Title: Many-Body Localization: Stability and Instability,
Abstract: Rare regions with weak disorder (Griffiths regions) have the potential to
spoil localization. We describe a non-perturbative construction of local
integrals of motion (LIOMs) for a weakly interacting spin chain in one
dimension, under a physically reasonable assumption on the statistics of
eigenvalues. We discuss ideas about the situation in higher dimensions, where
one can no longer ensure that interactions involving the Griffiths regions are
much smaller than the typical energy-level spacing for such regions. We argue
that ergodicity is restored in dimension d > 1, although equilibration should
be extremely slow, similar to the dynamics of glasses.