Classical and quantum butterfly effect in nonlinear vector mechanics

Abstract

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 κ ∼ 4 √ λT /N with temperature T , number of degrees of freedom N , and coupling constant λ.