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Chaos in hyperscaling violating Lifshitz theories

T0 review · 1 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper derives the butterfly velocity for charged hyperscaling-violating Lifshitz theories, shows it is reproduced by entanglement wedge reconstruction, and maps how it depends on entropy and charge.

desk verdict The shockwave v_B formula for charged HVL theories and its thermodynamic form are likely correct and worth citing, but the advertised entanglement-wedge cross-check has a factor error as printed and needs a correction before the paper's central claim can be taken at face value. read the letter →

arxiv 2411.09667 v2 pith:J5G45HJE submitted 2024-11-14 hep-th gr-qcnlin.CDquant-ph

Nikesh Lilani This is my paper · ORCID
classification hep-thgr-qcnlin.CDquant-ph
keywords holographicquantumchaosbutterflyvelocityhyperscalingviolationLifshitzholographyout-of-time-orderedcorrelatorshockwaveanalysisentanglementwedgethermodynamicdictionary
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper derives a closed-form expression for the butterfly velocity $v_B$, the speed at which a local perturbation scrambles information across a thermal quantum system, for charged hyperscaling-violating Lifshitz (HVL) holographic theories. It obtains the same formula by shockwave analysis of the out-of-time-ordered correlator and by entanglement wedge reconstruction. Using a thermodynamic dictionary, it rewrites $v_B$ solely in terms of boundary quantities: entropy per central charge $\tilde{S}$, charge per central charge $\tilde{Q}$, the Lifshitz exponent $z$, and the hyperscaling-violating parameter $\theta$. The paper then maps the parameter dependence, finding non-monotonicity in $z$ for $\tilde{S}<1$, non-monotonicity in $\theta$ when charge is present, and monotonic growth with $\tilde{S}$ and decay with $|\tilde{Q}|$ inside the region where temperature is positive, the null energy condition holds, and $v_B$ is subluminal. These features matter because they tie a diagnostic of quantum chaos directly to boundary thermodynamics and theory data.

What carries the argument

The load-bearing mechanism is the shockwave picture of operator growth in a two-sided black hole: an early perturbation is blue-shifted into a thin shell localized on the horizon, and the transverse shift $h(x)$ of infalling geodesics is fixed by the Einstein equation; the butterfly velocity is read from the exponential decay rate of $h(x)$ in $x$, $v_B=2\pi/(\beta\chi)$. A second, independent route uses the near-horizon profile of the Ryu-Takayanagi surface, whose growth rate gives the same $v_B$. The remaining ingredient is the holographic thermodynamic dictionary, which converts the bulk parameters $r_h$ and $q$ into the boundary ratios $\tilde{S}$ and $\tilde{Q}$, so that $v_B$ is expressed without reference to bulk integration constants.

What would settle it

Compute $v_B$ for a fixed charged HVL geometry by an independent method that avoids the near-horizon and large-x approximations, such as a direct numerical solution of the full shockwave equation or a pole-skipping computation, and compare with Eq. (42) at the same $\{z,\theta,\tilde{S},\tilde{Q}\}$; a discrepancy at any permitted point, or a numerical maximum of $v_B(z)$ at a different location than Eq. (42) predicts, would falsify the central claim.

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Extended reading notes

Core claim

The central discovery is the butterfly-velocity formula for charged HVL black branes, $$$v_B^{2}$=\frac{$r_h^{{2(z-1)}}$(d-1+z-\$\theta$)+$q^{2}$ $r_h^{{-2(d-1-\theta)}}$(3-d-z+\$\theta$)}{2(d-1-\$\theta$)},$$ with $r_h$ the horizon radius, $q$ the charge parameter, $d$ the boundary dimension, $z$ the dynamical critical exponent, and $\theta$ the hyperscaling-violating parameter. The same expression follows from a near-horizon extremal-surface (entanglement wedge) computation, so the shockwave and entanglement pictures agree. Through the dictionary, this becomes an expression in boundary thermodynamics: $$$v_B^{2}$=\frac{\tilde{S}^{\frac{2(z-1)}{d-\$\theta$-1}}(d-1+z-\$\theta$)(d-\$\theta$-1)-(\tilde{Q}/\tilde{S})^2}{2(d-\$\theta$-1)^2},$$ where $\tilde{S}$ is entropy per central charge and $\tilde{Q}$ is charge per central charge. The paper's substantive claim is that the scrambling speed is a non-trivial function of these thermodynamic ratios: it develops a maximum in $z$ in the entropy-poor regime $\tilde{S}<1$, a maximum in $\theta$ once $\tilde{Q}\neq 0$, while remaining monotonically increasing in $\tilde{S}$ and decreasing in $|\tilde{Q}|$ over all allowed parameters.

Load-bearing premise

The load-bearing premise is that the null energy condition, together with positive temperature and subluminal butterfly speed, is a sufficient filter for physically consistent holographic parameters; if that filter admits or excludes the wrong region, the reported maxima and monotonicity statements could shift.

Editorial extensions

If this is right

  • If Eq. (42) is correct, the butterfly velocity in HVL theories is not fixed by $z$ alone; the entropy-to-central-charge ratio determines whether increasing $z$ speeds up or slows down scrambling.
  • The matching of the shockwave and entanglement-wedge derivations gives a consistency check that the same scrambling speed governs both the OTOC and the growth of the entanglement wedge.
  • Writing $v_B$ in terms of $\tilde{S}$ and $\tilde{Q}$ means the thermodynamic data of the boundary theory determine chaos properties, so measurements of one constrain the other.
  • Charge always reduces the scrambling speed (by the magnitude of $\tilde{Q}$) in the allowed parameter region, and the paper conjectures this may be universal across holographic models.
  • For the uncharged case $q=0$, $v_B$ coincides with the entanglement growth velocity at saturation, linking chaos to thermalization speed.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One extension not pursued in the paper is the spherical and hyperbolic horizon topology; if the monotonicity structure survives there, the non-monotonic points may mark sharp transitions in scrambling behavior across theories.
  • The conjectured universality of charge slowing scrambling could be tested by computing $v_B$ in other charged holographic models, such as those with momentum relaxation or higher-derivative corrections, and checking whether the decrease with $|\tilde{Q}|$ persists.
  • The clean dependence on $\tilde{S}^{2(z-1)/(d-\theta-1)}$ suggests a scaling relation between scrambling speed and entropy density; a sharper formulation might look for a speed bound in the non-relativistic regime, which the paper does not address.
  • Because the non-monotonicities in $v_B$ do not coincide with those in temperature, identifying the microscopic quantity that controls the switch from increasing to decreasing $v_B$ could be a productive next step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The manuscript computes the out-of-time-order correlator butterfly velocity v_B for charged hyperscaling-violating Lifshitz (HVL) black branes using shockwave analysis, obtaining the closed-form expression in Eq. (42). It then attempts an independent derivation via entanglement wedge reconstruction, claims agreement in Eq. (56), and uses the thermodynamic dictionary of [38] to express v_B in terms of boundary quantities, yielding Eq. (64). Section VI analyzes the dependence of v_B on the dynamical exponent z, the hyperscaling-violating parameter theta, the entropy ratio S-tilde, and the charge ratio Q-tilde, subject to positivity of temperature, the null energy condition, and subluminality.

Significance. If the results hold, Eq. (42) is a useful explicit formula for the butterfly velocity in a broad class of holographic theories with Lifshitz and hyperscaling-violating scaling, and Eq. (64) gives a concrete link between chaos and boundary thermodynamics. The shockwave computation follows a standard and mostly transparent route, and the neutral AdS limit is correctly reproduced. The monotonicity observations, especially the entropy-dependent non-monotonicity in z, are interesting and would be falsifiable predictions. The paper is self-contained in its main shockwave derivation, and the final formulas are parameter-free in the sense that no fitted parameter is recycled.

major comments (1)
  1. [Section IV, Eqs. (50)–(56)] The entanglement-wedge derivation is algebraically inconsistent as printed. Extremizing the expansion (50) gives u''(r) + (d-2)u'(r)/r - [f'_o (d-1-theta)/2] u(r) = 0, not Eq. (51), which has f'_o/[2(d-1-theta)] in the last term. Consequently Eq. (55) should read mu^2 = f'_o (d-1-theta)/2. With the printed Eq. (55), combining with Eq. (54) yields v_B^2 = 2 pi T (d-1-theta); in the neutral AdS-Schwarzschild limit (d=4, theta=0, z=1, q=0, r_h=1, T=1/pi) this gives v_B^2 = 6, a factor (d-1)^2 = 9 larger than the correct value v_B^2 = d/[2(d-1)] = 2/3. Eq. (56) is correct and does follow after the above fix, and it then matches the shockwave result Eq. (45). Because the abstract and Section IV explicitly advertise the entanglement-wedge computation as a cross-check of the shockwave result, this inconsistency must be corrected; as printed, the advertised matching is unsupported.
minor comments (5)
  1. [Abstract and Section VI] The abstract states that v_B varies non-monotonically with z for S-tilde < 1, but the text in Section VI qualifies this: for d=4, theta=0, Q-tilde=0, the non-monotonicities lie inside the permissible region only for approximately 0.688 < S-tilde < 0.877. The abstract should be amended to include this qualification.
  2. [Throughout] There are several typographical errors, for example "boudary" (page 2), "Lyapunaov" (page 7), "theoreies" (page 2), and "relevants" (page 14). A careful proofread is needed.
  3. [Section IV, Eq. (50)] The symbol r-tilde is used for the boundary spatial radial coordinate, which is easily confused with the entropy ratio S-tilde introduced later in Section V. Consider using a different notation, such as rho.
  4. [Section VI] All numerical claims, including the range 0.688 < S-tilde < 0.877, are computed for d=4. This is stated in the captions of Figures 3–6, but it should also be stated prominently in the main text of Section VI, since the range may depend on the boundary dimension.
  5. [End of Section II A] The sentence "Note that the black hole solution is only valid for theta < d-1 and d-theta+z-3 > 0" appears before the derivation of the NEC constraints. It would be clearer to state whether these validity conditions come from [25] or are derived here.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the shockwave and entanglement-wedge velocities are independent computations from the same metric, and the thermodynamic dictionary is used only to reparametrize bulk parameters; the printed matching inconsistency in Eqs. (54)-(56) is an algebraic issue, not circularity.

full rationale

The paper's central derivation chain is self-contained against external benchmarks and does not reduce to its own inputs. The shockwave butterfly velocity, Eq. (42), is obtained from the Einstein-equation shockwave equation, Eq. (32), through the near-horizon metric functions A(0), B(0), and ∂u∂vB(0); no data are fitted and no parameter is recycled as a prediction. The entanglement-wedge computation is a separate route: it starts from the induced metric, Eq. (48), extremizes the area functional near the horizon, Eq. (51), and extracts vB from the decay rate μ of the near-horizon RT profile, Eqs. (52)-(56). Because both computations use the same bulk background, their agreement is a genuine cross-check rather than a circular reduction. The as-printed step from Eq. (55) to Eq. (56) is internally inconsistent with Eq. (54), as the text itself would give a factor (d−1−θ)^2 mismatch; this is a correctness/typographical defect in the exposition, not a circularity, and it does not make the shockwave result depend on the entanglement-wedge result. The thermodynamic dictionary of Ref. [38] is used only to replace the bulk parameters rh and q by S-tilde and Q-tilde via Eqs. (60)-(61); this is an algebraic substitution, not a fitted input, and the authors of [38] are not the present author. The AdS/uncharged limit is checked against Ref. [39], providing an external benchmark. The only self-reference is the mention of the author's upcoming work in Sec. VI, which is used to motivate a conjecture about charge suppressing scrambling, not to derive any equation. Null-energy-condition constraints are imported from the external solution paper [25] and are stated assumptions rather than derived outputs. No load-bearing step reduces by definition to its inputs, and no prediction is forced by self-citation. Score 0 is therefore appropriate.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central derivation uses no fitted data and introduces no new entities. The paper's v_B formula depends on the HVL background of [25], the standard shockwave/OTOC dictionary, the Mezei-Stanford entanglement wedge method, and the thermodynamic dictionary of [38]. The entanglement wedge section appears internally inconsistent, so the ledger flags that cross-check as an unverified domain assumption rather than as a free parameter.

assumptions (6)
  • domain assumption The null energy condition serves as a sufficient condition for a physically consistent holographic dual in the semiclassical limit.
    Stated in Section II A and used to derive the constraints (12)-(14) and to define the permissible parameter region in Section VI.
  • domain assumption The charged HVL black hole metric (1)-(2) obtained in [25] is the correct holographic dual, with validity restricted to theta < d-1 and d-theta+z-3 > 0.
    Invoked throughout Sections II to VI; all subsequent formulas depend on this background.
  • standard math Near the horizon, the tortoise coordinate is dominated by the logarithmic term r* approximately log(r-r_h)/(f'(r_h) r_h^{z+1}), which leads to r approximately r_h - uv.
    Used in Section III, Eqs. (24)-(26); this is a standard near-horizon expansion, exact only in the shockwave-localized limit.
  • domain assumption The standard shockwave and OTOC dictionary applies to HVL theories, so the OTOC growth is governed by the geodesic shift with lambda_L = 2 pi T and v_B = 2 pi/(beta chi).
    Used in Section III following [26, 27, 45]; the paper does not rederive this dictionary.
  • domain assumption The thermodynamic dictionary of [38] correctly maps bulk horizon radius and charge parameter to boundary entropy and charge per central charge for planar charged HVL black holes, with Z_o held constant.
    Used in Section V, Eqs. (57)-(64); the monotonicity analysis relies on this mapping.
  • domain assumption Entanglement wedge reconstruction and the near-horizon RT surface growth rate give the butterfly velocity for isotropic planar geometries.
    Used in Section IV following [45]; this method is imported as established and is the source of the cross-check claimed in the paper.

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Pith. "Pith review of Chaos in hyperscaling violating Lifshitz theories." pith.science (2026). https://pith.science/paper/J5G45HJE

@misc{pith2026241109667,
  author       = {Pith},
  title        = {Pith review of: Chaos in hyperscaling violating Lifshitz theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J5G45HJE}},
  note         = {Machine review of arXiv:2411.09667}
}
abstract

We holographically study quantum chaos in hyperscaling-violating Lifshitz (HVL) theories (with charge). Specifically, we present a detailed computation of the out-of-time ordered correlator (OTOC) via shockwave analysis in the bulk HVL geometry with a planar horizon topology. We also compute the butterfly velocity ($v_{B}$) using the entanglement wedge reconstruction and find that it matches the result obtained from the shockwave analysis. Using a recently developed thermodynamic dictionary for HVL theories, we express $v_B$ purely in terms of boundary thermodynamic variables. Furthermore, we analyze in detail the behavior of $v_{B}$ with respect to the dynamical critical exponent ($z$), hyperscaling-violating parameter ($\theta$), entropy (more precisely, the ratio of entropy to the central charge, $\tilde{S}$), and charge (more precisely, the ratio of charge to the central charge, $\tilde{Q}$). Interestingly, $v_B$ varies non-monotonically with $z$ for $\tilde{S} < 1$, whereas it increases monotonically with $z$ for $\tilde{S} \geq 1$. Additionally, $v_B$ varies non-monotonically with $\theta$ for non-zero charge. Moreover, $v_B$ monotonically increases with $\tilde{S}$ and decreases with $\tilde{Q}$ for all allowed values of $z$ and $\theta$. All these features are reported for combinations /{$z$, $\theta$, $\tilde{S}$, $\tilde{Q}$/} for which the temperature is positive, the null energy condition is satisfied, and $v_B$ is not superluminal. Unpacking the non-monotonicities in $v_B$ can offer interesting insights into these theories.

Figures

Figures reproduced from arXiv: 2411.09667 by the authors.

Figure 1
Figure 1. FIG. 1: The Penrose diagram, explaining the effect of the shockwave created near the horizon. The [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The growth of the entanglement wedge (shaded in yellow) as the particle, originating from [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

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  1. Comparative study of the butterfly velocity in holographic QCD models at finite temperature and chemical potential

    hep-th 2025-05 conditional novelty 4.0 of 10

    Using three independent holographic methods, the authors obtain matching butterfly velocities for four QCD-like models and find a universal increase with temperature and decrease with chemical potential.

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