Stationary State of Integrable Systems in Matrix Product Form

Abstract

Proposed is a method to construct a stationary state of one-dimensional integrable systems in the form of products of matrices. This is a generalization of the so-called “matrix product ansatz (MPA)”. The key idea is that the matrices are chosen to constitute the Zamolodchikov-Faddeev algebra (ZF-algebra) for the R-matrix and the K-matrix, which are the solutions of the Yang-Baxter equation (YBE) and the reflection equation (RE). It is shown that a matrix product state gives a simultaneous stationary state of commuting operators which are expressed in terms of the R-matrices and the K-matrices. As an example, a solution for the isotropic Heisenberg spin chain with non-diagonal boundary fields is given. The connection to the conventional MPA is clarified. Applications to other models and the relationship to the algebraic Bethe ansatz (ABA) are also discussed.