Multiscale entanglement clusters at the many-body localization phase transition

We study numerically the formation of entanglement clusters across the many-body localization phase transition. We observe a crossover from strong many-body entanglement in the ergodic phase to weak local correlations in the localized phase, with contiguous clusters throughout the phase diagram. Critical states close to the transition have a structure compatible with fractal or multiscale-entangled states, characterized by entanglement at multiple levels: small strongly entangled clusters are weakly entangled together to form larger clusters. The critical point therefore features subthermal entanglement and a power-law distributed cluster size, while the localized phase presents an exponentially decreasing cluster distribution. These results are consistent with some of the recently proposed phenomenological renormalization-group schemes characterizing the many-body localized critical point, and may serve to constrain other such schemes.