Many-body localization transition: Schmidt gap, entanglement length, and scaling

Many-body localization has become an important phenomenon for illuminating a potential rift between nonequilibrium quantum systems and statistical mechanics. However, the nature of the transition between ergodic and localized phases in models displaying many-body localization is not yet well understood. Assuming that this is a continuous transition, analytic results show that the length scale should diverge with a critical exponent. >= 2 in one-dimensional systems. Interestingly, this is in stark contrast with all exact numerical studies which find nu similar to 1. We introduce the Schmidt gap, new in this context, which scales near the transition with an exponent nu > 2 compatible with the analytical bound. We attribute this to an insensitivity to certain finite-size fluctuations, which remain significant in other quantities at the sizes accessible to exact numerical methods. Additionally, we find that a physical manifestation of the diverging length scale is apparent in the entanglement length computed using the logarithmic negativity between disjoint blocks.