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

The transition between many-body localized states and the delocalized thermal states is an eigen-state phase transition at finite energy density outside the scope of conventional quantum statistical mechanics. In this work we investigate the properties of the transition by studying the behavior of the entanglement 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. The many-body eigenstates are obtained by exact diagonalization of a disordered quantum spin chain under twisted boundary conditions to reduce the finite-size effect. We present a scaling theory based on the assumption that the transition is continuous and use the subsystem size $L_A /\xi$ as the scaling variable, where $\xi$ is the correlation length. We show that this scaling theory provides an effective description of the critical behavior and that the entanglement entropy follows the thermal volume law at the transition point. We extract the critical exponent governing the divergence of $\xi$ upon approaching the transition point. We also study the participation entropy in the spin-basis of the domain wall excitations and show that the transition point and the critical exponent agree with those obtained from finite size scaling of the entanglement entropy. Our findings suggest that the many-body localization transition in this model is continuous and describable as a localization transition in the many-body configuration space.