Classical and quantum butterfly effect in nonlinear vector mechanics

We establish the correspondence between the classical and quantum butterfly effects in nonlinear vector mechanics with the broken $O(N)$ symmetry. On one hand, we analytically calculate the out-of-time ordered correlation functions and the quantum Lyapunov exponent using the augmented Schwinger-Keldysh technique in the large-$N$ limit. On the other hand, we numerically estimate the classical Lyapunov exponent in the high-temperature limit, where the classical chaotic behavior emerges. In both cases, Lyapunov exponents approximately coincide and scale as $\kappa \approx 1.3 \sqrt[4]{\lambda T}/N$ with temperature $T$, number of degrees of freedom $N$, and coupling constant $\lambda$.