Title: How a small quantum bath can thermalize long localized chains,
Abstract: We investigate the stability of the many-body localized (MBL) phase for a
system in contact with a single ergodic grain, modelling a Griffiths region
with low disorder. Our numerical analysis provides evidence that even a small
ergodic grain consisting of only 3 qubits can delocalize a localized chain, as
soon as the localization length exceeds a critical value separating localized
and extended regimes of the whole system. We present a simple theory,
consistent with the arguments in [Phys. Rev. B 95, 155129 (2017)], that assumes
a system to be locally ergodic unless the local relaxation time, determined by
Fermi's Golden Rule, is larger than the inverse level spacing. This theory
predicts a critical value for the localization length that is perfectly
consistent with our numerical calculations. We analyze in detail the behavior
of local operators inside and outside the ergodic grain, and find excellent
agreement of numerics and theory.