Temporal fluctuations of correlators in integrable and chaotic quantum systems

TL;DR

This paper establishes bounds on the decay of temporal fluctuations of correlators in quantum systems, showing exponential decay in chaotic and integrable systems without degeneracies, but polynomial decay in noninteracting models with degeneracies.

Contribution

It provides a unified framework for understanding how temporal fluctuations decay in different quantum systems, challenging their use as chaos indicators.

Findings

Exponential decay of fluctuations in chaotic and integrable systems without degeneracies.

Polynomial decay in noninteracting systems with degeneracies.

Decay of fluctuations is not a reliable chaos metric.

Abstract

We provide bounds on temporal fluctuations around the infinite-time average of out-of-time-ordered and time-ordered correlators of many-body quantum systems without energy gap degeneracies. For physical initial states, our bounds predict the exponential decay of the temporal fluctuations as a function of the system size. We numerically verify this prediction for chaotic and interacting integrable spin-1/2 chains, which satisfy the assumption of our bounds. On the other hand, we show analytically and numerically that for the XX model, which is a noninteracting system with gap degeneracies, the temporal fluctuations decay polynomially with system size for operators that are local in the fermion representation and decrease exponentially in the system size for non-local operators. Our results demonstrate that the decay of the temporal fluctuations of correlators cannot be used as a reliable metric of chaos or lack thereof.