{"id":"747162e6-09f6-48e2-b256-3efd05ebb95a","arxiv_id":"2411.09667","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For charged hyperscaling-violating Lifshitz theories, the butterfly velocity is derived via shockwave and entanglement wedge methods, rewritten in entropy and charge per central charge, and shown to be non-monotonic in z for small entropy per central charge.","lead":"Using holography, the paper computes the butterfly velocity, the speed of information scrambling, for charged hyperscaling-violating Lifshitz quantum systems. It rewrites that speed using entropy and charge per central charge and finds how it depends on the theory's critical exponents.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Entanglement-wedge cross-check has an inconsistent intermediate step: Eq. (55) gives μ² inverted, so as printed Eqs. (54)-(55) yield v_B² a factor (d−1−θ)² too large in the AdS limit; Eq. (56) is correct only if Eq. (55) is a typo.","rationale":"The reader identified the entanglement-wedge section as the decisive weakness, citing a factor-of-nine discrepancy in the AdS limit, and returned a CONDITIONAL verdict. My independent check confirms that the printed chain Eqs. (54)-(55) does not imply Eq. (56): using μ² = f'_o/[2(d−1−θ)] gives v_B² = 2πT(d−1−θ), which is (d−1−θ)² times larger than Eq. (56). However, the correct extremization of the induced metric in Eq. (50) gives μ² = f'_o(d−1−θ)/2, and then Eq. (56) follows exactly. So the final cross-check result is right, and the flaw is a typographical error in the intermediate expression for μ² rather than a wrong physics result. The main shockwave-based formula (42) and the thermodynamic dictionary analysis (64) are unaffected. I therefore do not see a reason to change the reader's CONDITIONAL verdict: the paper is acceptable pending correction of Eq. (55) or removal of the cross-check claim. The reader's stated weakest assumption (NEC sufficiency for the permissibility region) is a legitimate physical caveat, but it is less concrete and less immediately load-bearing than the Eq. (55) inconsistency, which directly affects an advertised central result.","tokens_in":14315,"tokens_out":23098,"duration_ms":191116,"concrete_test":"Re-derive Eq. (51) from Eq. (50) by varying S_EE with √γ = r_F^θ r̃^{d−2}[1 − ε(2(u')²/f'_o + (d−1−θ)u²)]; the Euler-Lagrange equation is u'' + (d−2)u'/r̃ − (f'_o/2)(d−1−θ)u = 0, so μ² = (f'_o/2)(d−1−θ). Then check Eq. (54): v_B² = (2πT/μ)² = 2πT/(d−1−θ), reproducing Eq. (56). If instead one keeps Eq. (55) verbatim, the AdS-limit value is 6, not 2/3. Fix Eq. (55) or remove the claimed match.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim includes that the entanglement wedge method reproduces the shockwave result. As printed, the derivation does not: with r_h=1, f'_o=4πT, Eq. (55) gives μ² = f'_o/[2(d−1−θ)] = 2πT/(d−1−θ). Combining with Eq. (54), v_B = 2πT/μ yields v_B² = 2πT(d−1−θ), whereas Eq. (56) gives v_B² = 2πT/(d−1−θ). In the AdS limit (θ=0, z=1, q=0, d=4), these differ by a factor (d−1)² = 9. The near-horizon extremization leading to Eq. (51)/(55) appears to have dropped a factor of (d−1−θ) in the u² term of Eq. (50); the correct coefficient is μ² = f'_o(d−1−θ)/2, which makes Eq. (56) follow. Thus the advertised matching is unsupported as written, although the final formula (56) and the shockwave formula (42) may both be correct. This is the most load-bearing issue because the abstract and Section IV explicitly promise this cross-check.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14579,"tokens_out":17788,"duration_ms":146167,"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":[{"comment":"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.","section":"Section IV, Eqs. (50)–(56)"}],"minor_comments":[{"comment":"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.","section":"Abstract and Section VI"},{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Section IV, Eq. (50)"},{"comment":"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.","section":"Section VI"},{"comment":"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.","section":"End of Section II A"}],"recommendation":"major_revision","confidential_remarks":"The entanglement-wedge algebra is the only substantive obstacle I found; the shockwave computation and the final formulas (42) and (56) appear sound. If the authors correct Eqs. (51) and (55) and adjust the surrounding text, the paper would be suitable for publication. I do not see grounds for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read for Eq. (42) and Eq. (64). The shockwave computation of v_B for charged HVL black holes is a genuine generalization of the uncharged, theta=0 result in [39], and the thermodynamic dictionary from [38] gives a compact expression in terms of S-tilde and Q-tilde that will be useful for people working on non-relativistic holography. The parameter scan is careful: the author restricts to positive T, NEC, and subluminal v_B, and finds genuinely new qualitative features, including non-monotonicity in z at small entropy and non-monotonicity in theta with nonzero charge. The main formula is likely correct; it reduces to AdS-Schwarzschild in the appropriate limit and the derivation follows the standard shockwave method. Citations look appropriate, with [39] as the right benchmark.\n\nThe problem is Section IV. The abstract promises that the entanglement wedge method reproduces the shockwave result. As printed, it does not. Eq. (55) gives mu^2 = f'_o / [2(d-1-theta)], but the correct coefficient to match Eqs. (54) and (56) is mu^2 = f'_o (d-1-theta)/2. In the AdS limit (d=4, theta=0, z=1, q=0, rh=1), Eqs. (54)+(55) give v_B^2 = 6, while Eq. (56) and the shockwave result give 2/3. The factor is (d-1)^2 = 9. This looks like a typo in Eq. (55) or in the near-horizon expansion leading to Eq. (51). Either way, the cross-check as written is unsupported. The shockwave result does not depend on this section, so the paper is not fatally undermined, but the abstract's 'matches' claim needs a corrected derivation or removal.\n\nThe softer issue is the use of the NEC as a sufficiency condition and the 'permissible' region in Section VI. The qualitative claims are all qualified as within this region, so if the region is mis-specified the claims shift. I don't see a sign of that, but the author should state clearly that 'permissible' is a working criterion, not a proven consistency bound.\n\nThe author also includes a self-referential note about 'upcoming works' and a conjecture that charge universally slows scrambling. That is clearly labeled as a conjecture, so I won't hold it against the paper, though it is out of place in a formal manuscript.\n\nWho is this for? Anyone computing scrambling speeds in non-relativistic holographic models. The paper deserves a serious referee; it is a legitimate computation with a concrete error in one section. I would send it to review with instructions that the entanglement wedge section be repaired or excised and the abstract adjusted accordingly.","headline":"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.","tokens_in":15159,"tokens_out":4376,"would_cite":true,"duration_ms":38630,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["holographic quantum chaos","butterfly velocity","hyperscaling violation","Lifshitz holography","out-of-time-ordered correlator","shockwave analysis","entanglement wedge","thermodynamic dictionary"],"falsifier":"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.","tokens_in":14059,"feed_emoji":"🦋","tokens_out":9999,"duration_ms":80334,"temperature":0.7,"pith_summary":"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.","feed_headline":"Butterfly speed depends on entropy in Lifshitz holography","feed_subtitle":"A formula ties the butterfly velocity to entropy and charge, and reveals when faster scrambling reverses.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the charged HVL black brane solution with planar horizon used throughout the shockwave and entanglement computations.","marker":"[25]"},{"why":"Provides the localized-shockwave method that yields the OTOC and the shift equation for h(x).","marker":"[27]"},{"why":"Gives the thermodynamic dictionary used to express v_B in terms of S-tilde and Q-tilde.","marker":"[38]"},{"why":"Shows how the butterfly velocity follows from entanglement wedge growth for isotropic planar geometries, the method used in Section IV.","marker":"[45]"},{"why":"Supplies the previous result to which the theta=0, q=0 limit of Eq. (42) reduces.","marker":"[39]"},{"why":"Establishes the entanglement growth velocity to which v_B is compared in the uncharged case.","marker":"[40]"}],"fun_headline_variants":["Butterfly velocity shows non-monotonic z-run in Lifshitz chaos","Butterfly speed rises with entropy in Lifshitz chaos","Butterfly speed peaks in z at low entropy in Lifshitz chaos","Entropy and charge set butterfly speed in Lifshitz holography","Charged Lifshitz chaos: butterfly velocity ties to entropy and charge"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Butterfly velocity shows non-monotonic z-run in Lifshitz chaos","Butterfly speed rises with entropy in Lifshitz chaos","Butterfly speed peaks in z at low entropy in Lifshitz chaos","Entropy and charge set butterfly speed in Lifshitz holography","Charged Lifshitz chaos: butterfly velocity ties to entropy and charge"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000909,"raw_usage":{"total_tokens":4020,"prompt_tokens":1173,"completion_tokens":2847,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":789,"completion_tokens_details":{"reasoning_tokens":2749}},"tokens_in":789,"tokens_out":2847,"duration_ms":20821,"temperature":1.0,"reasoning_tokens":2749,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T20:26:49.069573+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Roberts, D","cited_arxiv_id":null,"evidence_quote":"Provides the localized-shockwave method that yields the OTOC and the shift equation for h(x)."}],"review_version":1}