Derivation of the matrix product ansatz for the Heisenberg chain from the algebraic Bethe ansatz

We derive a matrix product representation of the Bethe ansatz state for the XXX and XXZ spin- 1/2 Heisenberg chains using the algebraic Bethe ansatz. In this representation, the components of the Bethe eigenstates are expressed as traces of products of matrices which act on H-bar, the tensor product of auxiliary spaces. By changing the basis in H-bar, we derive explicit finite-dimensional representations for the matrices. These matrices are the same as those appearing in the recently proposed matrix product ansatz by Alcaraz and Lazo (2006 J. Phys. A: Math. Gen. 39 11335) apart from normalization factors. We also discuss the close relation between the matrix product representation of the Bethe eigenstates and the six-vertex model with domain wall boundary conditions (Korepin 1982 Commun. Math. Phys. 86 391) and show that the change of basis corresponds to a mapping from the six-vertex model to the five-vertex model.