{"id":"0158320e-65bb-4d31-9a07-d0580a121f8b","arxiv_id":"1909.02578","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In the Dicke and Lipkin-Meshkov-Glick models, exponential OTOC growth can be caused by classically unstable stationary points even when the surrounding classical dynamics is regular.","lead":"This paper shows that a widely used quantum chaos signal, the exponential growth of out-of-time-order correlators, can appear in systems whose classical motion is completely regular. The cause is an unstable saddle point in the classical energy landscape, not chaos, which matters for ion-trap experiments that use this signal.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Dicke-model half of the claim rests on TWA at ω0=3, but the exact-vs-TWA benchmark is at ω0=0.649 in the chaotic regime, so the regular-regime exponential growth is not yet validated.","rationale":"The paper's central claim is that exponential FOTOC growth in regular regimes of both the Dicke and LMG models is caused by unstable stationary points. The LMG result is backed by exact evolution and appears sound. The Dicke result is the experimentally relevant and novel part; it is computed with TWA at j=500 in Fig. 3. The only exact-vs-TWA check in SM Fig. S1 is at ω0=0.649, where the unstable point lies in a chaotic region, and separately at a regular point away from the unstable point. This does not validate TWA for the specific mechanism at play: quantum activation of an unstable point embedded in a regular region. TWA is semiclassical and could in principle overestimate the growth by letting sampled classical trajectories feel the unstable point while the exact quantum state would not spread as much. The paper explicitly states that the SM validation assures the use of TWA for larger j, so this is a load-bearing assumption at the exact parameter point where the headline phenomenon is demonstrated. This is a correctness risk, not a disagreement with consensus. I agree with the reader's weakest-assumption identification, and the recommended verdict is unchanged: the paper should remain conditional until the benchmark at ω0=3 is supplied.","tokens_in":19518,"tokens_out":5872,"duration_ms":62605,"concrete_test":"Perform exact diagonalization of the Dicke Hamiltonian at ω0=3, γ=0.66, ω=0.5 for j=100, the size used for exact calculations in the paper, using the converged-basis method described there, and compute σ_Q²(t)+σ_P²(t)+σ_q²(t)+σ_p²(t) for coherent states centered at O, A, B, and C. Compare the exponential slope 2Λ and the time window with the TWA result at the same j. If the slopes agree within Monte Carlo uncertainty, for example 10%, over the exponential window, the concern is resolved; if they do not, the Dicke-model claim in Fig. 3 lacks numerical support. A repeated comparison at j=200 would further establish finite-size stability.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The LMG model is supported by exact quantum evolution and is not the issue. The Dicke model carries the experimentally relevant part of the claim, and its central Fig. 3 is computed with the truncated Wigner approximation (TWA) at ω0=3, γ=0.66, ω=0.5, j=500. The only exact-vs-TWA validation offered in SM Fig. S1 is at ω0=0.649. At that parameter the unstable point sits inside a chaotic region, as the main text explicitly states, so the benchmark tests TWA in the chaotic regime. The regular-region benchmark in Fig. S1(b) uses a generic regular trajectory, not the unstable point or its immediate neighborhood. Neither test checks whether TWA reproduces the exact quantum FOTOC growth for states O, A, B, and C in Fig. 3. Because TWA replaces quantum evolution by classical trajectories sampled from the Wigner function, the exponential growth at O, A, B, and C could in principle be an artifact of sampled classical trajectories passing near the unstable point, while the exact quantum variance might grow differently. The validation in Fig. S1 also uses only σ_q², not the full sum σ_Q²+σ_P²+σ_q²+σ_p² used in Fig. 3. Since the abstract and title make a general claim and explicitly reference trapped-ion parameters, the Dicke component requires this check before the central claim is fully supported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript argues that exponential growth of out-of-time-ordered correlators is not by itself a reliable signature of classical chaos. For the LMG and Dicke models in parameter regimes where the classical limit is regular, the authors compute the Lyapunov exponent of an unstable stationary point from the Jacobian (Eqs. (3) and (6)) and compare it with the growth rate Λ extracted from the fidelity OTOC, defined as the variance of canonical coordinates. For the LMG model the comparison is made with exact quantum evolution; for the Dicke model it is made with the truncated Wigner approximation at large j. The main numerical finding (Fig. 3) is that coherent states centered not only at the unstable point but also at nearby points with zero classical Lyapunov exponents show an initial exponential FOTOC growth with rate 2Λ ≈ 2λ. The paper concludes that in these experimentally relevant systems the exponential OTOC growth is caused by unstable stationary points rather than by chaos.","tokens_in":19795,"tokens_out":6906,"duration_ms":77537,"significance":"If the Dicke part of the evidence is properly validated, this is a significant cautionary result for the current practice of identifying quantum chaos with OTOC scrambling. The analytical formulas are clear, the comparison with classical Lyapunov exponents is parameter-free and not obtained by fitting, and the LMG half of the claim is backed by exact quantum evolution. The paper also identifies a concrete experimental context, the trapped-ion Dicke simulator, where a positive quantum Lyapunov exponent can be observed in a regular classical regime. The main obstacle is the missing direct TWA benchmark at the regular-regime parameters used in the central Dicke figure; this is a fixable but load-bearing gap.","major_comments":[{"comment":"The validation of the truncated Wigner approximation against exact quantum evolution is performed only at ω0=0.649, where, as the main text states, the unstable point is immersed in a chaotic region. The central regular-regime Dicke result in Fig. 3(d) uses ω0=3, γ=0.66, ω=0.5, j=500. The benchmark in Fig. S1 also uses only σ_q², whereas the Dicke FOTOC in Figs. 2(b) and 3(d) is the four-term sum σ_Q²+σ_P²+σ_q²+σ_p². Because the Dicke model is the experimentally relevant part of the title and abstract claim, an exact-vs-TWA comparison is needed at ω0=3, for example at j=100, for the full sum and for the states O, A, B, and C, over the time window of exponential growth. Without this, the statement that the exact quantum evolution and the TWA match is not supported at the parameters where the regular-regime exponential behavior is demonstrated.","section":"SM III A, Fig. S1; main text 'Quantum-classical correspondence' and Fig. 3(d)"},{"comment":"The sentence 'given that the exact quantum evolution and the TWA match' is stronger than what the SM currently demonstrates, since the existing match is at ω0=0.649 and only for one quadrature. The statement should be qualified until the ω0=3 benchmark is supplied, and the Fig. 3 caption should state explicitly that panel (d) is computed with TWA while panel (b) is exact. This is a presentation issue that becomes load-bearing because it is used to justify the Dicke model's place in the paper's main conclusion.","section":"Discussion, first paragraph; Fig. 3 caption"}],"minor_comments":[{"comment":"The caption should state explicitly that the Dicke panel uses TWA while the LMG panel uses exact quantum evolution; the current wording 'the TWA is used' appears only in the body text.","section":"Fig. 2 caption"},{"comment":"Several equations in the Supplemental Material contain doubled norm bars such as '||e^{At}||'; this appears to be a LaTeX typo and should be corrected.","section":"SM Eq. (S5) and nearby text"},{"comment":"The phrase 'The choices of A, B, and C are done such that' should be 'The choices of A, B, and C are made such that'.","section":"Main text, 'Quantum activation of the instability'"},{"comment":"The exponential growth rates are described as 'exactly the same' on the basis of visual inspection; reporting the extracted slopes and the fitted time windows for each of the four states would make the claim quantitative and reproducible.","section":"Fig. 3(b) and 3(d)"}],"recommendation":"major_revision","confidential_remarks":"This is a well-written and potentially important paper. The LMG results and the analytical Lyapunov-exponent formulas are solid, and I do not see grounds for rejection. The missing TWA benchmark at the regular-regime Dicke parameters is straightforward to supply and should be requested before publication; once provided, the paper would be suitable for acceptance. The paper's cautionary message about OTOC growth as a stand-alone signature of quantum chaos is timely and fits the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the two things to know. The core claim—exponential FOTOC growth does not by itself imply chaos—is correct, and the LMG-model half of the paper is rock solid. The Dicke-model half is plausible but under-validated as written.\n\nWhat is new: the one-DOF critical-point mechanism was already in Hummel et al. and Rozenbaum et al. The extensions here are the two-DOF Dicke model at experimentally accessible parameters, the integrable LMG model, and the demonstration that generic neighboring coherent states, not just the unstable point, show the same exponential rate. The analytical Lyapunov formulas (Eqs. 3 and 6) are correctly worked out from the Jacobian eigenvalues, and the quantum LE is read from the FOTOC slope without any fitting and compared with the independently computed classical LE. No circularity.\n\nThe LMG result uses exact quantum evolution and matches λ perfectly. That alone establishes the phenomenon in a clean integrable system. The Dicke result, however, is computed with the truncated Wigner approximation (TWA) at ω0=3, γ=0.66, ω=0.5, j=500. The exact-vs-TWA validation in SM Fig. S1 is at ω0=0.649, a point the main text explicitly calls chaotic, and at a regular low-energy point; neither test covers the unstable point or its neighborhood at ω0=3. The validation also uses only σ_q², not the full sum σ_Q²+σ_P²+σ_q²+σ_p² used in Fig. 3. So the reader's concern is fair.\n\nHow serious is this? Not fatal. TWA is a standard method and typically reliable for short-time variance growth in regular regions, and the LMG section already carries the conceptual weight. But the specific quantitative claim for the Dicke model—the rate 2Λ≈2λ for points O, A, B, C—is not pinned down by the supplied numerics. A referee should ask for either an exact check at j=100 for ω0=3 or a TWA-vs-exact comparison for the full variance there. Adding error bars to Fig. 3 would also help, though that is minor.\n\nThe citation pattern is fine. The paper builds on [3,32] and cites its own [18] for the OTOC-classical-LE correspondence, which is independently supported. The text is honest about the open definition of quantum chaos and does not overstate.\n\nBottom line: this deserves a serious referee and should probably be published after a modest revision. If you work on OTOC-based diagnostics of chaos, read it and cite the caution.","headline":"Core claim is right but the Dicke-model numerics need one more validation pass at the reported parameters.","tokens_in":20383,"tokens_out":3189,"would_cite":true,"duration_ms":31931,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81Q50","37D45","81V80"],"pacs":["05.45.Mt","03.65.-w"],"model":"deepseek-v4-flash","headline":"Exponential growth of the fidelity OTOC occurs in the regular Dicke model and integrable LMG model because of unstable stationary points, not because of chaos.","keywords":["out-of-time-order correlators","quantum chaos","Dicke model","Lipkin-Meshkov-Glick model","Lyapunov exponents","unstable stationary points","excited-state quantum phase transition","fidelity OTOC"],"falsifier":"Perform an exact quantum simulation of the fidelity OTOC for the Dicke model at $\\omega_0=3$, $\\gamma=0.66$, $\\omega=0.5$, $j=500$, for coherent states centered at $Q=0.1,0.2,0.3$ at energy $E=-j\\omega_0$, and check whether the initial growth rate $2\\Lambda$ still matches $2\\lambda$ from Eq. (6); if no exponential window appears, the quantum-activation claim for the regular regime would fail.","tokens_in":19297,"feed_emoji":"⚛️","tokens_out":4635,"duration_ms":46229,"temperature":0.7,"pith_summary":"This paper challenges the widespread identification of exponential out-of-time-order-correlator (OTOC) growth with quantum chaos. It shows that the fidelity OTOC can grow exponentially in the Dicke model when the classical limit is regular, and in the classically integrable Lipkin-Meshkov-Glick model. The growth rate matches the positive classical Lyapunov exponent of an unstable stationary point even for initial states centered away from that point. The paper concludes that in experimentally accessible systems OTOC growth can signal the presence of unstable stationary points, not chaos, and so exponential OTOC growth is not by itself a reliable signature of quantum chaos.","feed_headline":"Exponential OTOC growth can occur without classical chaos","feed_subtitle":"In regular or integrable regimes, unstable stationary points alone produce positive quantum Lyapunov exponents in the Dicke and LMG models.","key_machinery":"The mechanism is the unstable stationary point of the classical Hamiltonian: the saddle point $(Q,P)=(0,0)$ in the LMG model and $(q,p,Q,P)=(0,0,0,0)$ in the Dicke model, whose Jacobian matrix has a positive real eigenvalue that defines the classical Lyapunov exponent $\\lambda$. In the quantum dynamics, an initially localized coherent state broadens into this unstable direction; the fidelity OTOC is computed as the variance of canonical operators, $\\sigma_Q^2+\\sigma_P^2$ for LMG and $\\sigma_Q^2+\\sigma_P^2+\\sigma_q^2+\\sigma_p^2$ for Dicke, whose initial growth rate is $2\\Lambda \\approx 2\\lambda$. Exact quantum evolution validates the truncated Wigner approximation for sizes where exact diagonalization is feasible, and the approximation is then used for larger $j$, where the correspondence improves as $j$ grows from 500 to 5000.","core_discovery":"The central claim is that positive quantum Lyapunov exponents, defined through the exponential growth rate of the fidelity OTOC, occur in quantum systems whose classical limits are regular or integrable. At an unstable stationary point the quantum exponent $\\Lambda$ coincides with the classical Lyapunov exponent $\\lambda$ in both models; more importantly, coherent states centered in the surrounding regular region exhibit the same initial exponential rate $2\\Lambda \\approx 2\\lambda$, even though those classical orbits have zero Lyapunov exponents. The exponential regime lasts shorter and saturates lower as the center moves away from the unstable point, but it remains present for generic nearby states. For the Dicke model this happens at parameters used in trapped-ion experiments, so those experiments may observe the effect of an unstable point rather than of classical chaos.","pith_inferences":["A testable extension would be to repeat the fidelity-OTOC measurement around hyperbolic fixed points of other collective models, such as the two-mode Bose-Hubbard dimer, where a similar exponential window should appear despite integrability.","The paper's mechanism implies that an OTOC growth-rate measurement alone cannot certify chaos; a practical diagnostic would combine the rate with level statistics or phase-space structure, a step the authors do not take.","Finite-size scaling is a natural next probe: one expects the duration of the exponential window to grow like $\\log j / \\lambda$, which could be checked numerically or in ion-trap experiments."],"forward_implications":["Exponential OTOC growth alone cannot certify quantum chaos, because it also occurs in regular and integrable systems around unstable stationary points.","Trapped-ion experiments on the Dicke model that measure the fidelity OTOC may observe exponential growth caused by an unstable stationary point rather than by classical chaos.","The quantum-classical correspondence between OTOC growth rate and classical Lyapunov exponent holds at the unstable point but breaks in its neighborhood, where quantum mechanics generates instability in a classically stable region.","The LMG model, being classically integrable, provides a simple platform to observe the same effect without the complications of a chaotic regime."],"supporting_citations":[{"why":"Provides the earlier argument that non-chaotic quantum systems can show early-time exponential instabilities, which this paper extends to the Dicke and LMG models.","marker":"[3]"},{"why":"Establishes the baseline relation between OTOC exponential growth rate and the classical Lyapunov exponent in chaotic systems.","marker":"[15]"},{"why":"Supplies the quantum and classical Lyapunov exponent analysis for atom-field interaction systems, including the Dicke model.","marker":"[18]"},{"why":"Defines the fidelity OTOC and provides the trapped-ion experimental parameters used in this paper.","marker":"[25]"},{"why":"Shows via semiclassical quantization that OTOCs can grow exponentially in critical one-degree-of-freedom systems, a result the paper extends to two-degree-of-freedom models.","marker":"[32]"},{"why":"Describes the trapped-ion implementation of the Dicke model that motivates the experimentally relevant parameters.","marker":"[42]"},{"why":"Provides the LMG model context and the formula for its stationary-point Lyapunov exponent used here.","marker":"[44]"},{"why":"Gives the classical chaos analysis of atom-field systems, including the Lyapunov exponent map shown in Fig. 1.","marker":"[67]"},{"why":"Supplies the truncated Wigner approximation method used for the large-$j$ Dicke model calculations.","marker":"[83]"},{"why":"Provides the Monte Carlo trajectory sampling procedure used to implement the truncated Wigner approximation.","marker":"[84]"}],"fun_headline_variants":["Quantum chaos signs without classical chaos","Unstable points mimic quantum chaos in regular systems","OTOC growth can occur in integrable models","Positive Lyapunov exponents from unstable points, not chaos"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The Dicke-model results in the regular regime rely on the truncated Wigner approximation, and its validation against exact quantum evolution is performed at a different parameter value where the unstable point lies inside a chaotic region, so the accuracy at the regular-regime parameters is assumed rather than directly demonstrated.","fun_headline_variants_meta":{"raw":{"variants":["Quantum chaos signs without classical chaos","Unstable points mimic quantum chaos in regular systems","OTOC growth can occur in integrable models","Positive Lyapunov exponents from unstable points, not chaos"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000141,"raw_usage":{"total_tokens":1116,"prompt_tokens":852,"completion_tokens":264,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":468,"completion_tokens_details":{"reasoning_tokens":206}},"tokens_in":468,"tokens_out":264,"duration_ms":3444,"temperature":1.0,"reasoning_tokens":206,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:46:18.541596+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Perform an exact quantum simulation of the fidelity OTOC for the Dicke model at $\\omega_0=3$, $\\gamma=0.66$, $\\omega=0.5$, $j=500$, for coherent states centered at $Q=0.1,0.2,0.3$ at energy $E=-j\\omega_0$, and check whether the initial growth rate $2\\Lambda$ still matches $2\\lambda$ from Eq. (6); if no exponential window appears, the quantum-activation claim for the regular regime would fail.","supporting_citations":[{"cited_title":"Sensitivity of quantum information to environment perturbations measured with a non-local out-of-time-order correlation function","cited_arxiv_id":"1808.04375","evidence_quote":"Shows via semiclassical quantization that OTOCs can grow exponentially in critical one-degree-of-freedom systems, a result the paper extends to two-degree-of-freedom models."},{"cited_title":"Dicke-model simulation via cavity-assisted Raman tran- sitions,","cited_arxiv_id":null,"evidence_quote":"Describes the trapped-ion implementation of the Dicke model that motivates the experimentally relevant parameters."},{"cited_title":"In our case, however, the statistics is Poisso- nian","cited_arxiv_id":null,"evidence_quote":"Provides the LMG model context and the formula for its stationary-point Lyapunov exponent used here."},{"cited_title":"Comparative quantum and semiclassical analysis of 7 atom-field systems. ii. Chaos and regularity,","cited_arxiv_id":null,"evidence_quote":"Gives the classical chaos analysis of atom-field systems, including the Lyapunov exponent map shown in Fig. 1."},{"cited_title":"Phase space representation of quantum dynamics,","cited_arxiv_id":null,"evidence_quote":"Provides the Monte Carlo trajectory sampling procedure used to implement the truncated Wigner approximation."}],"review_version":1}