Analytic solution of the domain-wall nonequilibrium stationary state

We consider the out-of-equilibrium dynamics generated by joining two domains with arbitrary opposite magnetizations. We study the stationary state which emerges by the unitary evolution via the spin-1/2 $XXZ$ Hamiltonian, in the gapless regime, where the system develops a stationary spin current. Using the generalized hydrodynamic approach, we present a simple formula for the space-time profile of the spin current and the magnetization exact in the limit of long times. As a remarkable effect, we show that the stationary state has a strongly discontinuous dependence on the strength of interaction as confirmed by the exact analytic expression of the Drude weight that we compute. These features allow us to give a qualitative estimation for the transient behavior of the current which is in good agreement with numerical simulations. Moreover, we analyze the behavior around the edge of the magnetization profile, and we argue that, unlike the $XX$ free-fermionic point, interactions always prevent the emergence of a Tracy-Widom scaling.

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