Logarithmic current fluctuations in nonequilibrium quantum spin chains

We study zero-temperature quantum spin chains, which are characterized by a nonvanishing current. For the XX model starting from the initial state parallel to center dot up arrow up arrow up arrow down arrow down arrow down arrow center dot > 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 subleading constant term. We then argue that the logarithmic growth remains valid for the XXZ model in the critical region.