  We study the statistical and dynamical aspects of a translation-invariant
Hamiltonian, without quench disorder, as an example of the manifestation of the
phenomenon of many-body localization. This is characterized by the breakdown of
thermalization and by information preservation of initial preparations at long
times. To realize this, we use quasi-periodic long-range interactions, which
are now achievable in high-finesse cavity experiments, to find evidence
suggestive of a divergent time-scale in which charge inhomogeneities in the
initial state survive asymptotically. This is reminiscent of a glassy behavior,
which appears in the ground-state of this system, being also present at
infinite temperatures.
