Abstract
We discuss a generalised version of Sklyanin's Boundary Quantum Inverse Scattering Method applied to the spin-1/2, trigonometric sl(2) case, for which both the twisted-periodic and boundary constructions are obtained as limiting cases. We then investigate the quasi-classical limit of this approach leading to a set of mutually commuting conserved operators which we refer to as the trigonometric, spin-1/2 Richardson–Gaudin system. We prove that the rational limit of the set of conserved operators for the trigonometric system is equivalent, through a change of variables, rescaling, and a basis transformation, to the original set of trigonometric conserved operators. Moreover, we prove that the twisted-periodic and boundary constructions are equivalent in the trigonometric case, but not in the rational limit.
Highlights
In 1988 Sklyanin proposed the Boundary Quantum Inverse Scattering Method [39]
Boundary constructions are equivalent in the trigonometric case, but not in the rational limit
In this work we have studied the spin-1/2 Richardson–Gaudin system as the quasi-classical limit of a formulation provided by a generalised Boundary Quantum Inverse Scattering Method (BQISM)
Summary
In 1988 Sklyanin proposed the Boundary Quantum Inverse Scattering Method [39]. Based on the Yang–Baxter Equation [4, 30, 49] and the reflection equations [10], this formalism permits the construction of one-dimensional quantum systems with integrable boundary conditions, and the derivation of associated exact Bethe Ansatz solutions. It has since been clarified that Richardson’s solution for the s-wave model, and the conserved operators, may be obtained as the quasi-classical limit of the twisted-periodic rational sl(2) transfer matrix of the Quantum Inverse Scattering Method [27, 46] with generic inhomogeneities. The quasi-classical limit of the Boundary Quantum Inverse Scattering Method was studied by Sklyanin in [38], prior to his more well-known publication [39] Adopting this approach, several authors have implemented constructions to produce generalised versions of Richardson–Gaudin systems [11, 12, 21, 41, 43]. We confirm in the Appendix that the equivalences hold at the level of eigenvalue expressions for the conserved operators
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