Many-Body Quantum Interference and the Saturation of Out-of-Time-Order Correlators

TL;DR

This paper develops a path-integral method to analyze the full time evolution of out-of-time-order correlators in many-body quantum systems, linking early growth to chaos and identifying quantum saturation mechanisms.

Contribution

It introduces a novel path-integral approach to study OTOC dynamics across different regimes, revealing the quantum interference effects responsible for saturation.

Findings

OTOCs grow exponentially up to the Ehrenfest time related to Lyapunov exponent

Beyond Ehrenfest time, quantum interference causes OTOC saturation

Method applies to both large-N and semiclassical limits, including quantum-chaotic systems

Abstract

Out-of-time-order correlators (OTOCs) have been proposed as sensitive probes for chaos in interacting quantum systems. They exhibit a characteristic classical exponential growth, but saturate beyond the so-called scrambling or Ehrenfest time $\tau_{\rm E}$ in the quantum correlated regime. Here we present a path-integral approach for the entire time evolution of OTOCs for bosonic $N$-particle systems. We first show how the growth of OTOCs up to $\tau_{\rm E} = (1/\lambda) \log N$ is related to the Lyapunov exponent $\lambda$ of the corresponding chaotic mean-field dynamics in the semiclassical large-$N$ limit. Beyond $\tau_{\rm E}$, where simple mean-field approaches break down, we identify the underlying quantum mechanism responsible for the saturation. To this end we express OTOCs by coherent sums over contributions from different mean-field solutions and compute the dominant many-body interference term amongst them. Our method further applies to the complementary semiclassical limit $\hbar \rightarrow 0$ for fixed $N$, including quantum-chaotic single- and few-particle systems.