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We investigate the stability of an Anderson localized chain to the inclusionof a single finite interacting thermal seed. This system models the effects ofrare low-disorder regions on many-body localized chains. Above a thresholdvalue of the mean localization length, the seed causes runaway thermalizationin which a finite fraction of the orbitals are absorbed into a thermal bubble.This `partially avalanched' regime provides a simple example of a delocalized,non-ergodic dynamical phase. We derive the hierarchy of length scales necessaryfor typical samples to exhibit the avalanche instability, and show that therequired seed size diverges at the avalanche threshold. We introduce a newdimensionless statistic that measures the effective size of the thermal bubble,and use it to numerically confirm the predictions of avalanche theory in theAnderson chain at infinite temperature.

Avalanche induced co-existing localised and thermal regions in disordered chains