  We review the physics of many-body localization in models with incommensurate
potentials. In particular, we consider one-dimensional quasiperiodic models
with single-particle mobility edges. Although a conventional perspective
suggests that delocalized states act as a thermalizing bath for the localized
states in the presence of of interactions, there is evidence that such systems
can display non-ergodicity. This is in part due to the fact that the
delocalized states do not have any kind of protection due to symmetry or
topology and are thus susceptible to localization. A study of non-interacting
incommensurate models shows that they admit extended, partially extended, and
fully localized many-body states. These models cannot thermalize dynamically
and remain localized upon the introduction of interactions. In particular, for
a certain range of energy, the system can host a non-ergodic extended (i.e.
metallic) phase in which the energy eigenstates violate the eigenstate
thermalization hypothesis (ETH) but the entanglement entropy obeys volume-law
scaling. The level statistics and entanglement growth also indicate the lack of
ergodicity in these models. The phenomenon of localization and non-ergodicity
in a system with interactions despite the presence of single-particle
delocalized states is closely related to the so-called "many-body proximity
effect" and can also be observed in models with disorder coupled to systems
with delocalized degrees of freedom. Many-body localization in systems with
incommensurate potentials (without single-particle mobility edges) have been
realized experimentally, and we show how this can be modified to study the the
effects of such mobility edges. Demonstrating the failure of thermalization in
the presence of a single-particle mobility edge in the thermodynamic limit
would indicate a more robust violation of the ETH.
