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arXiv:0807.4391 · Symmetry Integrability and Geometry Methods and Applications

Tridiagonal Symmetries of Models of Nonequilibrium Physics

Boyka Aneva

Published Jul 28, 2008• 5 citations
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Abstract

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.

DOI
10.3842/sigma.2008.056

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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.
We address the problem of exactly describing stochastic nonequilibrium systems that are widely used to model one-dimensional transport, in biology, traffic flow and others. We review the matrix product states ansatz to interacting multiparticle systems and its extension to a tridiagonal (generalized Onsager) algebra approach. The stationary probability distribution is expressed as a matrix product state with respect to a quadratic algebra defined by the dynamics of the process. The states involved in the matrix dements 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. The importance of the boundary conditions manifests itself in the fact that the boundary operators are 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 for particular values of the structure, constants. Previously known representations, both infinite dimensional and finite dimensional ones, are recovered within the tridiagonal framework. The boundary Askey-Wilson and. tridiagonal symmetry is the deep algebraic property of driven diffusive systems allowing for the exact solvability in the steady state and the exact description of the stochastic dynamics.
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.
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.
The sine-Gordon model and affine Toda field theories on the half-line, on the one hand, the XXZ spin chain with nondiagoual boundary terms, and interacting many-body lattice systems with a flow, on the other, have a common characteristic. They possess nonlocal conserved boundary charges, generating the Askey-Wilson algebra, a coideal subalgebra of the bulk quantized affine symmetry. We argue that the boundary Askey-Wilson symmetry is the deep algebraic property allowing for integrability of the physical system in consideration.