Does scrambling equal chaos?

TL;DR

This paper demonstrates that exponential growth of out-of-time order correlators (OTOCs), often associated with chaos, can occur in integrable systems due to unstable fixed points, challenging the traditional link between scrambling and chaos.

Contribution

It establishes that scrambling can result from local fixed point dynamics, providing a bound on the OTOC Lyapunov exponent and distinguishing it from chaos.

Findings

Exponential OTOC growth can occur in integrable models.

A lower bound on the OTOC Lyapunov exponent depends on local fixed point properties.

Scrambling can be dominated by local fixed point dynamics, not chaos.

Abstract

Focusing on semiclassical systems, we show that the parametrically long exponential growth of out-of-time order correlators (OTOCs), also known as scrambling, does not necessitate chaos. Indeed, scrambling can simply result from the presence of unstable fixed points in phase space, even in a classically integrable model. We derive a lower bound on the OTOC Lyapunov exponent which depends only on local properties of such fixed points. We present several models for which this bound is tight, i.e. for which scrambling is dominated by the local dynamics around the fixed points. We propose that the notion of scrambling be distinguished from that of chaos.