Abstract
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.
Highlights
Gaudin systems have been a subject of study for almost half a century
In Appendix D we present the key formula in the proof of the offshell action of the generating function of the trigonometric Gaudin Hamiltonians with boundary terms
As we have shown in [38] the trigonometric s (2) Gaudin Hamiltonians with the boundary terms are obtained as the residues of the generating function τ (λ) at poles λ = ±αm
Summary
Gaudin systems have been a subject of study for almost half a century. Gaudin originally introduced them as a quasi-classical limit of the Heisenberg spin chains [1,2,3]. We faced serious obstacles in pursuing these goals: the straightforward approach adapted from the rational case was failing to produce either KZ solutions or norm/scalar product formulas It was not before we noticed and employed a combination of freedom in defining the local realization of Gaudin algebra generators and a freedom in defining the effective creation operators for the Bethe vectors, that we could achieve our established objectives. The two formulations are equivalent, the motivation for making the present choice is related to the forementioned freedom in the local realization of the new set of generators of the generalized trigonometric s (2) Gaudin algebra This approach yields a neat form of the off-shell action of the generating function on the Bethe vectors while at the same time enables the quest for the solutions to the corresponding Knizhnik-Zamolodchikov equations. In Appendix D we present the key formula in the proof of the offshell action of the generating function of the trigonometric Gaudin Hamiltonians with boundary terms
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