Algebraic Bethe ansatz for the XXX chain with triangular boundaries and Gaudin model

TL;DR

This paper develops an algebraic Bethe ansatz approach for the XXX Heisenberg spin chain with triangular boundary conditions, deriving explicit spectra, Bethe equations, and extending results to Gaudin models via quasi-classical limits.

Contribution

It introduces a complete algebraic Bethe ansatz framework for the XXX chain with triangular boundaries and connects it to Gaudin models through quasi-classical analysis.

Findings

Explicit Bethe vectors with simple off-shell transfer matrix action

Derivation of the spectrum and Bethe equations for the model

Extension to Gaudin Hamiltonians via quasi-classical limit

Abstract

We implement fully the algebraic Bethe ansatz for the XXX Heisenberg spin chain in the case when both boundary matrices can be brought to the upper-triangular form. We define the Bethe vectors which yield the strikingly simple expression for the off shell action of the transfer matrix, deriving the spectrum and the corresponding Bethe equations. We explore further these results by obtaining the off shell action of the generating function of the Gaudin Hamiltonians on the Bethe vectors through the so-called quasi-classical limit.