Universal Logarithmic Scrambling in Many Body Localization

Out of time ordered correlator (OTOC) is recently introduced as a powerful diagnose for quantum chaos. To go beyond, here we present an analytical solution of OTOC for a non-chaotic many body localized (MBL) system, showing distinct feature from quantum chaos and Anderson localization (AL). The OTOC is found to fall only if the nearest distance between the two operators being shorter than $\xi\ln t$, where $\xi$ is dimensionless localization length. Thereafter, we found an universal power law decay of OTOC as $2^{-\xi\ln t}$, implying an universal logarithmic growth of second R\'{e}nyi entropy, where $\xi$ plays the role of information scrambling rate. A relation between butterfly velocity and scrambling rate is found.