  We introduce and study a toy model for anomalous transport and Griffiths
effects in one dimensional quantum disordered isolated systems near the
Many-Body Localization (MBL) transitions. The model is constituted by a
collection of 1d tight-binding chains with on-site random energies, locally
coupled to a weak GOE-like perturbation, which mimics the effect of thermal
inclusions due to delocalizing interactions by providing a local broadening of
the Poisson spectrum. While in absence of such a coupling the model is
localized as expected for the one dimensional Anderson model, increasing the
coupling with the GOE perturbation we find a delocalization transition to a
conducting one driven by the proliferation of quantum avalanches which does not
fit the standard paradigm of Anderson localization. In particular an
intermediate Griffiths region emerges, where exponentially distributed
insulating segments coexist with a few, rare resonances. Typical correlations
decay exponentially fast, while average correlations decay as stretched
exponential and diverge with the length of the chain, indicating that the
conducting inclusions have a fractal structure and that the localization length
is broadly distributed at the critical point. This behavior is consistent with
a Kosterlitz-Thouless-like criticality of the transition. Transport and
relaxation are dominated by rare resonances and rare strong insulating regions,
and show anomalous behaviors strikingly similar to those observed in recent
simulations and experiments in the bad metal delocalized phase preceding MBL.
In particular, we find sub-diffusive transport and power-laws decay of the
return probability at large times, with exponents that gradually change as one
moves across the intermediate region. Concomitantly, the a.c. conductivity
vanishes near zero frequency with an anomalous power-law.
