Subleading Bounds on Chaos

TL;DR

This paper derives new constraints on quantum chaos indicators, extending the MSS bound, and establishes a sharp upper limit on the Lyapunov exponent for subleading chaos corrections.

Contribution

It introduces an infinite set of bounds on out-of-time-order correlators, including subleading constraints that refine the understanding of chaos saturation.

Findings

MSS bound is part of an infinite hierarchy of constraints.

Exact saturation of MSS bound cannot persist over any finite time.

Sharp bound on the Lyapunov exponent for subleading chaos corrections.

Abstract

Chaos, in quantum systems, can be diagnosed by certain out-of-time-order correlators (OTOCs) that obey the chaos bound of Maldacena, Shenker, and Stanford (MSS). We begin by deriving a dispersion relation for this class of OTOCs, implying that they must satisfy many more constraints beyond the MSS bound. Motivated by this observation, we perform a systematic analysis obtaining an infinite set of constraints on the OTOC. This infinite set includes the MSS bound as the leading constraint. In addition, it also contains subleading bounds that are highly constraining, especially when the MSS bound is saturated by the leading term. These new bounds, among other things, imply that the MSS bound cannot be exactly saturated over any duration of time, however short. Furthermore, we derive a sharp bound on the Lyapunov exponent $\lambda_2 \le \frac{6\pi}{\beta}$ of the subleading correction to maximal chaos.