Algebraic Bethe ansatz for open XXX model with triangular boundary matrices

We consider open XXX spins chain with two general boundary matrices submitted to one constraint, which is equivalent to the possibility to put the two matrices in a triangular form. We construct Bethe vectors from a generalized algebraic Bethe ansatz. As usual, the method also provides Bethe equations and transfer matrix eigenvalues.

💡 Research Summary

The paper addresses the integrable open XXX spin‑½ chain with the most general integrable boundary conditions, represented by two reflection (K‑) matrices. In the standard algebraic Bethe ansatz (ABA) framework, exact diagonalisation is straightforward only when the K‑matrices are diagonal or satisfy very restrictive relations; otherwise one must resort to coordinate Bethe ansatz, functional relations, or more elaborate algebraic constructions. The authors identify a single, physically transparent constraint that allows both boundary matrices to be brought simultaneously to a triangular form by a similarity transformation. This “triangularizability” condition is equivalent to a proportionality relation between certain matrix elements of the two K‑matrices and is the only extra assumption beyond integrability.

With the triangular form in hand, the authors rebuild the double‑row monodromy matrix for the open chain. Because the K‑matrices are now upper (or lower) triangular, the monodromy matrix itself acquires an upper‑triangular structure when acting on the reference (pseudovacuum) state. Consequently, the creation operator B(u) can be defined exactly as in the periodic case, but with a modest modification that incorporates the off‑diagonal boundary contributions. The authors introduce a generalized B‑operator, prove that it satisfies the same RTT‑type commutation relations with the diagonal operators A(u) and D(u), and construct Bethe vectors as ordered products of B‑operators acting on the vacuum: \