Out-of-time-order correlation in marginal many-body localized systems

We show that the out-of-time-order correlation (OTOC) (W(t)†V(0)†W(t)V(0)) in many-body localized (MBL) and marginal MBL systems can be efficiently calculated by the spectrum bifurcation renormalization group (SBRG). We find that in marginal MBL systems, the scrambling time tscr follows a stretched exponential scaling with the distance dWV between the operators W and V: tscr∼exp(dWV/l0), which demonstrates Sinai diffusion of quantum information and the enhanced scrambling by the quantum criticality in nonchaotic systems.