Characterizing many-body localization by out-of-time-ordered correlation

The out-of-time-ordered (OTO) correlation is unconventional. It is a key quantity for quantifying quantum chaoticity and has been recently used in the investigation of quantum holography. On the other hand, it is difficult to reveal and understand the special features of many-body localization (MBL) by conventional quantities and methods. Here, the authors demonstrate that OTO correlation can provide a pertinent and comprehensive description for MBL. It readily reveals the special logarithmic dynamics of MBL, distinguishes MBL sharply from Anderson localization as well as ergodic states, and allows accurate extraction of localization lengths for MBL. The divergence of the localization length at a finite interaction predicts an ergodic-MBL transition. Furthermore, the infinite-time ``thermal'' fluctuation of the OTO correlation is zero (finite) in the ergodic (MBL) phase and thus can be considered as an order parameter for the ergodic-MBL transition, through which the transition can be identified and characterized. Specifically, the critical point and related critical exponents can be calculated.