Many-body localization for the random XXZ spin chain in fixed energy intervals

A key signature of MBL (many-body localization) is the slow rate at which information spreads. It is shown that the infinite random Heisenberg XXZ spin-$\frac{1}{2}$ chain exhibits slow propagation of information (logarithmic light cone) in any arbitrary but fixed energy interval. The relevant parameter regime, which covers both weak interaction and strong disorder, is determined solely by the energy interval.