  Here we study the phase diagram of the Aubry-Andre-Harper model in the
presence of strong interactions as the strength of the quasiperiodic potential
is varied. Previous work has established the existence of many-body localized
phase at large potential strength; here, we find a rich phase diagram in the
delocalized regime characterized by spin transport and unusual correlations. We
calculate the non-equilibrium steady states of a boundary-driven strongly
interacting Aubry-Andre-Harper model by employing the time-evolving block
decimation algorithm on matrix product density operators. From these steady
states, we extract spin transport as a function of system size and
quasiperiodic potential strength. This data shows spin transport going from
superdiffusive to subdiffusive well before the localization transition;
comparing to previous results, we also find that the transport transition is
distinct from a transition observed in the speed of operator growth in the
model. We also investigate the correlation structure of the steady state and
find an unusual oscillation pattern for intermediate values of the potential
strength. The unusual spin transport and quantum correlation structure suggest
multiple dynamical phases between the much-studied thermal and
many-body-localized phases.
