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We consider the non-equilibrium time evolution of piecewise homogeneousstates in the XXZ spin-1/2 chain, a paradigmatic example of an interactingintegrable model. The initial state can be thought as the result of joiningchains with different global properties. Through dephasing, at late times thestate becomes locally equivalent to a stationary state which explicitly dependson position and time. We propose a kinetic theory of elementary excitations andderive a continuity equation which fully characterizes the thermodynamics ofthe model. We restrict ourselves to the gapless phase and consider cases wherethe chains are prepared: 1) at different temperatures; 2) in the ground stateof two different models; 3) in the "domain wall" state. We find excellentagreement (any discrepancy is within the numerical error) between theoreticalpredictions and numerical simulations of time evolution based on TEBDalgorithms. As a corollary, we unveil an exact expression for the expectationvalues of the charge currents in a generic stationary state.

Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents