# Finite-size scaling of out-of-time-ordered correlators at late times

## TL;DR

This paper investigates how out-of-time-ordered correlators behave at late times in finite quantum many-body systems, revealing that their residual values scale differently depending on energy conservation, thus offering insights into quantum chaos.

## Contribution

It demonstrates that the late-time saturation value of out-of-time-ordered correlators scales polynomially with system size when energy is conserved, contrasting with exponential decay in non-conserving systems.

## Key findings

- Residual correlator value scales as inverse polynomial with system size when energy is conserved.
- Numerical simulations support analytical predictions about late-time behavior.
- Provides new understanding of quantum chaos signatures in finite systems.

## Abstract

Chaotic dynamics in quantum many-body systems scrambles local information so that at late times it can no longer be accessed locally. This is reflected quantitatively in the out-of-time-ordered correlator of local operators, which is expected to decay to zero with time. However, for systems of finite size, out-of-time-ordered correlators do not decay exactly to zero and in this paper we show that the residual value can provide useful insights into the chaotic dynamics. When energy is conserved, the late-time saturation value of the out-of-time-ordered correlator of generic traceless local operators scales as an inverse polynomial in the system size. This is in contrast to the inverse exponential scaling expected for chaotic dynamics without energy conservation. We provide both analytical arguments and numerical simulations to support this conclusion.