Instability of many-body localized systems as a phase transition in a nonstandard thermodynamic limit

The many-body localization (MBL) phase transition is not a conventional thermodynamic phase transition. Thus, to define the phase transition, one should allow the possibility of taking the limit of an infinite system in a way that is not the conventional thermodynamic limit. We explore this for the so-called avalanche instability due to rare thermalizing regions in the MBL phase for systems with quenched randomness in two cases: for short-range interacting systems in more than one spatial dimension and for systems in which the interactions fall off with distance as a power law. We find an unconventional way of scaling these systems so that they do have a type of phase transition. Our arguments suggest that the MBL phase transition in systems with short-range interactions in more than one dimension (or with sufficiently rapidly decaying power laws) is a transition where entanglement in the eigenstates begins to spread into some typical regions: The transition is set by when the avalanches start. Once this entanglement gets started, the system does thermalize. From this point of view, the much-studied case of one-dimensional MBL with short-range interactions is a special case with a different, and in some ways more conventional, type of phase transition.