Out-of-equilibrium systems have nowadays an important role in 1d statistical physics. Although an equilibrium state obviously doesn’t exist for such systems, one looks for a steady state (that is stationary in time). It is defined as the zero-eigenvalue eigenstate of the Markov matrix that describe the evolution of the system. Its exact computation is at the core of many researches.
In some cases, the matrix product state ansatz (matrix ansatz for short) allows to compute this steady state. However no general approach for this ansatz is known. On the other hand, many 1d statistical models appear to be integrable, which allows to get eigenstates of the Markov matrix through Bethe ansatz. The goal of this presentation is to show how integrability gives a natural framework to construct the matrix ansatz for 1d systems with boundaries.
It can be done on very general grounds, allowing to construct the matrix ansatz when it is not known, and also to define new models and/or to find boundary conditions ‘adapted’ to the model under consideration. We will illustrate the technique on some examples.

* Eric Ragoucy, Laboratoire d’Annecy-le-Vieux de Physique Théorique, Annecy, France