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Journal of Statistical Mechanics Theory and Experiment

Thesl(2|1)(2)Gaudin magnet with diagonal boundary terms

A. Lima-Santos

Published Jul 16, 2009• 7 citations
Publisher

Abstract

This work is concerned with the quasi-classical limit of the boundary quantum inverse scattering method for the twisted sl (2|1) (2) vertex model with diagonal K -matrices. In this limit Gaudin’s Hamiltonians with diagonal boundary terms are presented and diagonalized.
DOI
10.1088/1742-5468/2009/07/p07025

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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.
Following Sklyanin's proposal in the periodic case, we derive the generating function of the Gaudin Hamiltonians with boundary terms. Our derivation is based on the quasi-classical expansion of the linear combination of the transfer matrix of the XXX Heisenberg spin chain and the central element, the so-called Sklyanin determinant. The corresponding Gaudin Hamiltonians with boundary terms are obtained as the residues of the generating function. By defining the appropriate Bethe vectors which yield strikingly simple off shell action of the generating function, we fully implement the algebraic Bethe ansatz, obtaining the spectrum of the generating function and the corresponding Bethe equations.