Multiscale entanglement clusters at the many-body localization phase transition

We numerically study the formation of entanglement clusters across the many-body localization transition. We observe a crossover from strong many-body entanglement in the ergodic phase to weak local correlations in the localized phase, with continuous 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. Upon entering the localized phase, the power-law distribution seems to persist with a varying power that crosses over into a stretched exponent before eventually becoming exponential deep in the localized phase. These results are in agreement with some of the recently proposed phenomenological renormalization-group schemes characterizing the many-body localized critical point, and serve to constrain other such schemes.