  We consider the isotropic XY quantum spin chain in a random external field in
the $z$ direction, with single site distributions given by i.i.d. random
variables times the critical decaying envelope $j^{-1/2}$. Our motivation is
the study of many-body localization. We investigate transport properties in
terms of polynomial Lieb-Robinson (PLR) bounds. We prove a zero-velocity PLR
bound for large disorder strength $\lambda$ and for small $\lambda$ we show a
partial converse, which suggests the existence of non-trivial transport in the
model.
