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We construct exact steady states of unitary non-equilibrium time evolution inthe gapless XXZ spin-1/2 chain where integrability preserves ballistic spintransport at long-times. We characterize the quasi-local conserved quantitiesresponsible for this feature and introduce a computationally effective way toevaluate their expectation values on generic matrix product initial states. Weemploy this approach to reproduce the long-time limit of local observables inall quantum quenches which explicitly break particle-hole or time-reversalsymmetry. We focus on a class of initial states supporting persistent spincurrents and our predictions remarkably agree with numerical simulations atlong times. Furthermore, we propose a protocol for this model whereinteractions, even when antiferromagnetic, are responsible for the unboundedgrowth of a macroscopic magnetic domain.

Non-equilibrium spin transport in integrable spin chains: persistent currents and emergence of magnetic domains