  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.
