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 $V$: $t_{\text{scr}}\sim \exp (\sqrt{d_{WV}\mathrm{/}{l}_0})$, which demonstrates Sinai diffusion of quantum information and the enhanced scrambling by the quantum criticality in non-chaotic systems.