Out-of-Time Ordered Correlator for a Chaotic Many-Body Quantum System

TL;DR

This work derives the large-time behavior of the out-of-time-ordered correlator (OTOC) for a chaotic many-body quantum system by employing a random-matrix ensemble within an energy window of width $Δ$ and a universal Hamiltonian parametrization. The ensemble averages of the OTOC $C(t)$ and its symmetrized form ${\cal F}(t)$ reveal a Gaussian temporal decay with a characteristic time scale $t \sim \hbar / Δ$, and the analysis leads to the conjecture $Δ = ħ λ_{max}$, linking quantum spectral fluctuations to the classical Lyapunov exponent. Consequently, the long-time OTOC is governed by the dimensionless product $λ_{max} t$, implying a universal temporal structure across chaotic systems within the semiclassical regime. The study also clarifies that ${\cal F}(t)$ is largely an auxiliary quantity, while providing detailed comparisons to prior results and highlighting the conditions under which the Gaussian decay emerges. These results offer a concrete framework connecting quantum chaos indicators to classical chaos measures via spectral statistics and energy-width scaling.

Abstract

Using the parametric representation of a chaotic many-body quantum system derived earlier, we calculate explicitly the large-time dependence and asymptotic value of the out-of-time correlator (OTOC) of that system. The dependence on time $t$ is determined by $Δt / \hbar$. Here $Δ$ is the energy correlation width within which the Bohigas-Giannoni-Schmit conjecture applies. We conjecture that $Δ$ is universally related to the leading Ljapunov coefficient of the corresponding classical system by $Δ= \hbar λ_{\max}$. Then the large-time behavior of OTOC is given by the dimensionless parameter $λ_{\max} t$.