Scrambling dynamics across a thermalization-localization quantum phase transition

We study quantum information scrambling, specifically the growth of Heisenberg operators, in large disordered spin chains using matrix product operator dynamics to scan across the thermalization-localization quantum phase transition. We observe ballistic operator growth for weak disorder, and a sharp transition to a phase with sub-ballistic operator spreading. The critical disorder strength for the ballistic to sub-ballistic transition is well below the many body localization phase transition, as determined from finite size scaling of energy eigenstate entanglement entropy in small chains. In contrast, at the eigenstate localization transition, the operator dynamics changes from a power-law behaviour to logarithmic, which is hard to detect from our numerics. These data are discussed in the context of a universal form for the growing operator shape and substantiated with a simple phenomenological model of rare regions.