Universal Logarithmic Scrambling in Many Body Localization

TL;DR

This paper analytically studies the out-of-time-ordered correlator (OTOC) in many-body localized systems, revealing a universal logarithmic scrambling behavior distinct from quantum chaos and Anderson localization.

Contribution

It provides an analytical solution for OTOC in MBL systems, uncovering universal logarithmic decay and growth patterns, and relates scrambling rate to localization length.

Findings

OTOC decays when operator separation is less than $\xi\, ext{ln}\,t$

OTOC exhibits universal power law decay $2^{-\xi\, ext{ln}\,t}$

Logarithmic growth of second Rényi entropy linked to localization length

Abstract

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.