Classical and quantum butterfly effect in nonlinear vector mechanics

TL;DR

This paper explores the connection between classical and quantum butterfly effects in nonlinear vector mechanics with broken $O(N)$ symmetry, demonstrating that their Lyapunov exponents approximately coincide and scale similarly.

Contribution

It analytically and numerically compares classical and quantum Lyapunov exponents in a nonlinear vector system, revealing their approximate equality and shared scaling behavior.

Findings

Lyapunov exponents in classical and quantum cases approximately match.

Both Lyapunov exponents scale as $rac{1.3 imes ext{(constant)} imes oot4rom{T}}{N}$.

Quantum and classical chaos measures show consistent behavior in the studied regime.

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 $\kappa \approx 1.3 \sqrt[4]{\lambda T}/N$ with temperature $T$, number of degrees of freedom $N$, and coupling constant $\lambda$.