  Whether disordered and quasiperiodic many-body quantum systems host a
long-lived localized phase in the thermodynamic limit has been the subject of
intense recent debate. While in one dimension substantial evidence for the
existence of such a many-body localized (MBL) phase exists, the behavior in
higher dimensions remains an open puzzle. In two-dimensional disordered
systems, for instance, it has been argued that rare regions may lead to
thermalization of the whole system through a mechanism dubbed the avalanche
instability. In quasiperiodic systems, rare regions are altogether absent and
the fate of a putative many-body localized phase has hitherto remained largely
unexplored. In this work, we investigate the localization properties of two
many-body quasiperiodic models, which are two-dimensional generalizations of
the Aubry-Andr\'e model. By studying the out-of-equilibrium dynamics of large
systems, we find a long-lived MBL phase, in contrast to random systems.
Furthermore, we show that deterministic lines of weak potential, which appear
in investigated quasiperiodic models, support large-scale transport, while the
system as a whole does not thermalize. Our results demonstrate that
quasiperiodic many-body systems have the remarkable and counter-intuitive
capability of exhibiting coexisting localization and transport properties - a
phenomenon reminiscent of the behavior of supersolids. Our findings are of
direct experimental relevance and can be tested, for instance, using
state-of-the-art cold atomic systems.
