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

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 $\bar{\mathcal{H}}$, the tensor product of auxiliary spaces. By changing the basis in $\bar{\mathcal{H}}$, 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.

24 pages; minor typos are corrected

References in corpus (8)

• Matrix product states represent ground states faithfully
• Entanglement entropy of excited states
• Hidden Grassmann Structure in the XXZ Model II: Creation Operators
• Hidden Grassmann structure in the XXZ model
• Hidden Grassmann Structure in the XXZ Model III: Introducing Matsubara direction
• Derivation of a Matrix Product Representation for the Asymmetric Exclusion Process from Algebraic Bethe Ansatz
• Generalization of the matrix product ansatz for integrable chains
• Dynamical correlation functions of the XXZ model at finite temperature

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