Chiral coordinate Bethe ansatz for phantom eigenstates in the open XXZ spin-$\frac12$ chain

TL;DR

This paper develops a generalized coordinate Bethe ansatz for the open XXZ spin-1/2 chain, specifically addressing phantom eigenstates and their invariant subspaces, revealing new structural and symmetry properties.

Contribution

It introduces a novel Bethe ansatz framework for phantom eigenstates, including chiral shock vectors and symmetry analysis, expanding understanding of boundary effects in integrable models.

Findings

Eigenstates expanded in chiral shock vectors with symmetrical structure

Bulk and boundary scattering matrices derived consistent with other methods

Demonstrated the approach with simple case examples

Abstract

We construct the coordinate Bethe ansatz for all eigenstates of the open spin-$\frac12$ XXZ chain that fulfill the phantom roots criterion (PRC). Under the PRC, the Hilbert space splits into two invariant subspaces and there are two sets of homogeneous Bethe ansatz equations (BAE) to characterize the subspaces in each case. We propose two sets of vectors with chiral shocks to span the invariant subspaces and expand the corresponding eigenstates. All the vectors are factorized and have symmetrical and simple structures. Using several simple cases as examples, we present the core elements of our generalized coordinate Bethe ansatz method. The eigenstates are expanded in our generating set and show clear chirality and certain symmetry properties. The bulk scattering matrices, the reflection matrices on the two boundaries and the BAE are obtained, which demonstrates the agreement with other approaches. Some hypotheses are formulated for the generalization of our approach.