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

Abstract

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 $\ensuremath{\nu}\ensuremath{\ge}2$ in one-dimensional systems. Interestingly, this is in stark contrast with all exact numerical studies which find $\ensuremath{\nu}\ensuremath{\sim}1$. We introduce the Schmidt gap, new in this context, which scales near the transition with an exponent $\ensuremath{\nu}>2$ 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.