Avalanche induced coexisting localized and thermal regions in disordered chains

We investigate the stability of an Anderson localised chain to the inclusion of a single finite interacting thermal seed. This system models the effects of rare low-disorder regions on many-body localised chains. Above a threshold value of the mean localisation length, the seed causes runaway thermalisation in which a finite fraction of the orbitals are absorbed into a thermal bubble. This `partially avalanched' regime provides a simple example of a delocalised, non-ergodic dynamical phase. We derive the hierarchy of length scales necessary for typical samples to exhibit the avalanche stability, and show that the required seed size diverges at the avalanche threshold. We introduce a new dimensionless statistic that measures the effective size of the thermal bubble, and use it to numerically confirm the predictions of avalanche theory in the Anderson chain at infinite temperature.