Free fermions in the repeated interactions scheme

We discuss collision models made of fermions colliding repeatedly with an extended fermionic system whose hamiltonian is quadratic. After a review of the general theory of collision models, we derive a general Lyapunov dynamical equation that governs the covariance matrix of the fermionic extended system. We solve explicitly, in a wide range of physical parameters, several situations of interest such as the case of a boundary driven chain or the replication of entanglement leading to a rainbow like state.