Fine-Grained Chaos in $AdS_2$ Gravity

TL;DR

<3-5 sentence high-level summary> The paper investigates higher-point out-of-time-order (OTO) correlators in the Schwarzian theory, which describes $AdS_2$ gravity coupled to matter and captures the low-energy dynamics of the SYK model. It identifies a class of observables, the maximally braided $2k$-point correlators $F_{2k}$, that grow exponentially with a Lyapunov exponent $\lambda_L=2\pi/\beta$ up to a longer scrambling time $\hat{u}_*^{(k)}=(k-1)\hat{u}_*$, revealing a hierarchy of chaotic time scales probing finer-grained quantum information. The results are obtained by analyzing backreaction of matter on the Schwarzian soft mode, organizing Euclidean $2k$-point calculations, and performing analytic continuation to Lorentzian times, with explicit results for $F_4$, $F_6$, and $F_8$ illustrating the pattern. The work suggests connections to unitary $k$-design and prompts further exploration of Lorentzian realizations, higher-dimensional generalizations, and potential experimental implications of this chaos hierarchy.

Abstract

Quantum chaos can be characterized by an exponential growth of the thermal out-of-time-order four-point function up to a scrambling time $\widehat{u}_*$. We discuss generalizations of this statement for certain higher-point correlation functions. For concreteness, we study the Schwarzian theory of a one-dimensional time reparametrization mode, which describes $AdS_2$ gravity and the low-energy dynamics of the SYK model. We identify a particular set of $2k$-point functions, characterized as being both "maximally braided" and "k-OTO", which exhibit exponential growth until progressively longer timescales $\widehat{u}^{(k)}_* = (k-1)\widehat{u}_*$. We suggest an interpretation as scrambling of increasingly fine-grained measures of quantum information, which correspondingly take progressively longer time to reach their thermal values.

Figures (2)

• Figure 1: General maximally braided $2k$-point correlator (first term obtained by expanding out commutators in $F_{2k}$): only diagrams of the type shown contribute to the connected correlator $F_{2k}$ at leading order in $\kappa$. The arrangement of insertions along the circle indicates the ordering in Euclidean time.
• Figure 2: Complex time contour representation of the maximally braided correlator. We show the first (and dominant) term in the expansion of commutators in $F_{2k}$. Lorentzian time runs horizontally. We depict the Lorentzian time configuration which is maximally OTO. Operators are separated by small imaginary times, which enforces the operator ordering along the contour.