On the boundaries of quantum integrability for the spin-1/2 Richardson-Gaudin system

TL;DR

This paper explores the boundaries of quantum integrability in the spin-1/2 Richardson-Gaudin system, analyzing boundary conditions, classical limits, and equivalences between different constructions.

Contribution

It extends Sklyanin's Boundary Quantum Inverse Scattering Method to the trigonometric case and establishes equivalences and differences with the rational limit.

Findings

The rational limit of conserved operators is equivalent to the trigonometric set after transformations.

Twisted-periodic and boundary constructions are equivalent in the trigonometric case.

The boundary constructions differ in the rational limit.

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.