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Pole-skipping organizes into Regge-like trajectories that carry the quantum Lyapunov exponent in generic chaotic systems, with stringy horizons staying sharp for each mode.

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2026-08-03 14:20 UTC pith:FN7VJO7R

load-bearing objection The Rindler spin-J pole-skipping formula is a real result, but the SYK-chain verification of the central conjecture fails arithmetically as written. the 4 major comments →

arxiv 2512.20700 v2 pith:FN7VJO7R submitted 2025-12-23 hep-th gr-qcquant-ph

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

Ping Gao , Hong Liu This is my paper
classification hep-th gr-qcquant-ph
keywords pole-skippingquantum chaosLyapunov exponentRegge trajectoriesRindler CFTSYK chainstringy horizonsholography
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper argues that pole-skipping—the special complex frequencies where lines of poles and zeros in a thermal retarded correlator intersect—is a universal feature of quantum chaotic systems, not just of maximally chaotic ones. In non-maximally chaotic systems, the pole-skipping points of few-body operators are claimed to organize into trajectories in the complex frequency–momentum plane, and the leading trajectory's intercept gives the quantum Lyapunov exponent. In holographic settings, these trajectories are identified with the Regge trajectories of stringy excitations around a black hole horizon, so each string mode sees a sharp horizon even though the summed string spectrum may look fuzzy. The paper demonstrates this in two solvable examples—a Rindler CFT and the large-q SYK chain—and conjectures that it holds for general finite-temperature stringy black holes.

Core claim

In a Rindler CFT (a CFT on R^{1,d-1} restricted to the Rindler wedge, equivalent to a thermal CFT on R × H^{d-1}), the retarded two-point function of a spin-J primary of dimension Δ has pole-skipping points at (ω,k)_p.s. = (i(J−2n−1−s), i(Δ−d/2−s)) for n≤J and (i(−n−1−s), i(Δ−d/2+n−J−s)) for n≥J, plus the k→−k reflected family. The highest point for each operator sits at i(J−1, Δ−d/2), exactly on that operator's Regge trajectory. Since Regge trajectories in the (J, Δ) plane can be reconstructed from these points, the leading trajectory—and hence the non-maximal Lyapunov exponent λ = j_leading(0)−1—is determined by the collective pole-skipping data. The paper argues the same organization hold

What carries the argument

Pole-skipping points are intersections of pole lines and zero lines of a retarded Green's function in the complex frequency–momentum plane. In the Rindler CFT, the explicit Fourier transform of the conformal two-point function—with the spin-J hypergeometric polynomial—yields the full pole-skipping lattice; in the SYK chain, the effective large-q action gives two universal families. The bridge is Regge trajectories: analytic continuation in spin of the dimensions of exchanged operators. The central identity is Eq. (4): the highest pole-skipping point of a spin-J operator equals i(J−1, Δ−d/2), placing it on the Regge trajectory, so the leading trajectory's intercept λ is read off from pole-ski

Load-bearing premise

The paper assumes that the pole-skipping–Regge-trajectory correspondence proven in the vacuum Rindler CFT, and illustrated in the large-q SYK chain, continues to hold for interacting finite-temperature theories dual to stringy black holes; Section III states no such bridge is currently available there and the transfer is made by conjecture, with the SYK universality claim deferred to a companion paper.

What would settle it

Compute the full pole-skipping spectrum of a finite-temperature, finite-coupling holographic theory (or any non-maximally chaotic model) beyond the Rindler and SYK examples, and check whether the highest pole-skipping points at imaginary Matsubara frequencies lie on a single leading curve whose intercept equals the independently computed Lyapunov exponent. If the highest points scatter without a common envelope, or the envelope intercept differs from λ, the central conjecture fails. For the SYK chain specifically, the companion paper's universality claim can be tested by computing pole-skippin

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Pole-skipping can be used to extract the quantum Lyapunov exponent in systems that do not saturate the chaos bound, where the energy-density pole-skipping point alone carries no information about λ.
  • The full set of pole-skipping points over all operators encodes the Regge data of the theory; individual operators' pole-skipping points are fragments of this global structure.
  • The stringy horizon remains sharp for each bulk string mode: every higher-spin field has a well-defined pole-skipping point tied to a horizon symmetry, while the non-maximal Lyapunov exponent emerges only from the collective sum of all modes.
  • In the SYK chain, the leading pole-skipping trajectory reproduces the momentum-dependent Lyapunov exponent, giving a concrete non-conformal, non-spin example of the proposed organization.
  • Composite-operator correlators (time-ordered four-point functions) can reveal universal pole-skipping data without knowing the operator spectrum in advance.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the conjecture generalizes, pole-skipping could serve as a practical probe of quantum chaos in numerical simulations: scanning a retarded correlator's pole-zero crossings in the complex plane is often easier than direct OTOC measurement.
  • The paper's picture suggests 'horizon fuzziness' is an observable-dependent statement: single-string-mode probes see a sharp horizon and its symmetries; fuzziness appears only in sums over the Hagedorn tower. That distinction might be testable in toy stringy models by comparing single-mode versus summed correlators.
  • One might test the proposal in other solvable non-maximally chaotic models, e.g., SYK-like chains with different coupling patterns or low-dimensional non-conformal field theories, by checking whether upper envelopes of pole-skipping families always match independently computed Lyapunov exponents.
  • The exceptional pole-skipping points found in the composite-operator analysis of the Rindler CFT (off the Matsubara frequencies) may correspond to nonlocal intermediate operators; understanding them could sharpen the correspondence between pole-skipping and Regge trajectories.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 3 minor

Summary. Using Rindler CFT and the large-q SYK chain as examples, the paper argues that pole-skipping points of few-body operators in non-maximally chaotic systems form trajectories in the complex (omega,k) plane, and that the leading trajectory encodes the quantum Lyapunov exponent. In Rindler CFT it derives Eq. (3) for a spin-J primary, whose highest pole-skipping point (4) lies at i(J-1, Delta-d/2), on the operator's Regge trajectory, leading to the relation lambda = j_leading(0)-1 in Eq. (7). The paper then conjectures that in stringy black holes these trajectories coincide with Regge trajectories of bulk string excitations and that the horizon remains sharp for each mode. The same trajectory picture is claimed to be verified in the large-q SYK chain, where a leading trajectory (10) is asserted to be saturated by pole-skipping points.

Significance. The Rindler calculation is substantial, analytically explicit, and internally coherent; if correct, it provides a rare solvable handle on higher-spin pole-skipping and a clean conformal-field-theory derivation of the Regge/Lyapunov dictionary. The proposed reinterpretation of pole-skipping as tracking individual stringy modes is thought-provoking. However, the paper's only explicit non-conformal test of conjecture 1 is marred by an arithmetic inconsistency, and the completeness of the SYK pole-skipping set is delegated to an unpublished companion. These issues must be resolved before the broad claims can be accepted.

major comments (4)
  1. [IV, Eqs. (8)-(10)] Equation (8) and the saturation claim around Eq. (9) are inconsistent. For odd n, (8) gives h = n/lambda0 + 3/2 + 2k, so the upper bound in (9) is lambda0(h-1) = n + lambda0(1/2 + 2k) > n = Im(omega). For even n, h = n/lambda0 + 5/2 + 2k, so lambda0(h-2) = n + lambda0(1/2 + 2k) > n. Thus no point of family (8) saturates the bounds, contradicting the statement that each upper bound is saturated by infinitely many pole-skipping points. Consequently the identification of (10) as an envelope actually populated by pole-skipping points is unsupported. In addition, the n=1, k=0 point has h = 1/lambda0 + 3/2, which cannot equal h(0)=2 for lambda0 <= 1, so the claimed relation to the standard pole-skipping point (1) is not evident. Unless Eq. (8) or Eq. (9) contains a typo, the only explicit verification of conjecture 1 fails as written.
  2. [IV, around Eq. (11) and Ref. [42]] The identification of families (8) and (11) as the complete universal set of pole-skipping points for bilinear operators of the SYK chain is essential for the leading-trajectory claim, since other families could change the upper envelope. This completeness is stated to be proved in the unpublished companion [42]. The reader cannot check this load-bearing point from the present manuscript. The authors should either include the argument or clearly mark the completeness as a conjecture rather than a proved input.
  3. [II and S-I.B] The derivation of Eq. (3) deliberately keeps only zero lines coming from individual gamma functions and discards 'sporadic' pole-skipping points arising from linear combinations of terms, with the statement that these are non-universal. No systematic criterion is given for this separation. Since the conclusion that pole-skipping data reconstruct Regge trajectories is a statement about the full retarded correlator, the treatment of these extra points should be made precise. Otherwise the trajectory claim applies only to a hand-picked subset of pole-skipping points, which weakens the claimed universality.
  4. [III] The passage from the Rindler vacuum result to a finite-temperature, finite-coupling CFT on R x S^{d-1} is made by conjecture; the text explicitly states that the Regge bridge 'does not appear to be available' and then conjectures that items 1-4 apply to a stringy black hole horizon. The central assertion about generic non-maximally chaotic systems therefore rests on an unproven assumption. The paper should either soften the language or supply a concrete test, such as a near-horizon computation for a higher-spin string field or a finite-temperature conformal-block analysis, to make the conjectural step more controlled.
minor comments (3)
  1. [IV, Eq. (8)] The notation '(-)^n /2' is ambiguous. It should be written as (-1)^n/2, and the resulting odd/even expressions for h should be displayed explicitly to avoid the confusion that arises in the saturation check.
  2. [FIG. 2] The figure caption states that the blue and yellow curves are the two bounds in (9) and shows the pole-skipping points together with those curves. If, as the arithmetic in the main text indicates, no point saturates the bounds, the figure should be redrawn or annotated to make the actual relation between the points and the curves clear.
  3. [IV, text below Eq. (10)] The statement that the special point n=1, k=0 corresponds to the pole-skipping point (1) 'related to the maximal Lyapunov exponent' requires a derivation. From Eq. (8) this point has h = 1/lambda0 + 3/2, and it is not immediate how this maps to the energy-density pole-skipping point, especially since lambda0 < 1.

Circularity Check

2 steps flagged

SYK 'leading trajectory' recovers λ0 by construction, and the claimed universality of the pole-skipping families is deferred to an unpublished self-citation; the Rindler derivation itself is independent.

specific steps
  1. self definitional [Section IV, Eqs. (8)-(10)]
    "(ω, h(p))p.s. = (in, n/λ0 + 1 + (−)n/2 + 2k+ 1) (8) ... We thus identify the first line of (9) as giving the leading trajectory jleading(p) = (h(p)−1)λ0 + 1, which indeed recovers the Lyapunov exponent with jleading(0)−1 = λ0 (h(0) = 2). The large-qSYK chain thus verifies beautifully the conjecture 1."

    The 'leading trajectory' (10) is literally the first line of (9), an inequality obtained by algebra from (8), and (8) already contains λ0 as an explicit parameter. Substituting h(0)=2 into (10) gives jleading(0)-1=λ0 by definition, so 'recovering/extracting the Lyapunov exponent' is a rearrangement of the input, not an independent prediction. Moreover, by the paper's own formula no point of (8) saturates (9): for odd n, λ0(h-1)=n+λ0(1/2+2k)>n, and for even n, λ0(h-2)=n+λ0(1/2+2k)>n. Thus the claimed saturation is not realized, and the curve (10) is imposed using the already-known λ0 from (S72) rather than being an envelope of actual pole-skipping points.

  2. uniqueness imported from authors [Section IV, paragraph before Eq. (8)]
    "In [42], we show that the two families identified in [26] (shown in FIG. 2 and described below) are the only ones that are universal and independent of the specific choice of composite operator. They can therefore be identified with the full universal set of pole-skipping points associated with bilinear operators."

    The SYK verification of conjecture 1 depends on (8) and (11) being the complete universal set of pole-skipping points for bilinear operators. That uniqueness claim is not derived here but delegated to [42], an unpublished companion by the same authors. No external or machine-checkable support is given. If another family of pole-skipping points existed, the 'leading trajectory' (10) need not be the actual leading curve, so the central SYK test would be unsupported. This is a load-bearing self-citation rather than an independent mathematical fact.

full rationale

The Rindler-CFT part is not circular: Eq. (3)/(4) is obtained in S-I by Fourier-transforming the conformal two-point function (2), with no Lyapunov data as input, and Eq. (7) is standard conformal Regge theory supported by external references. The independent content of the paper rests there. The SYK chain section, however, contains a circular 'verification': Eq. (8) already contains λ0 as a parameter, and Eq. (10) is just the first line of (9), an algebraic inequality derived from (8). Therefore jleading(0)-1=λ0 follows by substitution (h(0)=2), so 'extracting the Lyapunov exponent from the TOC' is a rearrangement of the input. There is also an arithmetic tension, noted in the step above, that no point of (8) saturates (9), making the assignment of (10) as the envelope even more clearly an a priori choice using the known λ0. Finally, the universality of the two families (8),(11) is delegated to the authors' unpublished companion [42]; this self-citation is load-bearing for the SYK test. Since the central Rindler result is independent and the stringy-horizon generalization is explicitly conjectural, the circularity is partial rather than total: the claimed SYK verification reduces by construction, while the Rindler/Regge bridge retains independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The Rindler computation uses only conformal kinematics and standard pole/zero analysis; no parameter is fitted. In the SYK section, λ0 and η are physical model parameters from [26], not ad hoc fits. The conjectural extensions to stringy black holes and the unpublished universality claim in [42] are the main additional inputs. No new particles, forces, dimensions, or conserved quantities are postulated.

axioms (6)
  • domain assumption Lyapunov exponent of Rindler CFT is given by λ = j_leading(0) − 1 (Eq. 7), where j_leading is the analytic continuation of the leading Regge trajectory.
    Invoked in Section II after Eq. (4) to connect pole-skipping points to the Lyapunov exponent; standard conformal Regge theory, not re-derived here.
  • domain assumption Rindler wedge vacuum of a CFT is equivalent to a thermal state on R×H^{d−1} at inverse temperature β=2π.
    Used throughout Section II to treat vacuum two-point functions as thermal correlators and to define pole-skipping; standard Unruh/Rindler correspondence.
  • ad hoc to paper In the large-q SYK chain, the pole-skipping families (8) and (11) from [26] are the universal pole-skipping points of bilinear operators.
    Section IV and S-II: universality is deferred to the authors' unpublished companion [42]; without it, the SYK trajectory extraction lacks support.
  • ad hoc to paper The results for the Rindler CFT extend to a stringy black hole horizon: each bulk stringy field has pole-skipping at (4) and the set reconstructs bulk Regge trajectories.
    Section III states this as a conjecture ('we conjecture that the statements in items 1-4 above apply to a stringy black hole horizon as well'), not a derivation.
  • domain assumption Regge limit of CFT four-point functions maps to high-energy AdS scattering across the Rindler horizon; stringy Regge trajectories determine the non-maximal Lyapunov exponent.
    Section III, items 1-4 and surrounding text; relies on cited literature (e.g. [34,35,38]).
  • domain assumption The two-point function of a composite operator (S30) has pole-skipping points that reduce to those of the intermediate operators exchanged in the four-point function, up to ad hoc contributions.
    Section S-II A; this is the bridge used to extract universal pole-skipping data for bilinear operators in the SYK chain.

pith-pipeline@v1.3.0-alltime-deepseek · 28655 in / 21318 out tokens · 190034 ms · 2026-08-03T14:20:18.622242+00:00 · methodology

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read the original 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

Figures reproduced from arXiv: 2512.20700 by Hong Liu, Ping Gao.

Figure 1
Figure 1. Figure 1: FIG. 1. The overlay of Regge trajectories and the pole [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The pole-skipping points of the large [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. High-Order Pole-Skipping in Near-Extremal Holography

    hep-th 2026-07 conditional novelty 7.0

    In near-extremal holographic black holes, the n-th order pole-skipping momentum with mode index q becomes order-independent as T→0: k²_{n,q} → −m²h(r_h) + ½q(q−1)h(r_h)f″(r_h), with q identified as the AdS2 IR conform...

  2. Probing bulk geometry via pole skipping: from static to rotating spacetimes

    gr-qc 2026-04 unverdicted novelty 7.0

    Pole-skipping data encodes enough information to reconstruct the full metric of 3D rotating black holes and the radial functions of 4D separable rotating black holes, with Einstein equations becoming algebraic constra...

Reference graph

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