  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.
