Logarithmic current fluctuations in non-equilibrium quantum spin chains

We study zero-temperature quantum spin chains which are characterized by a non-vanishing current. For the XX model starting from the initial state |... + + + - - - ...> we derive an exact expression for the variance of the total spin current. We show that asymptotically the variance exhibits an anomalously slow logarithmic growth; we also extract the sub-leading constant term. We then argue that the logarithmic growth remains valid for the XXZ model in the critical region.