov equations (each corresponding to one of the Bethe states); ii) compact formulas for the on-shell norms of Bethe states; and iii) closed-form expressions for the off-shell scalar products of Bethe states.