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The transition between many-body localized states and the delocalized thermalstates is an eigen-state phase transition at finite energy density outside thescope of conventional quantum statistical mechanics. In this work weinvestigate the properties of the transition by studying the behavior of theentanglement entropy of a subsystem of size $L_A$ in a system of size $L > L_A$near the critical regime of the many-body localization transition. Themany-body eigenstates are obtained by exact diagonalization of a disorderedquantum spin chain under twisted boundary conditions to reduce the finite-sizeeffect. We present a scaling theory based on the assumption that the transitionis continuous and use the subsystem size $L_A /\xi$ as the scaling variable,where $\xi$ is the correlation length. We show that this scaling theoryprovides an effective description of the critical behavior and that theentanglement entropy follows the thermal volume law at the transition point. Weextract the critical exponent governing the divergence of $\xi$ uponapproaching the transition point. We also study the participation entropy inthe spin-basis of the domain wall excitations and show that the transitionpoint and the critical exponent agree with those obtained from finite sizescaling of the entanglement entropy. Our findings suggest that the many-bodylocalization transition in this model is continuous and describable as alocalization transition in the many-body configuration space.

Critical behaviors of the entanglement and participation entropy near the many-body localization transition in a disordered quantum spin chain