Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2512.20700 · Physical Review Letters

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

Ping Gao, Hong Liu

hep-thgr-qcquant-phSubmitted Dec 23, 2025• Announced Dec 25, 2025• 0 citations
v1 · Dec 23, 2025v2 · Jul 29, 2026
arXivHTMLPDF

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.

DOI
10.1103/cvjn-7kvp
Comments
7 pages plues supplementary materials

Related papers

Pole skipping away from maximal chaos

Changha Choi, Márk Mezei, Gábor Sárosi

Abstract Pole skipping is a recently discovered subtle effect in the thermal energy density retarded two point function at a special point in the complex (ω, p) planes. We propose that pole skipping is determined by the stress tensor contribution to many-body chaos, and the special point is at (ω, p)p.s.= $$ i{\lambda}^{(T)}\left(1,1/{u}_B^{(T)}\right) $$ iλT11/uBT , whereλ(T)= 2π/βand $$ {u}_B^{(T)} $$ uBT are the stress tensor contributions to the Lyapunov exponent and the butterfly velocity respectively. While this proposal is consistent with previous studies conducted for maximally chaotic theories, where the stress tensor dominates chaos, it clarifies that one cannot use pole skipping to extract the Lyapunov exponent of a theory, which obeysλ≤λ(T). On the other hand, in a large class of strongly coupled but non-maximally chaotic theories $$ {u}_B^{(T)} $$ uBT is the true butterfly velocity and we conjecture thatuB≤ $$ {u}_B^{(T)} $$ uBT is a universal bound. While it remains a challenge to explain pole skipping in a general framework, we provide a stringent test of our proposal in the large-qlimit of the SYK chain, where we determineλ, uB, and the energy density two point function in closed form for all values of the coupling, interpolating between the free and maximally chaotic limits. Since such an explicit expression for a thermal correlator is one of a kind, we take the opportunity to analyze many of its properties: the coupling dependence of the diffusion constant, the dispersion relations of poles, and the convergence properties of all order hydrodynamics.

arXiv:2010.08558•Journal of High Energy Physics•Feb 24, 2021•37 citations

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

Changha Choi, Felix M. Haehl, Márk Mezei +1 more

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.

arXiv:2301.05698•Journal of High Energy Physics•Mar 20, 2023•12 citations

Scattering strings off quantum extremal surfaces

Venkatesa Chandrasekaran, Thomas Faulkner, Adam Levine

Abstract We consider a Hayden & Preskill like setup for both maximally chaotic and sub-maximally chaotic quantum field theories. We act on the vacuum with an operator in a Rindler like wedge R and transfer a small subregion I of R to the other wedge. The chaotic scrambling dynamics of the QFT Rindler time evolution reveals the information in the other wedge. The holographic dual of this process involves a particle excitation falling into the bulk and crossing into the entanglement wedge of the complement to r = R\I. With the goal of studying the locality of the emergent holographic theory we compute various quantum information measures on the boundary that tell us when the particle has entered this entanglement wedge. In a maximally chaotic theory, these measures indicate a sharp transition where the particle enters the wedge exactly when the insertion is null separated from the quantum extremal surface for r. For sub-maximally chaotic theories, we find a smoothed crossover at a delayed time given in terms of the smaller Lyapunov exponent and dependent on the time-smearing scale of the probe excitation. The information quantities that we consider include the full vacuum modular energy R\I as well as the fidelity between the state with the particle and the state without. Along the way, we find a new explicit formula for the modular Hamiltonian of two intervals in an arbitrary 1+1 dimensional CFT to leading order in the small cross ratio limit. We also give an explicit calculation of the Regge limit of the modular flowed chaos correlator and find examples which do not saturate the modular chaos bound. Finally, we discuss the extent to which our results reveal properties of the target of the probe excitation as a “stringy quantum extremal surface” or simply quantify the probe itself thus giving a new approach to studying the notion of longitudinal string spreading.

arXiv:2108.01093•Journal of High Energy Physics•Aug 16, 2022•12 citations

Quantum chaos and pole skipping in two-dimensional conformal perturbation theory

Curtis T. Asplund, Sebastian Fischetti, Alexandra Miller +1 more

A bstract We analyze pole skipping of stress tensor two-point functions in two-dimensional quantum field theories perturbed away from conformality by a relevant deformation. The retarded two-point Green’s function can be formally computed in conformal perturbation theory, though it results in singular expressions. We propose a natural interpretation of these expressions and compute the resulting Green’s function to leading nontrivial order in the deformation. As a check of our results, we compare the Lyapunov exponents and butterfly velocities we find from our computed skipped poles to those obtained from both a leading-order conformal field theory analysis using Ward identities, as well as to a holographic gravitational dual perturbed by a massive scalar field; we find precise agreement. We comment on extensions to sub-leading order, where agreement with holographic expectations would no longer be expected.

arXiv:2509.18540•Journal of High Energy Physics•Sep 3, 2026•0 citations

Scrambling in hyperbolic black holes: shock waves and pole-skipping

Yongjun Ahn, Viktor Jahnke, Hyun-Sik Jeong +1 more

A bstract We study the scrambling properties of ( d + 1)-dimensional hyperbolic black holes. Using the eikonal approximation, we calculate out-of-time-order correlators (OTOCs) for a Rindler-AdS geometry with AdS radius ℓ , which is dual to a d -dimensional conformal field theory (CFT) in hyperbolic space with temperature T = 1 / (2 π ℓ ). We find agreement between our results for OTOCs and previously reported CFT calculations. For more generic hyperbolic black holes, we compute the butterfly velocity in two different ways, namely: from shock waves and from a pole-skipping analysis, finding perfect agreement between the two methods. The butterfly velocity v B ( T ) nicely interpolates between the Rindler-AdS result $$ {v}_B\left(T=\frac{1}{2\pi \ell}\right)=\frac{1}{d-1} $$ v B T = 1 2 π ℓ = 1 d − 1 and the planar result $$ {v}_B\left(T\gg \frac{1}{\ell}\right)=\sqrt{\frac{d}{2\left(d-1\right)}} $$ v B T ≫ 1 ℓ = d 2 d − 1 .

arXiv:1907.08030•Journal of High Energy Physics•Oct 1, 2019•71 citations

On systems of maximal quantum chaos

Mike Blake, Hong Liu

A remarkable feature of chaos in many-body quantum systems is the existence of a bound on the quantum Lyapunov exponent. An important question is to understand what is special about maximally chaotic systems which saturate this bound. Here we provide further evidence for the `hydrodynamic' origin of chaos in such systems, and discuss hallmarks of maximally chaotic systems. We first provide evidence that a hydrodynamic effective field theory of chaos we previously proposed should be understood as a theory of maximally chaotic systems. We then emphasize and make explicit a signature of maximal chaos which was only implicit in prior literature, namely the suppression of exponential growth in commutator squares of generic few-body operators. We provide a general argument for this suppression within our chaos effective field theory, and illustrate it using SYK models and holographic systems. We speculate that this suppression indicates that the nature of operator scrambling in maximally chaotic systems is fundamentally different to scrambling in non-maximally chaotic systems. We also discuss a simplest scenario for the existence of a maximally chaotic regime at sufficiently large distances even for non-maximally chaotic systems.

arXiv:2102.11294•Feb 22, 2021•0 citations
In this paper, we study pole-skipping in non-maximally quantum chaotic systems. Using Rindler conformal field theories and the large-qqq 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.
Abstract Pole skipping is a recently discovered subtle effect in the thermal energy density retarded two point function at a special point in the complex (ω, p) planes. We propose that pole skipping is determined by the stress tensor contribution to many-body chaos, and the special point is at (ω, p)p.s.= iλ(T)(1,1/uB(T)) i{\lambda}^{(T)}\left(1,1/{u}_B^{(T)}\right) iλ(T)(1,1/uB(T)​) iλT11/uBT , whereλ(T)= 2π/βand uB(T) {u}_B^{(T)} uB(T)​ uBT are the stress tensor contributions to the Lyapunov exponent and the butterfly velocity respectively. While this proposal is consistent with previous studies conducted for maximally chaotic theories, where the stress tensor dominates chaos, it clarifies that one cannot use pole skipping to extract the Lyapunov exponent of a theory, which obeysλ≤λ(T). On the other hand, in a large class of strongly coupled but non-maximally chaotic theories uB(T) {u}_B^{(T)} uB(T)​ uBT is the true butterfly velocity and we conjecture thatuB≤ uB(T) {u}_B^{(T)} uB(T)​ uBT is a universal bound. While it remains a challenge to explain pole skipping in a general framework, we provide a stringent test of our proposal in the large-qlimit of the SYK chain, where we determineλ, uB, and the energy density two point function in closed form for all values of the coupling, interpolating between the free and maximally chaotic limits. Since such an explicit expression for a thermal correlator is one of a kind, we take the opportunity to analyze many of its properties: the coupling dependence of the diffusion constant, the dispersion relations of poles, and the convergence properties of all order hydrodynamics.
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.
Abstract We consider a Hayden & Preskill like setup for both maximally chaotic and sub-maximally chaotic quantum field theories. We act on the vacuum with an operator in a Rindler like wedge R and transfer a small subregion I of R to the other wedge. The chaotic scrambling dynamics of the QFT Rindler time evolution reveals the information in the other wedge. The holographic dual of this process involves a particle excitation falling into the bulk and crossing into the entanglement wedge of the complement to r = R\I. With the goal of studying the locality of the emergent holographic theory we compute various quantum information measures on the boundary that tell us when the particle has entered this entanglement wedge. In a maximally chaotic theory, these measures indicate a sharp transition where the particle enters the wedge exactly when the insertion is null separated from the quantum extremal surface for r. For sub-maximally chaotic theories, we find a smoothed crossover at a delayed time given in terms of the smaller Lyapunov exponent and dependent on the time-smearing scale of the probe excitation. The information quantities that we consider include the full vacuum modular energy R\I as well as the fidelity between the state with the particle and the state without. Along the way, we find a new explicit formula for the modular Hamiltonian of two intervals in an arbitrary 1+1 dimensional CFT to leading order in the small cross ratio limit. We also give an explicit calculation of the Regge limit of the modular flowed chaos correlator and find examples which do not saturate the modular chaos bound. Finally, we discuss the extent to which our results reveal properties of the target of the probe excitation as a “stringy quantum extremal surface” or simply quantify the probe itself thus giving a new approach to studying the notion of longitudinal string spreading.
A bstract We analyze pole skipping of stress tensor two-point functions in two-dimensional quantum field theories perturbed away from conformality by a relevant deformation. The retarded two-point Green’s function can be formally computed in conformal perturbation theory, though it results in singular expressions. We propose a natural interpretation of these expressions and compute the resulting Green’s function to leading nontrivial order in the deformation. As a check of our results, we compare the Lyapunov exponents and butterfly velocities we find from our computed skipped poles to those obtained from both a leading-order conformal field theory analysis using Ward identities, as well as to a holographic gravitational dual perturbed by a massive scalar field; we find precise agreement. We comment on extensions to sub-leading order, where agreement with holographic expectations would no longer be expected.
A bstract We study the scrambling properties of ( d + 1)-dimensional hyperbolic black holes. Using the eikonal approximation, we calculate out-of-time-order correlators (OTOCs) for a Rindler-AdS geometry with AdS radius ℓ , which is dual to a d -dimensional conformal field theory (CFT) in hyperbolic space with temperature T = 1 / (2 π ℓ ). We find agreement between our results for OTOCs and previously reported CFT calculations. For more generic hyperbolic black holes, we compute the butterfly velocity in two different ways, namely: from shock waves and from a pole-skipping analysis, finding perfect agreement between the two methods. The butterfly velocity v B ( T ) nicely interpolates between the Rindler-AdS result vB(T=12πℓ)=1d−1 {v}_B\left(T=\frac{1}{2\pi \ell}\right)=\frac{1}{d-1} vB​(T=2πℓ1​)=d−11​ v B T = 1 2 π ℓ = 1 d − 1 and the planar result vB(T≫1ℓ)=d2(d−1) {v}_B\left(T\gg \frac{1}{\ell}\right)=\sqrt{\frac{d}{2\left(d-1\right)}} vB​(T≫ℓ1​)=2(d−1)d​​ v B T ≫ 1 ℓ = d 2 d − 1 .
A remarkable feature of chaos in many-body quantum systems is the existence of a bound on the quantum Lyapunov exponent. An important question is to understand what is special about maximally chaotic systems which saturate this bound. Here we provide further evidence for the `hydrodynamic' origin of chaos in such systems, and discuss hallmarks of maximally chaotic systems. We first provide evidence that a hydrodynamic effective field theory of chaos we previously proposed should be understood as a theory of maximally chaotic systems. We then emphasize and make explicit a signature of maximal chaos which was only implicit in prior literature, namely the suppression of exponential growth in commutator squares of generic few-body operators. We provide a general argument for this suppression within our chaos effective field theory, and illustrate it using SYK models and holographic systems. We speculate that this suppression indicates that the nature of operator scrambling in maximally chaotic systems is fundamentally different to scrambling in non-maximally chaotic systems. We also discuss a simplest scenario for the existence of a maximally chaotic regime at sufficiently large distances even for non-maximally chaotic systems.