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.

Figure 1

For systems in dimensions $d>1$ with short-range interactions. Upper left: Probability of a sample of size $L$ containing an avalanche, as a function of $J$ and $L$. Upper right: Probability of a typical degree of freedom being incorporated into an avalanche at time $t$, as a function of $J$ and $t$. Lower panel: Probability of a sample of size $L$ containing an avalanche vs rescaled coupling $J/J_c\left(L\right)$ at various $L$. Note the gradual sharpening of the transition as $L$ is increased. These plots illustrate the behavior of the functions Eqs. (12) and (15); however, the quantitative values other than $J/J_c$ are only schematic, as we have set various undetermined constants to unity.

Figure 2

Schematic illustration of the scenario discussed in the main text for MBL transitions $J_c\left(L\right)$ vs system size $L$ in one-dimensional systems with short-range interactions. At the smallest sizes, avalanches are rare and the transition occurs when typical regions are unstable to the proliferation of resonances (green line). At the largest sizes, the transition occurs when an avalanche is able to grow without bound (black line). As one increases the system size, one goes through a regime of intermediate sizes where the bottleneck for thermalization is having a rare region that is large enough to start an avalanche that can grow (red curve). In models with correlations in the disorder, the density of avalanches can be tuned separately from the typical localization length, allowing one in principle to slide the red curve horizontally (Sec. 6). Further, the black and green lines can be shifted relative to one another, e.g., for interacting particles, by tuning the relative strength of hopping and interactions.