We review some recent developments in the statistical mechanics of isolated
quantum systems. We provide a brief introduction to quantum thermalization,
paying particular attention to the `Eigenstate Thermalization Hypothesis'
(ETH), and the resulting `single-eigenstate statistical mechanics'. We then
focus on a class of systems which fail to quantum thermalize and whose
eigenstates violate the ETH: These are the many-body Anderson localized
systems; their long-time properties are not captured by the conventional
ensembles of quantum statistical mechanics. These systems can locally remember
forever information about their local initial conditions, and are thus of
interest for possibilities of storing quantum information. We discuss key
features of many-body localization (MBL), and review a phenomenology of the MBL
phase. Single-eigenstate statistical mechanics within the MBL phase reveals
dynamically-stable ordered phases, and phase transitions among them, that are
invisible to equilibrium statistical mechanics and can occur at high energy and
low spatial dimensionality where equilibrium ordering is forbidden.Comment: Updated to reflect recent development
