Algebraic Bethe ansatz for the trigonometric sl(2) Gaudin model with triangular boundary

TL;DR

This paper develops an algebraic Bethe ansatz approach for the trigonometric sl(2) Gaudin model with boundary conditions, deriving the spectrum and Bethe equations using Sklyanin's method and quasi-classical expansion.

Contribution

It extends the algebraic Bethe ansatz to the trigonometric sl(2) Gaudin model with boundary terms, providing explicit spectrum and Bethe equations.

Findings

Derived the generating function of Gaudin Hamiltonians with boundary terms.

Constructed Bethe vectors with simple off-shell action.

Obtained the spectrum and Bethe equations for the model.

Abstract

In the derivation of the generating function of the Gaudin Hamiltonians with boundary terms, we follow the same approach used previously in the rational case, which in turn was based on Sklyanin's method in the periodic case. Our derivation is centered on the quasi-classical expansion of the linear combination of the transfer matrix of the XXZ 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.