# Algebraic Bethe ansatz for the XXZ Heisenberg spin chain with triangular boundaries and the corresponding Gaudin model

## TL;DR

This paper develops an algebraic Bethe ansatz framework for the XXZ Heisenberg spin chain with triangular boundaries, providing explicit Bethe vectors, transfer matrix spectrum, and connections to Gaudin models.

## Contribution

It extends the algebraic Bethe ansatz to arbitrary spin XXZ chains with upper-triangular reflection matrices, including explicit Bethe vectors and off-shell actions.

## Key findings

- Explicit Bethe vectors for the XXZ chain with triangular boundaries.
- Simple off-shell transfer matrix action and spectrum expressions.
- Connection to Gaudin models via quasi-classical limit.

## Abstract

The implementation of the algebraic Bethe ansatz for the XXZ Heisenberg spin chain, of arbitrary spin-$s$, in the case, when both reflection matrices have the upper-triangular form is analyzed. The general form of the Bethe vectors is studied. In the particular form, Bethe vectors admit the recurrent procedure, with an appropriate modification, used previously in the case of the XXX Heisenberg chain. As expected, these Bethe vectors yield the strikingly simple expression for the off-shell action of the transfer matrix of the chain as well as the spectrum of the transfer matrix and the corresponding Bethe equations. As in the XXX case, the so-called quasi-classical limit gives the off-shell action of the generating function of the corresponding trigonometric Gaudin Hamiltonians with boundary terms.