Modified algebraic Bethe ansatz for XXZ chain on the segment – I: Triangular cases

The modified algebraic Bethe ansatz, introduced by Crampé and the author [8], is used to characterize the spectral problem of the Heisenberg XXZ spin-12 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-12 chain on the segment with two upper triangular boundaries.