Derivation of a Matrix Product Representation for the Asymmetric Exclusion Process from Algebraic Bethe Ansatz

TL;DR

This paper derives a generalized matrix product representation for the ASEP using algebraic Bethe Ansatz, expressing eigenvectors as traces of non-commuting operators and establishing their algebraic relations.

Contribution

It introduces a novel matrix product ansatz for ASEP derived from algebraic Bethe Ansatz, providing explicit finite-dimensional operator representations.

Findings

Eigenvectors expressed as traces of non-commuting operators

Operators generate a quadratic algebra with explicit relations

Finite dimensional representations of algebra generators provided

Abstract

We derive, using the algebraic Bethe Ansatz, a generalized Matrix Product Ansatz for the asymmetric exclusion process (ASEP) on a one-dimensional periodic lattice. In this Matrix Product Ansatz, the components of the eigenvectors of the ASEP Markov matrix can be expressed as traces of products of non-commuting operators. We derive the relations between the operators involved and show that they generate a quadratic algebra. Our construction provides explicit finite dimensional representations for the generators of this algebra.