{"id":"14c304c9-6f14-4779-adf3-3799c5266e37","arxiv_id":"2607.21583","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Long-time-averaged Krylov complexity in the Dicke model drops smoothly, then rises sharply across a coupling threshold, revealing a regular-to-chaotic dynamical crossover with distinct N-scaling.","lead":"Tuning the strength of the coupling between atoms and light in the Dicke model, the authors find that a measure of quantum 'complexity' first shrinks, then jumps sharply at a critical coupling. This offers a new, observable-based way to spot the boundary between regular and chaotic quantum dynamics.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Claimed nonanalytic complexity transition rests on finite-N, finite-time-window averages; no scaling or convergence analysis is provided to rule out a crossover artifact.","rationale":"The reader's weakest assumption identifies the exact load-bearing concern. The entire novelty claim—that Krylov complexity acts as an independent order parameter for a dynamical transition—depends on the sharp feature being intrinsic, not an artifact of the averaging protocol. The paper provides only three finite-N curves over a single time window, with no error bars, no transition-width scaling, and no statement of the boson truncation. This is especially acute because the regular regime's persistent oscillations (Fig. 2a) mean the finite-window average may depend on the window's phase relative to the oscillation period. The supplemental material itself shows that the first-maximum version of C_K gives a different g_tilde_c and that short-time observables change smoothly (Fig. S1), underscoring the sensitivity to the chosen diagnostic time. I do not see evidence of fabrication; the observation may well be a real crossover. But the central claim as worded is under-supported. The proposed finite-size/window/diagonal-ensemble check would settle whether the sharp change is a genuine transition or a crossover, so the reader's CONDITIONAL verdict is appropriate and unchanged.","tokens_in":18450,"tokens_out":4154,"duration_ms":45288,"concrete_test":"Perform exact diagonalization for N = 20, 30, 40, 60 at delta = Omega with the boson cutoff increased until Cbar_K is converged, and compute Cbar_K over the paper's window gt in [50, 200], longer windows gt in [100, 400] and [200, 800], and the exact infinite-time (diagonal ensemble) average. Extract the transition width Delta_g, e.g., the full width at half maximum of dCbar_K/dg_tilde near the steepest point, and track g_tilde_c and the dip depth versus N and window. If Delta_g does not decrease with N, or if the sharp feature shifts, weakens, or disappears in longer-window/diagonal-ensemble data, the claimed nonanalytic transition is not supported and the result should be reported as a sharp crossover.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that Cbar_K and Sbar_K undergo a sharp, nonanalytic change near g_tilde_c ~ 0.5 (abstract; main text: 'sharp (hence nonanalytic)'). For any finite N and fixed boson truncation, Cbar_K(g_tilde) is an analytic function of the Hamiltonian parameters, so nonanalyticity can only emerge in the N->infinity or t->infinity limit. The evidence is limited to N = 20, 30, 40 and a single time window gt in [50, 200] (Fig. 1), with no error bars, no study of window dependence, and no finite-size scaling of the transition width. This matters because the regular regime shows persistent oscillations of C_K(t) (Fig. 2a); a time average over a fixed window can be sensitive to oscillation phase and period, so the dip near g_tilde_c may be a window artifact. The toy model Eq. (3) has a genuine delocalization transition at eta = 1/2, but the authors state that fitted eta in the Dicke model stays below 1/2 and that disorder is decisive (Figs. S7, 5c-d), so the toy model does not independently establish the transition. Without demonstrating that the steepening sharpens with N and is stable in t, the central claim is an overclaim; a smooth crossover remains plausible.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This letter studies quench dynamics in the Dicke model through Krylov complexity. It claims that the long-time-averaged Krylov complexity C̅_K and Krylov entropy S̅_K undergo a sharp, nonanalytic transition as the dimensionless coupling g̃ is tuned, near g̃_c ≈ 0.5, separating a 'regular' confined oscillatory regime from a 'chaotic' spaghettified regime. Evidence is presented as finite-N data (N = 20, 30, 40), a complexity phase diagram in (g̃, Ω/δ), N-scaling C̅_K ∼ N^{3/2} from below and ∼ N^{5/2} above, and an interpretation in terms of competition between linear confinement, Rindler-like hopping, and deterministic disorder in Lanczos coefficients, supported by a solvable linear toy model.","tokens_in":18831,"tokens_out":7171,"duration_ms":71640,"significance":"If fully established, the result would be significant: Krylov complexity would provide a state-dependent, direct order parameter for dynamical transitions, with a transparent wave-packet picture and distinct finite-size scaling. The paper has notable strengths: it studies a realistic, experimentally relevant model; it carefully compares the original Lanczos coefficients with a smoothed version to isolate the role of deterministic disorder; and it provides an analytic eigenfunction solution of the toy model at η = 1/2. However, the central nonanalyticity claim is not yet supported by the evidence, and the current significance is conditional on closing that gap.","major_comments":[{"comment":"The claim of a 'sharp (hence nonanalytic)' transition is not established. The data are for N=20,30,40 with a single time average over gt∈[50,200]; no error bars, no stated boson-number truncation, and no finite-size scaling of the transition width are given. For fixed N and fixed truncation, C̅_K(g̃) is analytic in g̃, so nonanalyticity must be demonstrated via the N→∞ and/or t→∞ limit. The statement that the transition 'gets sharper for larger N' is qualitative. A scaling collapse of dC̅_K/dg̃ (peak height and width vs N) and a convergence check in time are needed to rule out a crossover that saturates. The toy model Eq. (3) does not fill this gap: Fig. S7 shows η(g̃) stays below 1/2 and gives 'no hint' of the C̅_K transition.","section":"Transition in complexity / Fig. 1(a,c)"},{"comment":"The long-time average is taken over gt∈[50,200], but the regular regime shows persistent oscillations of C_K(t). The averaged value can depend on the chosen window and may shift with oscillation phase/period as g̃ varies; the dip near g̃_c could be a window artifact. Please show for representative g̃ in both regimes that C̅_K(T) converges as T increases (e.g., varying the window and checking for a plateau), and provide estimates of the sampling fluctuations. This is essential because C̅_K is used as the order parameter for the transition.","section":"Fig. 2(a), Fig. S3, definition of C̅_K"},{"comment":"The claimed N^{3/2}/N^{5/2} scaling is not quantitative. Only three N values are used for a power-law fit, and the Supplemental Material explicitly states that neither N^{3/2} nor N^{5/2} 'fully captures the N dependence' and leaves the scaling coefficients for future work. If the scaling behavior is meant to corroborate the transition, the exponents and their uncertainties must be determined with a systematic finite-size analysis; otherwise the claim should be softened to 'consistent with' a crossover, which would change the paper's central message.","section":"Supplemental Material 'SCALING OF C_K', Fig. S4(b)"}],"minor_comments":[{"comment":"The caption lists panel (e) for the Krylov entropy S̅_K, but the figure contains panels (a)–(d). The entropy panel should be labeled (d), not (e).","section":"Fig. 1 caption"},{"comment":"The cutoff in the bosonic Hilbert space used for the finite-N simulations is not stated. This is needed to reproduce the numerics and to confirm that the results are not affected by truncation.","section":"Numerical methods / reproducibility"},{"comment":"In the Rindler-Krylov toy model, the time axis in Fig. 5(a)-(b) is not defined with units (e.g., in units of 1/a_1). Adding the time unit would improve clarity.","section":"Fig. 5(a)-(b)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within scope for cond-mat.quant-gas and the numerical observation is potentially interesting, but the central transition claim currently exceeds the finite-size and finite-time evidence. I recommend major revision rather than rejection because the missing scaling/convergence analysis could in principle be supplied, and the analytic toy-model result is a useful piece of work. I would not accept the manuscript in its present form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a look, but read the headline claim with a grain of salt. The numerical finding is new and probably real: in the Dicke model, the long-time-averaged Krylov complexity C̄_K is non-monotonic in the coupling, dips near g̃≈0.5, and then rises sharply, with different N-scaling on the two sides (N^{3/2} below, N^{5/2} above). If that holds up, it gives people studying quantum thermalization and chaos a new, state-based observable that doesn't depend on picking the right local probe.\n\nThe paper earns credit for effort and honesty. It maps the complexity over a (g̃, Ω/δ) plane, checks other initial states, and compares against level statistics, Lyapunov exponents, and entanglement; it doesn't oversell those comparisons. The Krylov-space wave-packet picture—confinement from roughly linear a_k, Rindler-like hopping from b_k, and deterministic 'disorder' in the Lanczos coefficients—is clear, and the toy model of strictly linear coefficients is solved carefully. The authors also explicitly note that the fitted η stays below 1/2 and that the toy model alone gives no hint of the transition. That is fair, and the screening of the smoothed model (disorder removed) shows disorder is genuinely load-bearing.\n\nThe soft spot is the word 'nonanalytic.' For any finite N and any fixed boson truncation, C̄_K(g̃) is an analytic function of g̃; nonanalyticity can only emerge in an N→∞ or t→∞ limit. The data show the curve steepening with N, but there is no finite-size scaling of the transition width, no convergence study of the time average, no error bars, and the regular regime is oscillatory—so the fixed window gt∈[50,200] could be picking up phase sensitivity. The authors jump from 'sharp' to 'hence nonanalytic' without supplying the scaling evidence that would justify the word. That is an overclaim, not a fatal flaw; a sharp crossover would still be interesting, and the N^{3/2} collapse on the small-g̃ side is suggestive. The lack of released code/data and unspecified boson truncation also make independent verification slower than it should be.\n\nWho this is for: the Krylov-complexity community and the Dicke/cavity-QED crowd. It deserves peer review—the question is timely, the observation is novel, and the presentation is honest—but the referee should push for either a softened claim (crossover) or a real finite-size/time-window scaling analysis. I would not desk-reject this; I'd send it out with that instruction.","headline":"A new numerical observation about Krylov complexity in the Dicke model, but the 'nonanalytic transition' claim is stronger than the finite-N, finite-time evidence supports.","tokens_in":19334,"tokens_out":3346,"would_cite":true,"duration_ms":34249,"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":"In the Dicke model, long-time-averaged Krylov complexity and entropy change sharply at a critical coupling, separating regular oscillatory dynamics from chaotic, spaghettified dynamics.","keywords":["Krylov complexity","Dicke model","quantum quench dynamics","dynamical phase transition","quantum chaos","Lanczos coefficients","wave packet dynamics","light-matter interaction"],"falsifier":"Compute C̄_K and S̄_K for N = 50, 80, 100 (and, if possible, trapped-ion data at N up to 100) while extending the time average to gt ∈ [200, 1000]. If the slope change at g̃_c remains smooth, the transition width stops shrinking with N, or the N^{3/2}/N^{5/2} scalings fail to persist, the central claim of a sharp complexity transition is refuted, leaving only a crossover.","tokens_in":18256,"feed_emoji":"⚛️","tokens_out":9132,"duration_ms":84038,"temperature":0.7,"pith_summary":"The paper argues that Krylov complexity, a measure of how an evolving quantum state spreads through Hilbert space, can detect a genuine dynamical transition in a realistic many-body model. In the Dicke model of two-level atoms coupled to a cavity mode, the long-time-averaged Krylov complexity and its Shannon entropy first decrease smoothly as the dimensionless coupling g̃ increases, then change slope abruptly near g̃_c ≈ 0.5 and rise steeply into a fluctuating regime. The authors interpret this as a transition between regular dynamics, where the Krylov wave packet is confined and oscillates, and chaotic dynamics, where the packet is stretched and destroyed by Rindler-like hopping. The transition is accompanied by distinct system-size scalings, C̄_K ∼ N^{3/2} versus ∼ N^{5/2}, and coincides broadly with the onset of chaos inferred from level statistics and Lyapunov exponents. The paper positions Krylov measures as an independent, state-based order parameter for dynamical phase transitions, with a clean signal where traditional observables become noisy.","feed_headline":"A state-spread complexity measure snaps at g≈0.5 in Dicke quenches","feed_subtitle":"The jump separates confined, oscillating wave-packet dynamics from stretched, chaotic ones","key_machinery":"Krylov complexity: expand the time-evolving state in the Lanczos basis generated from the initial state; the Hamiltonian becomes a one-dimensional tight-binding chain with site energies a_k and hopping amplitudes b_k. C_K is the center of mass of the occupation probabilities along this chain, and S_K is their Shannon entropy. The paper's explanatory engine is the Rindler-Krylov toy Hamiltonian, a_k = k, b_k = ηk, where η is the slope ratio: linear onsite energy confines the particle, while positional hopping mimics Rindler spacetime, and η = 1/2 is a delocalization critical point. Deviations from this linear form — deterministic disorder δa_k, δb_k produced by the Lanczos procedure — create","core_discovery":"On the paper's own terms, the central discovery is that the time-averaged Krylov complexity C̄_K of the Dicke model after a quench from a spin-coherent initial state is a non-monotonic function of the dimensionless coupling g̃. Approaching from below, it decreases smoothly; near g̃_c ≈ 0.5 its slope changes suddenly, while the Krylov entropy S̄_K jumps, and above that point both quantities become large and fluctuate strongly with g̃. The same behavior is seen for several initial states and detunings, and the data collapse onto distinct power laws in the spin number N — C̄_K ∼ N^{3/2} on the regular side and ∼ N^{5/2} on the chaotic side. The paper identifies the mechanism as wave-packet comp","pith_inferences":["If the transition is truly nonanalytic, it should appear as a sharpening feature in a controlled finite-size scaling: computing C̄_K for N up to 100 or more and extending the time average beyond gt ∈ [50,200] would reveal whether the slope change converges to a step or saturates as a crossover.","The different N-scaling suggests C̄_K may quantify the classical simulation cost of a quench: below g̃_c the state is effectively confined to a small Krylov window, while above it the N^{5/2} growth signals a much larger, harder-to-simulate Hilbert-space sector.","One could test the Rindler picture directly by fitting Lanczos coefficients in other models and looking for a dynamical transition whenever the fitted η approaches 1/2, effectively turning η into a universal control parameter for complexity transitions.","The off-resonance spin-locked region of the phase diagram may be a distinct third regime worth probing experimentally, since it predicts extremely low complexity and essentially frozen spin dynamics for weak coupling away from resonance."],"forward_implications":["Krylov complexity and Krylov entropy can act as an independent, state-based diagnostic of dynamical phase transitions, complementing time-averaged magnetizations, entanglement entropy, and level statistics that become noisy near chaos.","The complexity phase diagram in the (g̃, Ω/δ) plane gives experiment a map: trapped-ion or cavity-QED realizations of the Dicke model can look for the regular, chaotic, and spin-locked regimes identified here.","The distinct scalings C̄_K ∼ N^{3/2} and ∼ N^{5/2} provide a quantitative signature that can be checked at larger N and in other models.","The spaghettification mechanism is not specific to the Dicke model: any Hamiltonian whose Lanczos coefficients approach a_k ∝ k, b_k ∝ ηk near η = 1/2 should show a similar complexity transition, so the criteria can be exported to other many-body systems.","Because level statistics and Lyapunov exponents give fuzzy or parameter-dependent boundaries, the paper positions C̄_K and S̄_K as the cleaner, directly observable order parameters for the transition."],"fun_headline_variants":["Krylov complexity snaps at g≈0.5 in Dicke quenches","Sudden jump in Krylov entropy at critical coupling g≈0.5","Dicke model complexity transition: slope breaks at g≈0.5","Operator growth complexity shows sharp transition in Dicke model","Complexity measure changes sharply at g≈0.5 in Dicke system"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the numerically computed finite-N, finite-time-averaged Krylov quantities (taken over gt ∈ [50,200]) are representative of the true long-time, thermodynamic limit, so that the sharpening with N is a genuine nonanalytic transition rather than a finite-size crossover that would saturate at larger N.","fun_headline_variants_meta":{"raw":{"variants":["Krylov complexity snaps at g≈0.5 in Dicke quenches","Sudden jump in Krylov entropy at critical coupling g≈0.5","Dicke model complexity transition: slope breaks at g≈0.5","Operator growth complexity shows sharp transition in Dicke model","Complexity measure changes sharply at g≈0.5 in Dicke system"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000629,"raw_usage":{"total_tokens":2771,"prompt_tokens":795,"completion_tokens":1976,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":539,"completion_tokens_details":{"reasoning_tokens":1879}},"tokens_in":539,"tokens_out":1976,"duration_ms":16049,"temperature":1.0,"reasoning_tokens":1879,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T06:59:50.183962+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute C̄_K and S̄_K for N = 50, 80, 100 (and, if possible, trapped-ion data at N up to 100) while extending the time average to gt ∈ [200, 1000]. If the slope change at g̃_c remains smooth, the transition width stops shrinking with N, or the N^{3/2}/N^{5/2} scalings fail to persist, the central claim of a sharp complexity transition is refuted, leaving only a crossover.","supporting_citations":[],"review_version":1}