Stationary states in a free fermionic chain from the quench action method

We employ the quench action method 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 nonequilibrium steady state 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.