Out‐of‐time‐order correlations in many‐body localized and thermal phases

We use the out‐of‐time‐order (OTO) correlators to study the slow dynamics in the many‐body localized (MBL) phase. We investigate OTO correlators in the effective (“l‐bit”) model of the MBL phase, and show that their amplitudes after disorder averaging approach their long‐time limits as power‐laws of time. This power‐law dynamics is due to dephasing caused by interactions between the localized operators that fall off exponentially with distance. The long‐time limits of the OTO correlators are determined by the overlaps of the local operators with the conserved l‐bits. We demonstrate numerically our results in the effective model and three other more “realistic” spin chain models. Furthermore, we extend our calculations to the thermal phase and find that for a time‐independent Hamiltonian, the OTO correlators also appear to vanish as a power law at long time, perhaps due to coupling to conserved densities. In contrast, we find that in the thermal phase of a Floquet spin model with no conserved densities the OTO correlator decays exponentially at long times. image