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

Abstract

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 $t_{\text{scr}}$ 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.

6 pages, 4 figures