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Conformal perturbation theory, with a distributional interpretation of singular integrals, computes the shifted pole-skipping point of a deformed two-dimensional CFT and matches the holographic butterfly velocity.

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2026-08-04 15:42 UTC pith:B5S5NIEI

load-bearing objection A serious, honest CFT-side computation of pole-skipping away from conformality, with a real holographic cross-check; the main open question is whether the distributional regulator is unique. the 1 major comments →

arxiv 2509.18540 v3 pith:B5S5NIEI submitted 2025-09-23 hep-th cond-mat.stat-mech

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

Curtis T. Asplund , Sebastian Fischetti , Alexandra Miller , David M. Ramirez This is my paper
classification hep-th cond-mat.stat-mech MSC 81T40 PACS 11.25.Hf05.45.Mt
keywords pole skippingconformal perturbation theorybutterfly velocityLyapunov exponenthomogeneous distributionsretarded Green's functiondeformed CFTquantum chaos
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.

The paper tries to establish that pole-skipping points of the retarded stress-tensor Green's function can be computed away from conformality using conformal perturbation theory, provided the singular integrals are interpreted as homogeneous distributions. If correct, this gives a non-holographic route to Lyapunov exponents and butterfly velocities in two-dimensional quantum field theories deformed by a relevant operator. The central check is precise agreement with a holographic calculation of the butterfly velocity for a massive scalar deforming the bulk geometry, at leading nontrivial order in the deformation. The authors find that for h=1/2 the pole-skipping velocity shifts by 3(pi^2-8) beta^2 lambda^2/(8c) below unity, while the frequency stays at 2 pi i/beta, and that near h=1 the shift vanishes to leading order.

Core claim

The paper claims that the leading nontrivial correction to the retarded stress-tensor two-point function of a 2D CFT deformed by a relevant scalar operator of weight h in (0,1) can be computed in conformal perturbation theory, provided the singular integrals are read as homogeneous distributions. For h=1/2, the corrected skipped pole sits at (omega*, k*) = (2 pi i/beta)(1+O(lambda^3), 1+3(pi^2-8) beta^2 lambda^2/(8c)+O(lambda^3)), so the pole-skipping velocity is v_PS = 1 - 3(pi^2-8) beta^2 lambda^2/(8c)+O(lambda^3). The frequency remains at the maximal-chaos value, while the spatial momentum is shifted. The authors verify this against a holographic computation of the butterfly velocity from

What carries the argument

The central object is the distributional extension of singular correlation functions: the complex-homogeneous distribution D_H(1/z^2), the unique extension preserving complex homogeneity, and the associated homogeneous distribution D(1/|x|) on the real line. These give a finite, physically motivated meaning to the divergent integrals in the conformal-perturbation expansion, converting them into contour integrals that can be evaluated in closed form for h=1/2 and near h=1.

Load-bearing premise

The singular integrals in the conformal-perturbation expansion are made finite by a specific distributional prescription—complex-homogeneous extensions for 1/z^2 and associated homogeneous extensions for 1/|x|—and if the physically correct contact-term scheme differs, the perturbed skipped poles would shift.

What would settle it

Compute the order-lambda^2 correction to the out-of-time-ordered correlator in the deformed CFT and extract the butterfly velocity; if it disagrees with v_PS from (3.41)-(3.42), the distributional scheme or the pole-skipping/chaos link is wrong. A simpler falsifier: evaluate the retarded Green's function with an alternative contact-term regularization, such as a different extension of 1/|x|, and check whether the skipped pole moves.

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

If this is right

  • Pole-skipping survives deformation of a CFT by a relevant operator: the skipped pole remains at the maximal-chaos frequency omega* = 2 pi i/beta to order lambda^2, while its wavevector is shifted by 3(pi^2-8) beta^2 lambda^2/(8c).
  • The shifted skipped pole yields a pole-skipping velocity that equals the holographic butterfly velocity, so the OTOC-based chaos data can be read off from the retarded Green's function without computing an OTOC.
  • Near a marginal deformation (h close to 1), the pole-skipping point is unperturbed to O(lambda^2 (h-1)^2), so the maximal-chaos structure is robust in that limit.
  • Because the leading correction is O(1/c) and universal, any theory sharing the same central charge and relevant deformation must agree at this order, including holographic duals.
  • The distributional regularization provides a concrete prescription for extracting chaos data in non-holographic, non-conformal two-dimensional QFTs.

Where Pith is reading between the lines

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

  • Editorial extension: the same distributional regularization should apply to other singular correlators, such as those of conserved currents or higher-spin operators, offering a systematic way to compute pole-skipping data in deformed CFTs without holography.
  • Editorial extension: the non-uniqueness of D(1/|x|) up to delta functions is harmless for the skipped-pole location, but it would affect other Green's-function observables; a physical principle fixing the contact-term ambiguity remains to be identified.
  • Editorial extension: the paper reports exact matching at h=1/2 and near h=1; testing intermediate h, where holography predicts a nontrivial velocity curve, with the CFT machinery would extend the check and could reveal whether the distributional scheme is correct for all h.
  • Editorial extension: a direct order-lambda^2 OTOC computation in the deformed CFT could confirm independently that Lyapunov and butterfly data match pole-skipping data, closing the logical circle outside holography.

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

1 major / 4 minor

Summary. The paper studies pole skipping in the retarded stress-tensor two-point function of a two-dimensional CFT perturbed by a relevant scalar primary of weight h in (0,1). Using conformal perturbation theory, the O(lambda^2) correction is expressed as integrals of bare CFT correlators. The singular T(u)O(z) insertions are interpreted with complex-homogeneous distributions, yielding a closed-form Euclidean Green's function for general h. For h=1/2 and h near 1, the manuscript Fourier transforms to Lorentzian frequency space and computes the perturbed skipped pole, Eqs. (3.41)-(3.42), and the pole-skipping velocity. It then computes the butterfly velocity in a scalar-deformed BTZ black hole and finds precise agreement at h=1/2 and consistency near h=1. The paper argues that this validates the distributional scheme and extends the pole-skipping/chaos connection away from conformality and away from holography.

Significance. If correct, the paper is significant: it computes a non-holographic, non-conformal correction to the pole-skipping point directly from CFT data and shows that it tracks the holographic butterfly velocity, supporting the pole-skipping/chaos connection beyond the usual holographic setting. The manuscript is careful and self-critical: it includes independent consistency checks (translation invariance, light-cone support, vanishing at h=1, cancellation of 1/epsilon divergences between pole and cut contributions), extensive appendices with the key integral evaluations, and an independent bulk calculation. The main caveat is that the distributional regularization is a postulate; the paper itself presents the holographic match as a posteriori validation. Because the result is a concrete, falsifiable coefficient for the O(lambda^2) shift, it should be of interest to the quantum-chaos and CFT communities, provided the regularization ambiguity is addressed.

major comments (1)
  1. [§3.1-§3.2 and §4.3, Eqs. (2.26), (3.5), (3.41)-(3.42)] The load-bearing step is the replacement of T(u)O(z) poles by the unique complex-homogeneous distribution D^H(1/z^2). Uniqueness within the homogeneous class does not fix the physical contact-term ambiguity: any other extension of 1/z^2 differs by delta functions and derivatives supported at the insertion point. Because these delta terms sit inside the double integral in (2.26)/(3.5), they do not merely add a local polynomial in u-v; after integrating over the second insertion they can produce non-polynomial functions whose Fourier transforms may be singular at omega=k, precisely where the zero and pole surfaces are intersected to find the skipped pole. The argument in §3.4 that an additive N(omega,k) cannot move the skipped pole applies to the later 1/|x| Fourier ambiguity (3.26), not to the earlier 1/z^2 ambiguity. The paper should either prove that such contact terms are fixed or abse
minor comments (4)
  1. [§3.4] The notation for omega2 and k2 is confusing: in Eqs. (3.38)-(3.40) these are dimensionless coefficients, while the physical values reappear only in (3.41)-(3.42). Please define the notation explicitly and keep the distinction throughout.
  2. [§5] Typo: "skipped skipped poles" should be "skipped poles".
  3. [Eq. (3.11) and Appendix B] The variables zeta and bar-zeta are used in (3.11) but only zeta is defined in (B.6). Please define bar-zeta (or state explicitly that it is the independent conjugate light-cone combination) to avoid ambiguity.
  4. [§3.3, Eq. (3.15)] The split into Fpole and Fcut is natural, but it would help the reader if the text stated explicitly that the 1/epsilon divergences between (3.18) and (3.19) cancel before the epsilon -> 0 limit is taken; the current text says this but only after displaying the two singular expressions.

Circularity Check

0 steps flagged

No circularity: the O(λ²) pole-skipping shift is computed from CFT correlation data and independently cross-checked against a holographic butterfly-velocity calculation; the distributional regularization is an explicitly flagged assumption, not a fitted input or self-citation chain.

full rationale

The paper's central result, the O(λ²) shift of the pole-skipping point, is obtained by evaluating the conformal-perturbation-theory integral I(u,v) using CFT Ward identities and the scalar two-point function, then Fourier transforming and solving for the intersection of the zero and pole surfaces. No parameter is fitted to the final skipped-pole location. The unperturbed Green's function G0 is taken from [28] (one of the present authors) and [50], but this is background input for the unperturbed pole, not the new O(λ²) shift, and it does not contain or assume the target result. The holographic butterfly velocity is computed independently from the bulk Einstein-scalar equations and the standard shock-wave formula, with the scalar source normalization fixed by the standard holographic dictionary; it is not derived from the CFT pole-skipping computation. The distributional interpretation of singular integrals (Section 3.1) is a genuine assumption: the paper explicitly says it is 'natural' and that its physical correctness is verified only a posteriori by holographic agreement. This is an acknowledged limitation and a possible source of scheme-dependence, but it is not circular, because the alternative contact-term schemes are not used to define the result and the calculation is not constructed so that the answer is encoded in the input. The paper is self-contained and externally benchmarked against holography, so the circularity score is 0.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

All quantities (lambda, h, beta, c) are inputs of the problem; no constants are fitted to data. The divergences are regulated by logarithmic subtractions whose coefficients cancel between pole and cut contributions. No new particles, forces, dimensions, or conserved quantities are introduced; the distributional regulators are mathematical tools, not physical entities.

axioms (5)
  • domain assumption The deformed theory is S = S_0 + lambda integral O, with O a scalar Virasoro primary of weight h in (0,1), and ordinary perturbation theory is valid at O(beta) timescales.
    Section 2.1, equations (2.2)-(2.10); central setup, not derived, and RG improvement is explicitly neglected.
  • ad hoc to paper Singular T(u)O(z) correlators are extended by the unique complex-homogeneous distribution D_H(1/z^2), and real-line Fourier transforms use the associated homogeneous distribution D(1/|x|).
    Section 3.1 and Appendix A; the entire finite evaluation of G^E_2 depends on this scheme, whose physical correctness is verified only a posteriori by holography.
  • domain assumption The unperturbed retarded Green's function G^L_0 and the pole-skipping matching procedure of [28] are valid.
    Equation (3.33) and Section 3.4; [28] is by a co-author, but it supplies background, not the target result.
  • domain assumption Pole-skipping points of the stress tensor map to Lyapunov exponent and butterfly velocity via lambda_L = -i omega* and v_B = omega*/k* in equation (1.2), as established for holographic theories in [41].
    Introduction, equation (1.2); the paper does not prove this relation for non-holographic theories and explicitly leaves OTOC computation to future work.
  • domain assumption The holographic dictionary: G_N = 3/(2c), m^2 = 4h(h-1), and the shock-wave butterfly velocity formula v_B^2 = f'(r0)/s'(r0) from [55].
    Section 4; used only for the cross-check, not for the CFT derivation.

pith-pipeline@v1.3.0-alltime-deepseek · 33321 in / 16076 out tokens · 127263 ms · 2026-08-04T15:42:28.096090+00:00 · methodology

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read the original abstract

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.

discussion (0)

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Forward citations

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