  When a generic quantum system is prepared in a simple initial condition, it
typically equilibrates toward a state that can be described by a thermal
ensemble. A known exception are localized systems which are non-ergodic and do
not thermalize, however local observables are still believed to become
stationary. Here we demonstrate that this general picture is incomplete by
constructing product states which feature periodic high-fidelity revivals of
the full wavefunction and local observables that oscillate indefinitely. The
system neither equilibrates nor thermalizes. This is analogous to the
phenomenon of weak ergodicity breaking due to many-body scars and challenges
aspects of the current MBL phenomenology, such as the logarithmic growth of the
entanglement entropy. To support our claim, we combine analytic arguments with
large-scale tensor network numerics for the disordered Heisenberg chain. Our
results hold for arbitrarily long times in chains of 160 sites up to machine
precision.
