Derivation of Matrix Product Ansatz for the Heisenberg Chain from Algebraic Bethe Ansatz

TL;DR

This paper derives a matrix product representation for Bethe eigenstates of the Heisenberg chain using algebraic Bethe ansatz, linking it to the six-vertex model and providing explicit finite-dimensional matrices.

Contribution

It introduces a new derivation of the matrix product ansatz for the Heisenberg chain from algebraic Bethe ansatz, connecting it to vertex models and explicit matrix representations.

Findings

Matrix product representation of Bethe states derived

Explicit finite-dimensional matrices obtained

Connection established between vertex models and basis change

Abstract

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.