Extremal Chaos

TL;DR

The paper identifies an extremally chaotic OTOC $F_{\rm ext}(t)$ that uniquely saturates all saturable subleading chaos bounds, extending maximal chaos beyond leading Lyapunov growth. It derives a Källen–Lehmann–type spectral representation for OTOCs, $F_d-F(t)=\int_{t_0}^{\infty}dt'\mathcal F_{\rm ext}(t;t')\rho(t')$, and shows that all analytic completions of maximal chaos are small deformations of extremal chaos via narrow spectral densities $\rho(t)$. Non-analyticities of $F_{\rm ext}$ on the boundary are interpreted as distributions, regularized by an $i\epsilon$ prescription to yield physical, analytic OTOCs; late-time tails of $\rho(t)$ can modify asymptotics, linking to long-time behavior seen in Schwarzian and large-$c$ CFTs. The results offer a unified framework to study analytic completions, with implications for quantum gravity, holography, and models like SYK, and suggest that chaos in gravity-like theories is governed by a universal extremal structure perturbatively deformed by small spectral tails.

Abstract

In maximally chaotic quantum systems, a class of out-of-time-order correlators (OTOCs) saturate the Maldacena-Shenker-Stanford (MSS) bound on chaos. Recently, it has been shown that the same OTOCs must also obey an infinite set of (subleading) constraints in any thermal quantum system with a large number of degrees of freedom. In this paper, we find a unique analytic extension of the maximally chaotic OTOC that saturates all the subleading chaos bounds which allow saturation. This extremally chaotic OTOC has the feature that information of the initial perturbation is recovered at very late times. Furthermore, we argue that the extremally chaotic OTOC provides a Källen-Lehmann-type representation for all OTOCs. This representation enables the identification of all analytic completions of maximal chaos as small deformations of extremal chaos in a precise way.

Figures (7)

• Figure 1: The extremally chaotic OTOC $F_{\rm ext}(t)$ coincides with the maximally chaotic OTOC (red dashed line) only before the effective time scale $t_{\rm eff}$. The extremally chaotic OTOC has a minimum at $t=t_{\rm eff}$, irrespective of the scrambling time $t_*$. After $t=t_{\rm eff}$, the OTOC grows monotonically approaching the factorized value (also the initial value) $F_d$. So, $F_{\rm ext}(t)$ is bounded by the factorized correlator $F_d$ for real $t$ even after the scrambling time.
• Figure 2: Extremal chaos in physical systems. The extremally chaotic OTOC (dashed red) is non-analytic on the boundary: $\hbox{Im}\ t=\pm \beta/4$. This non-analyticity can be removed by a standard $i\epsilon$-shift. The associated regularized OTOC, which is shown in blue, differs only slightly from the unregularized OTOC. However, a late-time long-tailed correction to the density function $\rho(t>t_\Lambda)=\rho_\infty e^{-\frac{2 \pi }{\beta}t}$ does change the asymptotic behavior of the OTOC (shown in brown). Various relevant time scales are also shown in the figure: $t_d$= dissipation time $\sim \beta$, $t_0$= factorization time, $t_{\rm eff}=$ effective (or a gap) time scale, $t_*=$ scrambling time, and $t_\Lambda>t_*$ is a cut-off scale above which late-time corrections can become important. In the presence of a late-time long-tailed correction, a physical system can be extremally chaotic only approximately up to the cut-off scale $t_\Lambda$.
• Figure 3: $F(t)$, as a function of complex $t$, is analytic in the shaded blue region. However, the extremal OTOC (\ref{['eq:intro']}) has singularities at $t=t_{\rm eff}\pm i\beta/4$. These singularities can be removed by a simple $i \epsilon$-shift.
• Figure 4: Density functions (\ref{['eq:inversion']}) associated with various chaotic systems are shown here schematically. The dashed black line represents the bound (\ref{['rho:bound']}). Figure (a): The blue line is the density function for a period of Lyapunov growth with $0<\lambda_L< \frac{2\pi}{\beta}$. This density function becomes inconsistent with the bound (\ref{['rho:bound']}) for large $t$, where correction terms must become significantly large so that the density function decays faster. Figure (b): The blue line represents a typical density function associated with a period of maximal chaos. Over the same duration of time the density function $\rho(t)\approx 0$, however, it starts to grow around some time scale $t=t_{\rm eff}$. The red line represents the delta-function density associated with the extremally chaotic OTOC.
• Figure 5: The regularized density functions (a) and the associated OTOCs (b) are shown for various values of $\epsilon$. For the plot, we have chosen $\beta=2\pi, F_d=1, c_1=2, t_{\rm eff}=50$ and $t_*=51$. For this set of parameters, the bound (\ref{['epsilon:bound']}) requires that $\epsilon \ge 0.1415$. The dashed black lines represent the unregularized OTOC (\ref{['eq:intro']}). The regularized OTOC differs only slightly from the unregularized OTOC even when $\epsilon=1$.