Matrix-product ansatz as a tridiagonal algebra

Abstract

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.