Modified algebraic Bethe ansatz for XXZ chain on the segment - I - triangular cases

TL;DR

This paper applies a modified algebraic Bethe ansatz to analyze the spectral problem of the XXZ spin-1/2 chain with triangular boundary conditions, providing conjectures for eigenvalues and eigenvectors and a formula for Bethe vectors.

Contribution

It introduces a modified algebraic Bethe ansatz approach for the XXZ chain with triangular boundaries and conjectures spectral properties, including eigenvalues, eigenvectors, and Bethe vectors.

Findings

Eigenvalues and eigenvectors characterized by Bethe roots with additional terms.

Conjectured spectral solutions based on small chain analysis.

A factorized formula for Bethe vectors with two upper triangular boundaries.

Abstract

The modified algebraic Bethe ansatz, introduced by Cramp\'e and the author [8], is used to characterize the spectral problem of the Heisenberg XXZ spin-$\frac{1}{2}$ chain on the segment with lower and upper triangular boundaries. The eigenvalues and the eigenvectors are conjectured. They are characterized by a set of Bethe roots with cardinality equal to $N$ the length of the chain and which satisfies a set of Bethe equations with an additional term. The conjecture follows from exact results for small chains. We also present a factorized formula for the Bethe vectors of the Heisenberg XXZ spin-$\frac{1}{2}$ chain on the segment with two upper triangular boundaries.