Tridiagonal Symmetries of Models of Nonequilibrium Physics

TL;DR

This paper explores how boundary symmetries in nonequilibrium models, described via matrix product states, are characterized by tridiagonal algebras linked to Askey-Wilson polynomials, enabling exact solutions.

Contribution

It introduces a tridiagonal algebra framework for boundary symmetries in nonequilibrium models, connecting algebraic structures to boundary processes and their solvability.

Findings

Boundary operators generate a tridiagonal algebra.

Symmetric and totally asymmetric processes are limits of partially asymmetric ones.

Tridiagonal algebra provides a symmetry for boundary problems and exact solvability.

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.