Breakdown of the Generalized Gibbs Ensemble for Current-Generating Quenches

We establish a relation between two hallmarks of integrable systems: the relaxation towards the generalized Gibbs ensemble (GGE) and the dissipationless charge transport. We show that the former one is possible only if the so-called Mazur bound on the charge stiffness is saturated by local conserved quantities. As an example we show how a non-GGE steady state with a current can be generated in the one-dimensional model of interacting spinless fermions with a flux quench. Moreover, an extended GGE involving the quasilocal conserved quantities can be formulated for this case.