  Many-body localized (MBL) systems lie outside the framework of statistical
mechanics, as they fail to equilibrate under their own quantum dynamics. Even
basic features of MBL systems such as their stability to thermal inclusions and
the nature of the dynamical transition to thermalizing behavior remain poorly
understood. We study a simple model to address these questions: a two level
system interacting with strength $J$ with $N\gg 1$ localized bits subject to
random fields. On increasing $J$, the system transitions from a MBL to a
delocalized phase on the \emph{vanishing} scale $J_c(N) \sim 1/N$, up to
logarithmic corrections. In the transition region, the single-site eigenstate
entanglement entropies exhibit bi-modal distributions, so that localized bits
are either "on" (strongly entangled) or "off" (weakly entangled) in
eigenstates. The clusters of "on" bits vary significantly between eigenstates
of the \emph{same} sample, which provides evidence for a heterogenous
discontinuous transition out of the localized phase in single-site observables.
We obtain these results by perturbative mapping to bond percolation on the
hypercube at small $J$ and by numerical exact diagonalization of the full
many-body system. Our results imply the MBL phase is unstable in systems with
short-range interactions and quenched randomness in dimensions $d$ that are
high but finite.
