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 \(\lambda\).