  When pushed out of equilibrium, generic interacting quantum systems
equilibrate locally and are expected to evolve towards a locally thermal
description despite their unitary time evolution. Systems in which disorder
competes with interactions and transport can violate this expectation by
exhibiting many-body localization. The strength of the disorder with respect to
the other two parameters drives a transition from a thermalizing system towards
a non-thermalizing one. The existence of this transition is well established
both in experimental and numerical studies for finite systems. However, the
stability of many-body localization in the thermodynamic limit is largely
unclear. With increasing system size, a generic disordered system will contain
with high probability areas of low disorder variation. If large and frequent
enough, those areas constitute ergodic grains which can hybridize and thus
compete with localization. While the details of this process are not yet
settled, it is conceivable that if such regions appear sufficiently often, they
might be powerful enough to restore thermalization. We set out to shed light on
this problem by constructing potential landscapes with low disorder regions and
numerically investigating their localization behavior in the Heisenberg model.
Our findings suggest that many-body localization may be much more stable than
anticipated in other recent theoretical works.
