Effective description of sub-maximal chaos: stringy effects for SYK scrambling

TL;DR

This work develops a scramblon-based path-integral framework to describe quantum chaos across maximal and sub-maximal regimes. By starting from the Schwarzian theory and 2D CFT identity blocks and then moving to the large-q SYK dot and chain, it identifies nearly zero scramblon modes, derives their eikonal action, and constructs bilocal vertex functions that govern OTOCs. The approach reproduces known maximal-chaos results while providing a controlled description of sub-maximal scrambling, revealing stringy corrections to gravitational eikonal scattering and introducing momentum-dependent scrambling through the SYK chain. The findings point to a unified boundary description of chaos with bulk-stringy dynamics and suggest directions toward local effective theories and holographic interpretations of non-maximal scrambling.

Abstract

It has been proposed that the exponential decay and subsequent power law saturation of out-of-time-order correlation functions can be universally described by collective 'scramblon' modes. We develop this idea from a path integral perspective in several examples, thereby establishing a general formalism. After reformulating previous work on the Schwarzian theory and identity conformal blocks in two-dimensional CFTs relevant for systems in the infinite coupling limit with maximal quantum Lyapunov exponent, we focus on theories with sub-maximal chaos: we study the large-q limit of the SYK quantum dot and chain, both of which are amenable to analytical treatment at finite coupling. In both cases we identify the relevant scramblon modes, derive their effective action, and find bilocal vertex functions, thus constructing an effective description of chaos. The final results can be matched in detail to stringy corrections to the gravitational eikonal S-matrix in holographic CFTs, including a stringy Regge trajectory, bulk to boundary propagators, and multi-string effects that are unexplored holographically.

Figures (5)

• Figure 1: Generalized Schwinger-Keldysh contour ${\cal C}$ for the path integral calculation of the out-of-time-order correlators. The integration contour in the complex $t$ plane is parametrized by the real-valued contour time $s \in [0,4T+\beta]$. Infinitesimal contour separations by $i\delta$ serve as a regulator (which we only keep track of when it is needed for regularization).
• Figure 2: The support along the SK contour of forward and backward shocks, sourced by pairwise operator insertions at $\hat{t}_1=\hat{t}_2=0$ and $\hat{t}_3 = \hat{t}_4=T$.
• Figure 3: The bilocal version of figure \ref{['fig:shocks']}, as required for the large $q$ SYK model. Shaded regions show where the fluctuations $\delta_\pm g(s_1,s_2)$ have non-zero support. By symmetry, we can always focus on the triangle $s_1 > s_2$.
• Figure 4: Edges $E$ and vertices $V$ contributing non-trivial boundary terms to the off-diagonal quadratic action $\left\langle{\delta _-g, \mathcal{L} \delta_+ g}\right\rangle$. A similar picture applies to $\left\langle{\delta _+g, \mathcal{L} \delta_- g}\right\rangle$.
• Figure 5: Doubly logarithmic plot of the long-time OTOC \ref{['eq:FotocRes']} of the large $q$ SYK model for $N=10^{12}$, $v=0.8$, and some values of $\Delta$. The dotted green line shows the approximately exponential time dependence $\sim e^{vT}$, which is applicable for all values of $\Delta$ at early times. Other dotted lines approximate the subsequent 'Ruelle' regime up to ${\cal O}(\Delta^2)$, as shown in the second line of \ref{['eq:UexpandDelta']}. The approximation in the first line of \ref{['eq:UexpandDelta']} is not shown as it would be indistinguishable from the solid lines in the Ruelle regime. One can verify from the plot that 'scrambling' ends after a time of order $t_\text{scr} \sim \frac{1}{v} \log(N/q^2)$ and the time for the OTOC to deviate from 1 by an ${\cal O}(1)$ amount scales with $\Delta^{-1}$, c.f., \ref{['eq:largeqScr']}.