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Subleading Bounds on Chaos

Sandipan Kundu

TL;DR

The paper derives a dispersion relation for regularized thermal OTOCs, demonstrating that chaos bounds extend beyond the MSS bound and form an infinite hierarchy organized by OTOC moments. It introduces moments $\mu_J(t)$ that are positive, bounded, monotonically decreasing, and log-convex for $t \ge t_0$, from which the MSS bound ($\lambda_L \le 2\pi/\beta$) emerges as the leading constraint and subleading bounds (e.g., $\lambda_2 \le 6\pi/\beta$) follow in a hierarchical pattern. The framework unifies chaos bounds with conformal Regge constraints and Schwarzian theory, showing maximal chaos cannot be exact over any finite interval and that subleading corrections are generically required. These results provide a systematic, operator-based set of limits on scrambling in large-$N$ quantum systems with potential applications to holography and CFTs via Regge and pole-skipping analyses.

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 $λ_2 \le \frac{6π}β$ of the subleading correction to maximal chaos.

Subleading Bounds on Chaos

TL;DR

The paper derives a dispersion relation for regularized thermal OTOCs, demonstrating that chaos bounds extend beyond the MSS bound and form an infinite hierarchy organized by OTOC moments. It introduces moments that are positive, bounded, monotonically decreasing, and log-convex for , from which the MSS bound () emerges as the leading constraint and subleading bounds (e.g., ) follow in a hierarchical pattern. The framework unifies chaos bounds with conformal Regge constraints and Schwarzian theory, showing maximal chaos cannot be exact over any finite interval and that subleading corrections are generically required. These results provide a systematic, operator-based set of limits on scrambling in large- quantum systems with potential applications to holography and CFTs via Regge and pole-skipping analyses.

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 of the subleading correction to maximal chaos.

Paper Structure

This paper contains 16 sections, 33 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Chaos and OTOC
  3. Assumptions
  4. Analyticity and Positivity Conditions
  5. Maldacena-Shenker-Stanford Bound
  6. Dispersion Relation and Consistency Conditions
  7. Subleading Chaos Bounds
  8. Moments of the OTOC
  9. Bounds on Moments
  10. Positivity Condition
  11. Boundedness Condition
  12. Monotonicity Condition
  13. Log-Convexity Condition
  14. Bounds on the Rate of Change
  15. OTOC as a Sum Over Moments
  16. ...and 1 more sections

Figures (3)

  • Figure 1: Three time scales that are important for chaos are shown here schematically for thermal quantum systems with a large number of degrees of freedom and a simple Hamiltonian which is sufficiently local. These time scales are: $t_d$ = dissipation time $\sim \beta$, $t_0$ = factorization time, and $t_*=$ scrambling time.
  • Figure 2: The OTOC $F(t)$, as defined in (\ref{['eq:otoc']}), is analytic in the shaded blue region. Contours for deriving the dispersion relation (\ref{['OTOC:sumrule']}), where $t_0$ is the factorization time, are shown in red. The contour $C'$ surrounds a simple pole at $t'=t$ of the integral (\ref{['eq:start']}).
  • Figure 3: The contour $\gamma$ is used for deriving bounds on moments of the OTOC.