Scaling Theory of Entanglement at the Many-Body Localization Transition

We study the universal properties of eigenstate entanglement entropy across the transition between many-body localized (MBL) and thermal phases. We develop an improved real space renormalization group approach that enables numerical simulation of large system sizes and systematic extrapolation to the infinite system size limit. For systems smaller than the correlation length, the average entanglement follows a sub-thermal volume law, whose coefficient is a universal scaling function. Furthermore, the full distribution of entanglement follows a universal scaling form, and exhibits a bimodal structure that produces universal subleading power-law corrections to the leading volume-law. For systems larger than the correlation length, the short interval entanglement exhibits a discontinuous jump across the transition from fully thermal volume-law on the thermal side, to a pure area-law on the MBL side.