Out-of-Time-Order Correlation in Marginal Many-Body Localized Systems

We show that the out-of-time-order correlation (OTOC) $\langle W(t)^\dagger V(0)^\dagger W(t)V(0)\rangle$ 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 $t_\text{scr}$ follows a stretched exponential scaling with the distance $d_{WV}$ between the operators $W$ and $V$: $t_\text{scr}\sim\exp(\sqrt{d_{WV}/l_0})$, which demonstrates Sinai diffusion of quantum information and the enhanced scrambling by the quantum criticality in non-chaotic systems.