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 ${ \bar { \mathscr H}}$, the tensor product of auxiliary spaces. By changing the basis in ${ \bar { \mathscr H}}$, 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 [Alcaraz F C and Lazo M J 2006 { \it J. Phys. A: Math. Gen.} \textbf{ 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 V E 1982 { \it Commun. Math. Phys.}, \textbf{ 86} 391.] and show that the change of basis corresponds to a mapping from the six-vertex model to the five-vertex model.
Categories: cond-mat.stat-mech cond-mat.str-el hep-th math-ph math.MP quant-ph