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Does the integrability condition on the boundary parameters of the asymmetric exclusion process contradict the matrix product Ansatz

Boyka Aneva

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

This paper is due to the question whether with the constraint from the Bethe ansatz solution on the ASEP boundary parameters the matrix product ansazt fails to produce the stationary state. We discuss applicability of the matrix product ansatz from the point of view of the tridiagonal boundary algebra which reveals deep symmetry properties of the ASEP. We argue that the Bethe ansatz integrability constraint is the defining condition for a finite dimensional representation of the boundary algebra when the system reaches a steady state satisfying detailed balance.

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Derivation of a matrix product representation for the asymmetric exclusion process from the algebraic Bethe ansatz

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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.

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arXiv:1106.4712•Journal of Physics A Mathematical and Theoretical•Sep 14, 2011•60 citations
This paper is due to the question whether with the constraint from the Bethe ansatz solution on the ASEP boundary parameters the matrix product ansazt fails to produce the stationary state. We discuss applicability of the matrix product ansatz from the point of view of the tridiagonal boundary algebra which reveals deep symmetry properties of the ASEP. We argue that the Bethe ansatz integrability constraint is the defining condition for a finite dimensional representation of the boundary algebra when the system reaches a steady state satisfying detailed balance.
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.
In the matrix-product states approach to interacting multiparticle systems the stationary probability distribution is expressed as a matrix-product state with respect to a quadratic algebra determined by the dynamics of the process. The states involved in the matrix elements are determined by the boundary conditions. This reflects the intriguing feature of open systems that the bulk behaviour in the steady state strongly depends on the boundary rates. Led by the importance of the boundary conditions we consider the boundary operators as generators of a tridiagonal algebra whose irreducible modules are the Askey–Wilson polynomials. The matrices of the matrix-product ansatz obey the tridiagonal algebraic relations as well for particular values of the structure constants. This suggests the formulation of the steady-state properties in terms of noncommutative matrices generating a tridiagonal Askey–Wilson algebra. The previously known representations, both infinite dimensional and finite dimensional ones, are recovered within the tridiagonal framework.
We study the one-dimensional asymmetric simple exclusion process (ASEP) with generic open boundaries (including current-counting deformation), and obtain the exact solutions of this ASEP via the off-diagonal Bethe ansatz method. In particular, numerical results for the small size asymmetric simple exclusion process indicate that the spectrum obtained by the Bethe ansatz equations is complete. Moreover, we present the eigenvalue of the totally asymmetric exclusion process and the corresponding Bethe ansatz equations.
We study the boundary symmetries of models of nonequilibrium physics where the steady state behaviour strongly depends on the boundary rates.Within the matrix product state approach to many-body systems the physics is described in terms of matrices defining a noncommutative space with a quantum group symmetry.Boundary processes lead to a reduction of the bulk symmetry.We argue that the boundary operators of an interacting system with simple exclusion generate a tridiagonal algebra whose irreducible representations are expressed in terms of the Askey-Wilson polynomials.We show that the boundary algebras of the symmetric and the totally asymmetric processes are the proper limits of the partially asymmetric ones.In all three type of processes the tridiagonal algebra arises as a symmetry of the boundary problem and allows for the exact solvability of the model.