Probing Stringy Horizons with Pole-Skipping in Non-Maximal Chaotic Systems

TL;DR

The paper shows that pole-skipping points in non-maximally chaotic quantum systems form trajectories in complex frequency–momentum space, with the leading trajectory encoding the Lyapunov exponent. Using solvable models—the Rindler CFT and a large-$q$ SYK chain—it interprets these trajectories as Regge trajectories of stringy horizon excitations in a dual stringy black hole geometry. It then argues that pole-skipping reflects intrinsic horizon structure that persists into the stringy regime, with the leading horizon data captured by the leading Regge trajectory while the full tower of higher-spin states shapes the non-maximal chaos. The work further demonstrates, via the large-$q$ SYK chain, that the leading pole-skipping data can be extracted from universal bilinear operators, providing a robust, operator-independent probe of chaos and horizon physics at finite coupling.

Abstract

In this paper, we study pole-skipping in non-maximally quantum chaotic systems. Using Rindler conformal field theories and the large-$q$ SYK chain as illustrative examples, we argue that the pole-skipping points of few-body operators organize into trajectories in the complex frequency-momentum plane, with the leading trajectory encoding the quantum Lyapunov exponent. We further propose that these trajectories admit a natural interpretation as Regge trajectories of stringy excitations in a dual stringy black hole geometry. From this perspective, pole-skipping for an individual operator can be viewed as tracking the stringy horizon through the response of a single excitation. Our results suggest that pole-skipping reflects intrinsic properties of quantum chaotic systems and may be deeply connected to the structure of horizons in the stringy regime.

Figures (5)

• Figure 1: The overlay of Regge trajectories and the pole-skipping points. The horizon axis is $\Delta-d/2,\Im k$ and the vertical axis is $J-1,\Im\omega$. The blue and green curve together is the leading Regge trajectory; The yellow and red curves are two symmetric subleading Regge trajectories. Each dot on the trajectories is a physical operator with even spin, which seeds a family of pole-skipping points \ref{['eq:512-2']} (smaller dots with variant colors) below it. Pole-skipping for odd spins likewise forms trajectories, but since they are not involved in the leading trajectory, we suppress them to avoid cluttering the plot. The triangles are missing or decoupling zeros on a subleading trajectory Homrich:2022cfqHenriksson:2023cnh, where no physical operator exists at an integer spin, and are irrelevant to pole-skipping.
• Figure 2: The pole-skipping points of the large $q$ SYK chain on the $(\Im \omega,\Im p)$ plane with parameters $\eta=0.8$ and $\lambda_0=0.7$. The blue and yellow dots are \ref{['eq:480-3']} with odd and even $n$ respectively, and the green dots are \ref{['eq:459']}. The Pole-skipping points without pure imaginary $p$ are not included in this plot. The blue and yellow curves are the two bounds in \ref{['eq:522']} respectively.
• Figure S1: Pole-skipping of spin $J=3$ with parameter $d=5$ and $\Delta=6.7$. The bright lines are pole lines, and the dark lines are zero lines. The blue dots are the pole-skipping points (\ref{['eq:512-2']}), which are at the intersection of a pair of pole line and zero line. There are also a few sporadic crossings giving non-universal pole-skipping points other than the blue points.
• Figure S2: The contour change of $J$ integral from $\Gamma$ (spin $J$ integers) to $\Gamma'$ enclosing Regge trajectories $j_0(\mu),j_1(\mu),\cdots$ in the Sommerfeld-Watson resummation.
• Figure S3: The blue curve is the contour of $\text{i} t$ (left) and $\text{i} s$ (right) in (\ref{['eq:508-2']}). Red dots are poles of the gamma function.