Out-of-Time Ordered Correlator for a Chaotic Many-Body Quantum System

Using the parametric representation of a chaotic many-body quantum system derived earlier, we calculate explicitly the large-time dependence and asymptotic value of the out-of-time correlator (OTOC) of that system. The dependence on time $t$ is determined by $\mathrm{\Delta }t\mathrm{/}\mathrm{\hslash }$. Here $\mathrm{\Delta }$ is the energy correlation width within which the Bohigas-Giannoni-Schmit conjecture applies. We conjecture that $\mathrm{\Delta }$ is universally related to the leading Ljapunov coefficient of the corresponding classical system by $\mathrm{\Delta }=\mathrm{\hslash }{\lambda }_{\max }$. Then the large-time behavior of OTOC is given by the dimensionless parameter ${\lambda }_{\max }t$.