Out-of-Time-Ordered Correlator in Many-Body Localized Systems

In many-body localized systems, propagation of information forms a light cone that grows logarithmically with time. However, local changes in energy or other conserved quantities are expected to spread only within a finite distance. Is it possible to detect the logarithmic light cone using local observables, starting from thermal states or eigenstates? Here we show that this is indeed the case, but only in an out-of-time-ordered way. In particular, we numerically calculate various correlators in a prototypical model of many-body localization. While correlators of local operators A(t = 0) and B(t > 0) ordered in time sequence – e.g., A(t = 0)B(t > 0) – are not sensitive to the unbounded information propagation, those with out-of-time order – e.g., A(t = 0)B(t > 0)A(t = 0)B(t > 0) – can detect the logarithmic light cone. We demonstrate the connection between out-of-time-ordered correlators and the Lieb-Robinson bound in many-body localized systems, and explain how to saturate the latter by measuring the involved commutator on specially designed initial states. Moreover, out-of-time-ordered correlators also allow us to study the temperature dependence of the light cones and we observe an increase of propagation velocity with increasing temperature.