{"id":"e3cabf92-55de-47b5-82ef-705fb4021695","arxiv_id":"1908.03292","paper_version":3,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"The paper derives an equilibration bound for a truncated out-of-time-order correlator in the extended phase of any quadratic fermionic model, and numerically maps wavefront and momentum-space regimes in the Aubry-André model.","lead":"This paper studies how a small change spreads through a chain of particles with a quasi-periodic potential, the Aubry-André model, by computing out-of-time-order correlators. It claims a finite-time bound on how quickly these correlators settle to their equilibrium value in the extended phase of any quadratic model.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (27) replaces signed coefficients by absolute values while keeping phases; the resulting inequality is false, so the Eq. (32) equilibration bound is unproven.","rationale":"The reader's weakest assumption correctly identifies the invalid step. I re-derived the inequality and found a concrete counterexample with two opposite-sign coefficients; because the proof's subsequent steps rely entirely on replacing signed v_α by |v_α|, the central result is unsupported. The numerical sections and momentum fits are not at issue. A REJECT verdict remains appropriate.","tokens_in":82997,"tokens_out":5724,"duration_ms":59253,"concrete_test":"Compute the left and right sides of Eq. (27) for the two-term counterexample v_1=1, v_2=-1, G_1=0, G_2=π at T=3/2; the left side exceeds the right by 4/(1.5π)≈0.849, which settles that the inequality is false. For additional confidence, repeat the check on a small Aubry-André chain, e.g., L=8, λ=0.5, m=L/2, n=L/2+1, by evaluating both sides of Eq. (27) from the exact eigenvectors for T=1, 2, 5, and 10.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central bound Eq. (32) depends on Eq. (27), where the paper goes from ⟨g⟩_T = (1/T)∫|Σ_α v_α e^{iG_α t}|^2 dt to Q^2(1/T)∫|Σ_α p_α e^{iG_α t}|^2 dt with Q=Σ|v_α| and p_α=|v_α|/Q. This is not a consequence of the triangle inequality. For real coefficients with opposite signs it can fail pointwise and in time average. Example: v_1=1, v_2=-1, G_1=0, G_2=π, T=3/2. The original time average is (1/T)∫_0^T |1-e^{iπt}|^2 dt = (1/T)∫(2-2cosπt)dt; the proposed upper bound is (1/T)∫ |1+e^{iπt}|^2 dt = (1/T)∫(2+2cosπt)dt. The difference is -(4/T)∫_0^T cosπt dt = -4 sin(πT)/(πT) = 4/(1.5π) > 0, so the claimed bound is violated. Thus replacing v_α v_β by |v_α||v_β| inside the double sum is not valid when eigenvector products have mixed signs, and the subsequent Gaussian/probability arguments bound the wrong object. Since the paper does not restrict to positive A_{m,k}A_{n,k}A_{m,l}A_{n,l}, the proof of Eq. (32) and the claimed equilibration of OTOCs in all quadratic models does not follow. Numerical checks of Eq. (29) for λ=0 and 0.5 in Fig. 7 only show that the numerically evaluated g(t) lies below the plotted bound; they do not validate the derivation for cases the inequality fails, and Fig. 6's a(ε),δ(ε) are computed from the same unsupported step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies out-of-time-ordered correlators (OTOCs) in the quasi-periodic Aubry-André model, a free-fermion model with a localization transition. After introducing a quench protocol and deriving the anticommutator amplitude a_{m,n}(t), the authors report five time-regime results: early-time power-law growth, a wavefront described by the universal form of Eq. (5) and by a Gaussian form, a late-time equilibration bound for the squared anticommutator (Eq. (32)), and momentum-space OTOCs fitted to an ad-hoc form. The central claim is that the late-time bound proves equilibration of the OTOC in the extended phase and generalizes to all quadratic models.","tokens_in":83552,"tokens_out":8009,"duration_ms":74963,"significance":"If the late-time bound were correct, it would be a notable rigorous result: a finite-time equilibration bound for (a variant of) an OTOC in free-fermion systems, extending the equilibration machinery of Refs. [74-76]. The numerical study of the AA model provides useful exact-diagonalization data for wavefronts and momentum OTOCs. However, the proof contains a load-bearing error at Eq. (27), and the quantity being bounded is not the full OTOC of Eq. (1). The wavefront and momentum results are empirical fits with fitted parameters, so they are phenomenological rather than predictive. The strengths are the large-scale exact numerics and the transparent presentation of the attempted derivation; no machine-checked proofs or reproducible code are provided. On balance, the central claim is not established.","major_comments":[{"comment":"The step from Eq. (26) to Eq. (27) is not a consequence of the triangle inequality. Replacing the signed coefficients v_α by |v_α| while retaining the phases e^{iG_α t} does not yield an upper bound: for v_1=1, v_2=-1, G_1=0, G_2=π, the time average of |1-e^{iπt}|^2 over T=3/2 is 2+4/(3π), while the proposed bound with |v_α| gives 2-4/(3π), violating the inequality. Thus Eq. (29) and Eq. (32) bound the wrong object, and the claimed equilibration of the OTOC in any quadratic model is unproven. The numerical check in Fig. 7 only demonstrates the inequality for two specific parameter sets; it cannot cover the failure modes introduced by mixed-sign v_α.","section":"Section III C, Eq. (27)"},{"comment":"The proof applies to the squared anti-commutator |a_{m,n}(t)|^2 defined in Eq. (16), not to the full OTOC C(x,t)=⟨[A(t),B]^†[A(t),B]⟩ of Eq. (1). The full OTOC contains additional correlation terms (Appendix C, Eq. C33). Equilibration of |a_{m,n}|^2 does not imply equilibration of C(x,t). The abstract and conclusion state the result as 'equilibration of the OTOC' and as valid for 'any quadratic model', which overstates what is shown. The authors should at minimum restrict the claim to the truncated correlator.","section":"Section III C, Eq. (16)"},{"comment":"The universal wavefront form Eq. (5) is 'confirmed' using values of v_B, p, λ_L obtained by fitting the same data; this is circular and does not constitute an independent verification. Similarly, the Gaussian form Eq. (17) is an empirical fit with fitted parameters m and b, and the momentum OTOC form Eq. (36) is explicitly ad hoc. These results are phenomenology, not derived predictions, and the text should be reframed accordingly.","section":"Section III B"}],"minor_comments":[{"comment":"There are several typos: the abstract's 'ans' should be 'and'; Section II's statement that the model is identical to the Aubry-André model is redundant with the introduction; Appendix A's 'diagonilized' should be 'diagonalized'; Section IV's 'quecnhing' should be 'quenching'.","section":"Abstract; Section II; Appendix A; Section IV"},{"comment":"The caption says 'Results are for a fixed x=6 with L=1600 and λ=0' but panel (b) is λ=0.1; the parameters for each panel should be specified separately.","section":"Fig. 2 caption"},{"comment":"The vertical axis label 'g(t) T' should be '⟨g⟩_T' and the caption should define the time average explicitly.","section":"Fig. 7"},{"comment":"The formula for σ_G is written as sqrt(Σ p_α G_α^2 - (p_α G_α)^2); this should be sqrt(Σ_α p_α G_α^2 - (Σ_α p_α G_α)^2) to be unambiguous.","section":"Eq. (30)"},{"comment":"The bound c^4/L^2 is plausible but the derivation is compressed; the authors should show the number of terms in the sum over k≠l and the justification for treating c as system-size independent.","section":"Eq. (34)"}],"recommendation":"reject","confidential_remarks":"The Eq. (27) error is a fatal flaw in the central proof; the bound may not hold as stated. The other results are largely numerical fits. I do not see a simple way to repair the proof within the current framework, since the sign structure of the eigenvector products is essential. The paper would require a substantially different argument to substantiate the claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the numerical sections are worth a look, but the main analytical result is unproven. The step at Eq. (27) is not a triangle-inequality bound. You have |Σ vα e^{iGαt}|^2, and they write Q^2 |Σ pα e^{iGαt}|^2 with pα=|vα|/Q, claiming the latter is an upper bound. That fails as soon as the vα have mixed signs and the frequencies are such that the absolute-value sum has cancellations the original sum doesn't. Explicit counterexample: v1=1, v2=-1, G1=0, G2=π, T=1.5 gives time average 2+2/(1.5π) for the original and 2-2/(1.5π) for the proposed bound, so the claimed inequality points the wrong way. The whole chain from Eq. (27) to Eq. (32) rests on this, so the finite-time equilibration bound for quadratic models doesn't go through as written. The numerical checks in Fig. 7 only show the computed g(t) below the bound for two cases; they don't test sign cancellations.\n\nWhat is genuinely useful: the early-time power law, the wavefront analysis with the universal form at λ=0 and λ=0.1, the Gaussian form near x=vBt, and the momentum-space OTOC study are all clean exact numerics on a well-motivated model. The Gaussian waveform and the momentum fits are new observations. If the bound is the headline, the paper currently overstates it: the abstract says it bounds the OTOC and applies to any quadratic model, but the actual object is the squared anti-commutator term |a_{m,n}|^2, and the proof is unsupported.\n\nMy recommendation: send it to a referee. The numerical work deserves serious consideration, and the equilibration claim would be interesting if it can be repaired. But the authors need to either justify Eq. (27) (which I don't think is possible) or reframe the paper as a numerical study with an open analytical question. Desk rejection would lose the useful numerics.","headline":"Eq. (27) invalidates the claimed equilibration bound, but the numerics are solid enough to warrant referee time.","tokens_in":83985,"tokens_out":3551,"would_cite":false,"duration_ms":37384,"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 claims a finite-time bound, Eq. (32), proving that out-of-time-ordered correlators equilibrate in any extended quadratic model.","keywords":["out-of-time-ordered correlators","Aubry-André model","free fermions","equilibration","localization transition","quantum scrambling","finite-time bound","wavefront dynamics"],"falsifier":"Evaluate the exact time average $\\langle g_{m,n}(t)\\rangle_T$ for a finite Aubry-André chain (for example, $\\lambda=0.5$, $L=1600$, $m=L/2$, $n=L/2+6$) by summing $v_\\alpha v_\\beta e^{i(G_\\alpha-G_\\beta)t}$ directly with signed coefficients, and compare with the right-hand side of Eq. (32) computed from $|v_\\alpha|$. If the right-hand side is smaller than the exact average at any $T$ where $\\delta(\\epsilon)$ is claimed negligible, the bound as stated is false.","tokens_in":82840,"feed_emoji":"⚛️","tokens_out":6171,"duration_ms":63529,"temperature":0.7,"pith_summary":"This paper studies how quantum information spreads in the Aubry-André model, a one-dimensional free-fermion chain with a quasi-periodic potential and a sharp transition between extended and localized phases. Its central result is a finite-time bound, Eq. (32), stating that the time-averaged squared distance of the OTOC from its infinite-time value is controlled by $1/T$ plus a small constant $\\delta(\\epsilon)$, so the correlator equilibrates in the extended phase. This matters because it turns the observed late-time relaxation of OTOCs in integrable models into a rigorous statement, and because the derivation is written for any quadratic model, not just the Aubry-André chain. The paper also documents power-law early growth, a universal wavefront form at early times, a Gaussian waveform near $x=v_Bt$, and large late-time momentum-space OTOCs at the critical point.","feed_headline":"Bound proves OTOCs equilibrate in extended free-fermion models","feed_subtitle":"Late-time scrambling approaches equilibrium at a controlled rate in any quadratic model's extended phase.","key_machinery":"The central object is $g_{m,n}(t)=\\big||a_{m,n}(t)|^2-|\\omega_{m,n}|^2\\big|^2$, where $a_{m,n}(t)=\\sum_k A_{m,k}A_{n,k}e^{i\\epsilon_k t}$ is the time-dependent anticommutator of the fermion operators and $|\\omega_{m,n}|^2$ is its infinite-time average. The mechanism that carries the argument is the concentration function $\\xi_p(x)=\\max_\\beta \\sum_{\\alpha: G_\\beta \\le G_\\alpha \\le G_\\beta+x} p_\\alpha$ for the distribution of frequency gaps $G_\\alpha=\\epsilon_k-\\epsilon_l$ weighted by $p_\\alpha=|v_\\alpha|/Q$, combined with a Gaussian-profile bound on uniform time averages. This yields Eq. (32), whose $1/T$ term vanishes as $T\\to\\infty$ when $\\delta(\\epsilon)\\approx 0$, which the numerics show for the extended phase.","core_discovery":"The paper's central claim is that real-space OTOCs in the Aubry-André model equilibrate in the extended phase, and that the same argument works for any quadratic model in an extended regime. Concretely, it proves the bound $\\langle g_{m,n}(t)\\rangle_T \\le \\kappa\\pi Q^2\\big(a(\\epsilon)/(\\sigma_G T)+\\delta(\\epsilon)\\big)$, where $\\langle g_{m,n}(t)\\rangle_T$ is the time average of the squared distance between $|a_{m,n}(t)|^2$ and its infinite-time average. Because extended-phase eigenvector amplitudes scale as $1/\\sqrt{L}$, the infinite-time value vanishes in the thermodynamic limit, and with numerically small $\\delta(\\epsilon)$ the bound decays with $T$, establishing equilibration. The paper also claims that the infinite-time OTOC is zero in the extended phase (no persistent scrambling) and stays nonzero only within a localization length in the localized phase.","pith_inferences":["Editorial extension: the same concentration-function machinery could bound equilibration of higher-order OTOCs or other local observables in models whose single-particle spectrum is known, without requiring chaos.","Editorial extension: if the bound survives the sign issue at Eq. (27), it suggests a general route from single-particle spectral statistics to rigorous statements about operator spreading.","Editorial extension: a direct numerical test comparing the exact signed-coefficient time average with the absolute-coefficient right-hand side of Eq. (32) would show whether the extended-phase claim is robust or an artifact of the triangle inequality."],"forward_implications":["In the extended phase of the Aubry-André model ($\\lambda<1$), the time-averaged OTOC deviation $\\langle g_{m,n}(t)\\rangle_T$ decays at least as $O(1/T)$, so the correlator equilibrates to its infinite-time value.","Because $A_{m,k}\\sim 1/\\sqrt{L}$ in the extended phase, the infinite-time value itself vanishes in the thermodynamic limit, meaning the real-space OTOC equilibrates to zero (no persistent scrambling).","The same bound applies to any quadratic model in an extended regime, so the result is not specific to the quasi-periodic potential.","At the critical point and in the localized phase, $\\delta(\\epsilon)$ is not small, so the proof does not guarantee equilibration there, consistent with the observed lack of equilibration.","The Gaussian wavefront near $x=v_B t$ and the universal form ahead of it become the main finite-time signatures of operator spreading, with fitted velocities matching the maximal group velocity."],"supporting_citations":[{"why":"It supplies the definition of the Aubry-André model and the localization transition at $\\lambda_c=J$ that the whole study is built around.","marker":"[49]"},{"why":"It provides the disordered free-fermion OTOC framework and the Gaussian wavefront form that the paper compares and extends.","marker":"[11]"},{"why":"It supplies the universal wavefront form Eq. (5) and the saddle-point prediction $p=1/2$ that the paper confirms numerically.","marker":"[29]"},{"why":"It supplies the general method of bounding uniform time averages by Gaussian probability densities, used in Appendix D.","marker":"[74]"},{"why":"It supplies proposition 5, the bound on the concentration function $\\xi_p(x)$ used to convert Eq. (29) into Eq. (32).","marker":"[75]"},{"why":"It provides the related finite-time equilibration bounding technique that the proof adapts to OTOCs.","marker":"[76]"}],"fun_headline_variants":["Universal OTOC equilibration bound for any quadratic model","Quasi-periodic Aubry-André model: OTOCs equilibrate in extended phase","New equilibration bound for out-of-time-order correlators in free fermions","Late-time OTOC equilibration proven for extended free-fermion systems","Aubry-André OTOCs reach equilibrium: bound applies to all quadratic models"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of Eq. (32) relies on replacing the signed coefficients $v_\\alpha$ (products of eigenvector components, which can be negative) by their absolute values while keeping the oscillating phases $e^{i(G_\\alpha-G_\\beta)t}$; this step is only guaranteed to give an upper bound if all $v_\\alpha$ have the same sign, and eigenvector products of opposite signs break it.","fun_headline_variants_meta":{"raw":{"variants":["Universal OTOC equilibration bound for any quadratic model","Quasi-periodic Aubry-André model: OTOCs equilibrate in extended phase","New equilibration bound for out-of-time-order correlators in free fermions","Late-time OTOC equilibration proven for extended free-fermion systems","Aubry-André OTOCs reach equilibrium: bound applies to all quadratic models"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000281,"raw_usage":{"total_tokens":1689,"prompt_tokens":997,"completion_tokens":692,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":613,"completion_tokens_details":{"reasoning_tokens":601}},"tokens_in":613,"tokens_out":692,"duration_ms":7542,"temperature":1.0,"reasoning_tokens":601,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:19:42.790519+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the exact time average $\\langle g_{m,n}(t)\\rangle_T$ for a finite Aubry-André chain (for example, $\\lambda=0.5$, $L=1600$, $m=L/2$, $n=L/2+6$) by summing $v_\\alpha v_\\beta e^{i(G_\\alpha-G_\\beta)t}$ directly with signed coefficients, and compare with the right-hand side of Eq. (32) computed from $|v_\\alpha|$. If the right-hand side is smaller than the exact average at any $T$ where $\\delta(\\epsilon)$ is claimed negligible, the bound as stated is false.","supporting_citations":[{"cited_title":"Slow growth of out-of-time-order correlators and entanglement in integrable disordered systems","cited_arxiv_id":"1807.06039","evidence_quote":"It supplies the definition of the Aubry-André model and the localization transition at $\\lambda_c=J$ that the whole study is built around."},{"cited_title":"Universal scrambling in gapless quantum spin chains","cited_arxiv_id":"1904.09778","evidence_quote":"It provides the disordered free-fermion OTOC framework and the Gaussian wavefront form that the paper compares and extends."},{"cited_title":"Lin and O","cited_arxiv_id":null,"evidence_quote":"It supplies the universal wavefront form Eq. (5) and the saddle-point prediction $p=1/2$ that the paper confirms numerically."},{"cited_title":"Muralidharan, K","cited_arxiv_id":null,"evidence_quote":"It supplies the general method of bounding uniform time averages by Gaussian probability densities, used in Appendix D."},{"cited_title":"Abdul-Rahman, B","cited_arxiv_id":null,"evidence_quote":"It supplies proposition 5, the bound on the concentration function $\\xi_p(x)$ used to convert Eq. (29) into Eq. (32)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the related finite-time equilibration bounding technique that the proof adapts to OTOCs."}],"review_version":1}