  We suggest a new indicator of quantum chaos based on the logarithmic
out-of-time-order correlator. On the one hand, this indicator correctly
reproduces the average classical Lyapunov exponent in the semiclassical limit
and directly links the definitions of quantum chaos and classical K-system. On
the other hand, it can be analytically calculated using the replica trick and
the Schwinger-Keldysh diagram technique on a $2n$-fold Keldysh contour. To
illustrate this approach, we consider several one-dimensional systems,
including the quantum cat map, and three paradigmatic large-$N$ models,
including the Sachdev-Ye-Kitaev model. Furthermore, we find that correlations
between replicas can reduce the magnitude of the Lyapunov exponent compared to
estimates based on conventional out-of-time-order correlators.

High Energy Physics - Theory;Statistical Mechanics;Chaotic Dynamics;Quantum Physics