⚛️ high-energy theory

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

This paper demonstrates that in non-maximally quantum chaotic systems, pole-skipping points of few-body operators form trajectories encoding the Lyapunov exponent, which can be interpreted as Regge trajectories of stringy excitations probing the horizon structure of a dual stringy black hole.

Original authors: Ping Gao, Hong Liu

Published 2026-07-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ping Gao, Hong Liu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, chaotic dance floor where particles are constantly bumping into each other, swapping energy, and scrambling their information. Physicists have long been obsessed with understanding how fast this dance gets messy. They use a special tool called the "Lyapunov exponent" to measure the speed of this chaos. Think of it like a stopwatch for disorder: a high number means the system is scrambling information incredibly fast, while a lower number means it's a bit more sluggish.

For a long time, scientists thought they only understood this dance floor when it was running at maximum speed (what they call "maximal chaos"). In these super-fast systems, there's a weird phenomenon called "pole-skipping." Imagine trying to find a specific spot on a map where a road (a "pole") and a river (a "zero") cross. Usually, they just pass each other. But in these chaotic systems, at a very specific, magical coordinate, the road and the river seem to cancel each other out perfectly, leaving a blank spot where the map should have a line. This "skipping" point was known to reveal the speed of the chaos, but only for the fastest systems.

The big question was: what happens in systems that aren't running at maximum speed? Do these magical skipping spots still exist? And if they do, do they still tell us about the chaos, or do they just disappear? This is where the story gets interesting, because the answer might involve the very fabric of space and time, and even the mysterious "horizons" of black holes.

In this paper, physicists Ping Gao and Hong Liu take a deep dive into these "non-maximal" chaotic systems—places where the chaos is real but not quite at the theoretical limit. They wanted to see if the "pole-skipping" trick still works and what it might be hiding.

To solve this puzzle, they used two very different "test labs." The first was a theoretical setup called a "Rindler CFT," which is like a simplified, flat version of the universe that still has all the right rules for chaos. The second was a "large-q SYK chain," a model made of interacting particles that acts like a one-dimensional string of beads. These models are special because they are "stringy," meaning they behave like the vibrating strings proposed by string theory, rather than just simple point particles.

The authors discovered that pole-skipping doesn't just happen at a single point in these slower systems. Instead, the skipping spots organize themselves into beautiful, winding paths, or "trajectories," across a complex map of frequency and momentum. It's as if the single magical spot from the old systems has stretched out into a glowing highway.

Here is the most exciting part: the authors found that the very top of this highway (the "leading trajectory") holds the secret to the system's speed of chaos. By tracing this path, they can extract the "Lyapunov exponent," which tells us exactly how fast the system scrambles information. This suggests that even in systems that aren't running at full speed, the chaos is still encoded in these skipping patterns.

But the story gets even stranger. The authors propose that these glowing highways aren't just mathematical tricks; they are actually the "Regge trajectories" of stringy excitations. In the language of string theory, particles are like vibrating strings, and their different vibration modes form families that look like these trajectories. The paper suggests that in a dual world (a holographic view where our universe is a projection of a higher-dimensional one), these trajectories represent the actual "Regge trajectories" of strings moving near a black hole's horizon.

This leads to a fascinating new perspective: the "horizon" of a black hole (the point of no return) might not be a fuzzy, blurry mess in the stringy regime, as some had feared. Instead, the authors argue that for each individual stringy particle, the horizon remains sharp and well-defined, governed by specific symmetries. The "fuzziness" only appears when you try to look at the whole crowd of strings at once.

The paper also looked at the "large-q SYK chain" model, which is a lattice of particles rather than a smooth space. Even here, without the usual rules of smooth geometry, they found the same pattern: the pole-skipping points formed trajectories. One family of these points traced out the main highway that revealed the system's Lyapunov exponent, while another family seemed to live in a "fake" temperature, behaving differently from the rest.

In short, Gao and Liu suggest that pole-skipping is a universal feature of chaotic systems, not just the fastest ones. It acts like a probe, allowing us to "track" the stringy horizon through the response of a single excitation. While they haven't proven this for every possible system in the universe, their results in these specific models strongly suggest that the structure of horizons in the stringy regime is more organized and sharp than previously thought, and that the "speed of chaos" is written into the very shape of these mathematical highways.

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

Problem Statement
While pole-skipping has been established as a diagnostic for quantum chaos in maximally chaotic systems (where the Lyapunov exponent $\lambda = 2\pi/\beta$ ), its behavior and physical interpretation in non-maximally chaotic systems remain unclear. In such systems, the Lyapunov exponent is sub-maximal ( $\lambda < 2\pi/\beta$ ), and previous studies (e.g., in the large- $q$ SYK chain) suggested that the pole-skipping location of the energy density might remain fixed at the maximal value, seemingly decoupled from the actual non-maximal chaos. Furthermore, the connection between pole-skipping and the structure of black hole horizons in the stringy regime (where higher-derivative corrections or stringy excitations are relevant) is not fully understood. The central question is whether pole-skipping is a universal feature of chaotic systems that can encode the non-maximal Lyapunov exponent and reveal the nature of stringy horizons.

Methodology
The authors employ two primary illustrative frameworks to analyze pole-skipping for few-body operators in non-maximally chaotic systems:

Rindler Conformal Field Theory (CFT): They study a CFT on $\mathbb{R}^{1,d-1}$ in its vacuum state, focusing on the right Rindler wedge. This setup maps to a thermal CFT on $\mathbb{R} \times \mathbb{H}^{d-1}$ at inverse temperature $\beta = 2\pi$ . They analyze the pole-skipping of general higher-spin operators ( $O_{\mu_1 \dots \mu_J}$ ) with scaling dimension $\Delta$ and spin $J$ . By Fourier transforming the vacuum two-point function in Rindler coordinates, they derive the retarded correlator in momentum space to identify pole-skipping points.
Large- $q$ SYK Chain: They examine the large- $q$ limit of the Sachdev-Ye-Kitaev (SYK) chain, a model believed to be dual to a "string-type" gravity theory. They analyze the pole-skipping of composite operators (bilinear in Majorana fermions) to extract the universal set of pole-skipping points without needing to resolve individual operator spectra.

Key Contributions and Results

Organization into Trajectories: The authors demonstrate that pole-skipping points for few-body operators do not appear as isolated points but organize into trajectories in the complex frequency-momentum plane ( $\omega, k$ ).
- In the Rindler CFT, the pole-skipping points for a spin- $J$ operator are located at specific imaginary Matsubara frequencies. The highest pole-skipping point for each operator lies precisely on its Regge trajectory in the $(\Delta, J)$ plane.
- Collectively, the set of pole-skipping points across all operators reconstructs the full analytic Regge trajectories, including the leading Regge trajectory ( $j_{\text{leading}}(\mu)$ ).
Encoding the Lyapunov Exponent: The paper argues that the leading trajectory of these pole-skipping points encodes the quantum Lyapunov exponent.
- Specifically, the intercept of the leading trajectory at zero momentum, $j_{\text{leading}}(0)$ , determines the Lyapunov exponent via $\lambda = j_{\text{leading}}(0) - 1$ .
- In the Rindler CFT, this is derived from the conformal partial wave analysis where the Lyapunov exponent is determined by the exchange of an infinite tower of higher-spin operators.
- In the large- $q$ SYK chain, the authors identify a leading trajectory $j_{\text{leading}}(p) = (h(p)-1)\lambda_0 + 1$ (where $h(p)$ is a momentum-dependent function and $\lambda_0$ is the non-maximal Lyapunov exponent). The intercept at $p=0$ correctly recovers $\lambda_0$ , confirming that the collective pole-skipping data extracts the non-maximal chaos parameter.
Stringy Horizon Interpretation: The authors propose a holographic interpretation where these trajectories correspond to Regge trajectories of stringy excitations in a dual stringy black hole geometry.
- In the stringy regime, the bulk contains an infinite tower of higher-spin string modes. Each mode contributes individually to the scattering with an exponent $\lambda_J = J-1$ , but their collective sum yields the non-maximal Lyapunov exponent.
- Pole-skipping for an individual operator is interpreted as "tracking" the stringy horizon through the response of a single stringy excitation.
- Crucially, the paper argues that horizon symmetries persist in the stringy regime for each individual bulk field. The "fuzziness" of the horizon arises only when summing over the full Hagedorn density of modes or including interactions, not from the loss of symmetry for individual modes.
Composite Operators in SYK: For the SYK chain, which lacks conformal symmetry and a notion of spin, the authors show that analyzing composite operators allows the extraction of the universal set of pole-skipping points. This set reveals the leading trajectory and the associated Lyapunov exponent, even though individual operators cannot be separated. They also identify a second family of pole-skipping points associated with a "fake temperature" $T_{\text{f.d.}} = \lambda_0 / (2\pi)$ , suggesting a distinct physical mechanism (potentially related to a scalar-like operator).

Significance and Claims
The paper claims that pole-skipping is a universal feature of quantum chaotic systems, extending beyond the maximal chaos limit. Its primary significance lies in:

Reconstructing Chaos Data: It provides a method to extract the non-maximal Lyapunov exponent and the full Regge data of a system by analyzing the collective trajectories of pole-skipping points, rather than relying on the stress tensor alone.
Probing Stringy Horizons: It offers a new perspective on the structure of horizons in the stringy regime. The authors argue that horizons remain "sharp" for individual stringy fields due to underlying symmetries, and the apparent non-maximal chaos is a collective effect of the infinite tower of string modes.
Universality: The connection between pole-skipping trajectories and Regge data suggests a deep link between hydrodynamic/chaotic response and the fundamental spectrum of the theory (conformal or stringy), independent of specific model details.

The authors conclude that while the Lyapunov exponent is merely one piece of information extractable from pole-skipping, the phenomenon itself reflects intrinsic properties of the system's horizon structure and may be more fundamental than the exponent itself. They suggest that a microscopic understanding of this connection, potentially rooted in modular inclusions or algebraic structures, could yield further insights into quantum chaos and black hole dynamics.

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

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