Stationary states in a free fermionic chain from the Quench Action Method

We employ the Quench Action Method (QAM) for a recently considered geometrical quantum quench: two free fermionic chains initially at different temperatures are joined together in the middle and let evolve unitarily with a translation invariant Hamiltonian. We show that two different stationary regimes are reached at long times, depending on the interplay between the observation time scale T and the total length L of the system. We show the emergence of a non-equilibrium steady state (NESS) supporting an energy current for observation time T much smaller than the system size L. We then identify a longer time-scale for which thermalization occurs in a Generalized Gibbs Ensemble (GGE).