{"id":"5144f01f-701b-45cc-9982-b6b144815499","arxiv_id":"2501.09975","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The dynamic critical exponent of the 3D Z2 gauge model under Metropolis relaxational dynamics is z = 2.610(15), obtained from out-of-equilibrium finite-size scaling of the energy density.","lead":"By quenching the three-dimensional Z2 gauge model to its critical point and tracking the gauge-invariant energy density, the authors extract the dynamic critical exponent z = 2.610(15) for purely relaxational dynamics. The result sharpens earlier estimates and demonstrates a scaling method for topological transitions without a local order parameter.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The z estimate relies on an unverified time-scale separation; fits at Θ≈0.01 could be contaminated by slowly decaying regular-background corrections.","rationale":"The reader's weakest assumption identifies the same time-scale separation conjecture, and I agree that this is the most load-bearing point. The result is otherwise well supported by the quality of the collapses and the consistency across Υ values. The dominant systematic risk is not statistical noise—individual fit errors on z are ~0.001-0.003—but the correction model. The small-L fit (c) gives z≈2.59 while the large-L pure power-law fit gives z≈2.610; the 0.015 final error covers this spread, but only if the assumed correction exponents are correct. The unresolved discrepancy with the slow-crossing estimate z=2.70(3) of Ref. [59] adds further reason for caution, though it is not by itself disqualifying. A direct test of an explicit L^{-α/ν} correction would settle whether the central assumption of the method holds for the Z2 gauge model. Therefore I would keep the reader's conditional verdict unchanged.","tokens_in":16122,"tokens_out":9915,"duration_ms":105957,"concrete_test":"Perform a least-squares fit of the t(Ω,Υ,L) data used in Table I (same Ω and Υ values, same Lmin sets) to the ansatz t = a0 L^z (1 + a1 L^{-α/ν}), with α/ν=0.1747 fixed, and also to t = a0 L^z (1 + a1 L^{-α/ν} + a2 L^{-ω}). Compare the resulting z and goodness-of-fit with the reported fits (a) and (c). If the L^{-α/ν} coefficient is nonzero at a statistically significant level, or if z shifts by more than 0.015, the regular-background suppression assumed in Eq. (21) is not satisfied at the simulated Θ, and the central estimate is biased.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central estimate z=2.610(15) rests on Eq. (21), which asserts that for any fixed Θ>0 the subtracted energy density has scaling corrections O(L^{-ω}) because the analytic background thermalizes much faster than the critical modes. This time-scale separation is taken from Ref. [62] and is not independently verified for the Z2 gauge model. The fits that determine z use data at Θ≈0.01 (Table I), where residual crossover from the Θ→0 regime (Eq. 20) could still contribute an effective correction closer to L^{-α/ν} (α/ν≈0.17) than to L^{-ω} (ω≈0.83). The systematic spread in Table I—fit (c) with Lmin=16 gives z≈2.59 for Υ=2 while fit (a) with Lmin≥40 gives z≈2.610—is consistent with a slowly decaying correction not fully captured by the L^{-ω} and L^{-2} ansätze. If the regular contribution is not fully equilibrated at the simulated Θ, the fitted z is biased and the quoted uncertainty underestimates the systematic error.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the out-of-equilibrium critical relaxational dynamics of the three-dimensional Z2 lattice gauge model after an instantaneous quench to the critical point, using purely relaxational (single-spin-flip Metropolis) dynamics. The authors monitor the subtracted energy density Es(t)=E(t)-Ec and analyze its finite-size scaling within an out-of-equilibrium FSS framework developed in Ref. [62]. Their central result is the dynamic critical exponent z=2.610(15) for the 3D Z2 gauge universality class, extracted from fits of the time t at fixed values of the rescaled energy density Ω and fixed Υ=rL^{1/ν}, for lattice sizes up to L=100. The paper includes careful numerical checks: multiple fit ansätze, varying Lmin, comparisons of data collapse, cross-checks using time derivatives and differences of Ω, and an analysis of the small-Θ crossover behavior. The quoted z improves on earlier equilibrium and out-of-equilibrium estimates (z=2.55(6) and z=2.70(3)).","tokens_in":16338,"tokens_out":5633,"duration_ms":57344,"significance":"If the estimate z=2.610(15) is correct, it provides the most precise determination to date of the dynamic critical exponent for the purely relaxational dynamics of the 3D Z2 gauge model, and it demonstrates that the out-of-equilibrium energy-density method can be applied to gauge theories where no local order parameter exists. The numerical work is transparent and thorough: the paper reports many independent fits, checks the stability of z against the choice of Lmin and fit ansatz, and shows raw data collapse in Fig. 3. The main uncertainty is the theoretical input: the suppression of O(L^{-α/ν}) corrections at fixed Θ>0 rests on the time-scale separation conjecture of Ref. [62], which is not independently verified for the Z2 gauge model. That is a load-bearing assumption for the central estimate, but it is testable with additional fits, and the paper already contains the raw data and fit machinery needed to perform that test.","major_comments":[{"comment":"The central estimate z=2.610(15) rests on Eq. (21), which asserts that for any fixed Θ>0 the leading scaling corrections of the subtracted energy density are O(L^{-ω}) with ω≈0.83, rather than the equilibrium O(L^{-α/ν}) with α/ν≈0.17. This assertion is taken from Ref. [62], written by two of the present authors, and is not independently established for the Z2 gauge model. The fits that determine z use data at Θ≈0.01 (Table I). At this value the nonanalytic crossover of Eq. (20) is not negligible: Θ^{α/(νz)} ≈ 0.73 (using z=2.61), so a residual L^{-α/ν} contamination cannot be excluded a priori. The spread between fit (c) with Lmin=16 (z≈2.589(10) for Υ=2, Ω=7) and fit (a) with Lmin≥40 (z≈2.610(2)) is consistent with such a slowly decaying correction. To make the estimate robust, I request an explicit test: add an L^{-α/ν} term to the fit ansatz (or leave the correction exponent free) and show that z is stable; alternatively, analyze the fitted z as a function of Θ and demonstrate a platea in a regime where the Θ→0 crossover is already fully suppressed.","section":"III.B, Eq. (21) and Table I"},{"comment":"The quoted error z=2.610(15) is described as taking into account the range of results from different fits, but no precise selection criterion is given. The table entries themselves differ by about 0.020 (e.g., Υ=2, Ω=7: fit (c) with Lmin=16 gives 2.589(10), while fit (a) with Lmin=40 gives 2.610(2)). Since the central added value of this paper is the fourfold reduction of the uncertainty relative to the previous z=2.55(6), the systematic error budget must be explicit. Please define the set of fits included in the final estimate, report the scatter of those z values, and state whether the uncertainty due to the choice of correction ansatz is fully covered by the reported 0.015. If not, the error should be enlarged or the analysis should be restricted to fits with demonstrably controlled corrections.","section":"IV, Table I and Eq. (29)"}],"minor_comments":[{"comment":"The acceptance ratio of the Metropolis update at the critical point is stated to be about 2%; this is unusually low and may affect both the efficiency and the decorrelation of starting configurations. The statement that n≈0.2 L^z sweeps between trajectories provides 'almost decorrelated starting configurations' would benefit from a quantitative check, e.g., reporting the integrated autocorrelation time of the starting configurations.","section":"IV, paragraph after Eq. (23)"},{"comment":"In ansatz (c), the second correction is written as L^{-2}, while Eq. (24) has L^{-ω2} with ω2≈2.02. The approximation is reasonable, but the text should state explicitly that L^{-2} is used as a proxy for the lattice-anisotropy correction L^{-ω2}; currently the connection is only implicit.","section":"Eq. (24) and fits (26)-(28)"},{"comment":"The fit to a+b Θ^κ with κ=α/(νz) is performed only on the L=100 data. Since the text acknowledges that a proper L→∞ extrapolation is 'quite cumbersome', the current procedure is acceptable as a consistency check, but mentioning the known L-dependence of the fitted coefficients would help the reader assess the uncertainty of the crossover fit.","section":"Fig. 4"},{"comment":"The derivation of Ec from the dual Ising energy density is elegant, but the text does not discuss the possible influence of the finite-size corrections in the Ising simulations on the final Ec used in the subtraction. The quoted uncertainty of 3×10^-6 appears to include only statistical errors; a sentence on how systematic finite-size effects were estimated would be useful.","section":"II.B, Eq. (10)"}],"recommendation":"major_revision","confidential_remarks":"The paper is well within the scope of the journal and the numerical work is generally careful. My main concern is the reliance on the time-scale separation conjecture from the authors' own earlier work for the crucial suppression of O(L^{-α/ν}) corrections. This is a testable assumption, and I would like to see the requested fit with a free or L^{-α/ν} correction before publication, because the central estimate and its quoted precision depend on it. If the stability test passes, the paper would be a solid contribution. I have no concerns about novelty or overlap beyond the normal self-citation pattern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is simple: this paper gives z = 2.610(15) for the purely relaxational Metropolis dynamics of the 3D Z2 gauge model, roughly four times more precise than the previous equilibrium estimate and the first application of the out-of-equilibrium energy-density FSS method to a gauge model. That is a real advance, and the numerics are done carefully. The fits use multiple ansatze, varying Lmin, time derivatives, and differences of Upsilon; Fig. 3 shows good data collapse; the authors also pin down the critical energy density Ec to 10^-6 via duality with the Ising model. I believe the central number is defensible.\n\nThe soft spot is exactly where the reader put it: the method assumes that along the critical relaxational flow the analytic background thermalizes much faster than the critical modes, so that corrections at fixed Theta > 0 are O(L^-omega) with omega ~ 0.83 rather than O(L^-alpha/nu) with alpha/nu ~ 0.17. That time-scale separation is taken from the authors' own Ref. [62] and is not independently verified for the gauge model. The fits that drive the result sit at Theta ~ 0.01, close enough to the Theta = 0 crossover that a slowly decaying regular-background contribution could contaminate the fitted z. The spread in Table I is consistent with that worry: fit (c) with Lmin = 16 gives z ~ 2.59 for Upsilon = 2, while fit (a) with Lmin >= 40 gives 2.610. The quoted error of 0.015 covers this spread, so the estimate is not misleading, but the systematic uncertainty may be a touch larger than the error bar suggests.\n\nI do not think the self-citation is itself a problem. The framework was published and tested on Ising-like systems, and the dynamic exponent is extracted as a fit parameter, not imposed. But an independent check of the time-scale separation for the gauge model, or a preregistered fitting-window choice, would firm up the result. The lingering discrepancy with the slow-crossing estimate z = 2.70(3) also deserves attention, though it is not fatal to this paper.\n\nWho should read this: people working on critical dynamics of lattice gauge theories and on out-of-equilibrium FSS methods. It is a solid numerical contribution with an honest error bar and a clear methodological limitation. I would send it to peer review: the result is important enough and the analysis careful enough to warrant referee time, and the referee can push on the correction analysis. I would also cite it in my own work on gauge-theory critical dynamics.\n\nRecommendation: engage with it, but treat the exponent as conditional on the time-scale separation conjecture until that conjecture gets independent support.","headline":"A careful, more precise dynamic exponent for the 3D Z2 gauge model, built on a plausible but unproven time-scale separation from the authors' own framework; the quoted error is honest about the fit spread.","tokens_in":16927,"tokens_out":1386,"would_cite":true,"duration_ms":15285,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82B20","82B27","82C20","82C27"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that purely relaxational dynamics of the three-dimensional Z2 gauge model has dynamic critical exponent z = 2.610(15), and that this value follows from the out-of-equilibrium scaling of the energy density after an…","keywords":["dynamic critical exponent","Z2 gauge model","out-of-equilibrium finite-size scaling","critical relaxational flow","Metropolis dynamics","topological phase transition","energy density","critical slowing down"],"falsifier":"Fit $t(\\Omega,\\Upsilon,L)$ at a fixed small but nonzero $\\Theta$, such as $\\Theta \\approx 0.001$, with both $L^{z-\\omega}$ and $L^{z-\\alpha/\\nu}$ correction terms included; if the $L^{-\\alpha/\\nu}$ term is statistically significant at $\\Theta > 0$, the assumed separation of time scales fails and the reported $z$ is biased. A complementary check is to measure the relaxation time of short-range plaquette fluctuations directly: if it grows like $L^z$ instead of staying short, the central assumption collapses.","tokens_in":15872,"feed_emoji":"⏳","tokens_out":11505,"duration_ms":94559,"temperature":0.7,"pith_summary":"The paper studies what happens when the three-dimensional ${\\mathbb Z}_2$ gauge model is suddenly moved to its critical point and then evolved with purely local Metropolis updates. It claims that the time-dependent subtracted energy density obeys a clean out-of-equilibrium finite-size scaling, and that this scaling can be used to extract the dynamic critical exponent $z$. The central result is $z = 2.610(15)$, an uncertainty about four times smaller than that of the earlier equilibrium estimate $z = 2.55(6)$. This matters because the ${\\mathbb Z}_2$ gauge transition has no local order parameter, so the energy density is one of the few practical observables, and because it sharpens the contrast with the 3D Ising model, whose relaxational exponent is about $2.0245$.","feed_headline":"Quench dynamics pin the gauge model's critical slowdown at z=2.61","feed_subtitle":"Quench flow sharpens the dynamic exponent for a transition with no local order parameter.","key_machinery":"The load-bearing object is the out-of-equilibrium finite-size scaling of the subtracted energy density along the critical relaxational flow, expressed as $\\Omega(t,r,L) = L^{d-y_r} E_s(t,r,L) \\approx A(\\Theta,\\Upsilon)$ with $\\Theta = t L^{-z}$, $\\Upsilon = r L^{y_r}$, and $y_r = 1/\\nu$. The mechanism works because the short-ranged modes responsible for the regular analytic part of the energy density thermalize much faster than the critical modes; at fixed $\\Theta > 0$ this removes the equilibrium's slow $O(L^{-\\alpha/\\nu})$ corrections and leaves only $O(L^{-\\omega})$ corrections with $\\omega \\approx 0.83$. To extract $z$, the authors invert the monotonic relation between time and $\\Omega$ and fit $t(\\Omega,\\Upsilon,L)$ to $L^z$ times a scaling function, including correction terms $L^{z-\\omega}$ and $L^{z-2}$.","core_discovery":"The paper's central claim is that the purely relaxational dynamics of the three-dimensional ${\\mathbb Z}_2$ gauge model in its critical region is controlled by a single dynamic exponent $z = 2.610(15)$. Starting from thermalized configurations slightly below the critical coupling $K_c$, the authors quench to $K_c$ and follow the subtracted energy density $E_s(t) = E(t) - E_c$ under Metropolis link flips. They show that the rescaled quantity $\\Omega = L^{d-y_r} E_s$ collapses as a function of $\\Theta = t L^{-z}$ for lattice sizes up to $L = 100$, with leading finite-size corrections decaying as $L^{-\\omega}$ rather than the slower equilibrium corrections $L^{-\\alpha/\\nu}$. This out-of-equilibrium collapse is what lets them pin $z$. The estimate agrees with the equilibrium result $z = 2.55(6)$, reduces its error by roughly a factor of four, and leaves a mild tension with the slow-crossing value $z = 2.70(3)$.","pith_inferences":["Editorial inference: applying the same protocol to the time-derivative observable $dE/dt$, which does not require knowing $E_c$, could extend the usable fitting range and provide an independent cross-check of $z$.","Editorial inference: the central separation of time scales could be tested directly by measuring the two-time autocorrelation of individual plaquette energy terms; if their relaxation time grows as $L^0$ rather than $L^z$, the fitted $z$ is unbiased.","Editorial inference: if the method is applied to the three-dimensional Abelian-Higgs model and yields a $z$ close to $2.61$, that would suggest a common dynamic universality class for these gauge-type topological transitions; a different value would indicate that the dynamic universality class distinguishes the gauge group.","Editorial inference: the visible $O(L^{-2})$ corrections at small lattice sizes suggest that lattice-anisotropy operators dominate the approach to scaling; improved actions that suppress those operators could push reliable fits to smaller $L$."],"forward_implications":["The dynamic exponent of the three-dimensional ${\\mathbb Z}_2$ gauge model under local relaxational updates is $z = 2.610(15)$, roughly $0.6$ larger than the Ising value $z = 2.0245(15)$, confirming that the nonlocal duality between the two models does not map local relaxation dynamics.","The new estimate is consistent with and considerably more precise than the equilibrium estimate $z = 2.55(6)$, validating the out-of-equilibrium energy-density route for this universality class.","The mild disagreement with the slow-crossing estimate $z = 2.70(3)$ remains, indicating that different out-of-equilibrium protocols do not yet yield fully consistent numbers.","Because the approach works with a local gauge-invariant observable at fixed $\\Theta > 0$ without subtracting analytic backgrounds, it offers a practical route for other gauge models, such as the three-dimensional Abelian-Higgs model, whose continuous transitions also lack local order parameters."],"supporting_citations":[{"why":"Supplies the out-of-equilibrium energy-density scaling ansatz and the conjecture that regular backgrounds thermalize rapidly along the critical relaxational flow.","marker":"[62]"},{"why":"Provides the earlier equilibrium dynamic exponent z = 2.55(6) that the new estimate improves and agrees with.","marker":"[60]"},{"why":"Provides the earlier slow-crossing estimate z = 2.70(3) that the new result is compared against.","marker":"[59]"},{"why":"Gives the 3D Ising dynamic exponent z = 2.0245(15) used to highlight that gauge and spin dynamics differ.","marker":"[61]"},{"why":"Introduces the Z2 gauge model and its transition without local order parameters.","marker":"[5]"},{"why":"Supplies the precise Ising critical coupling Jc from which Kc is obtained through duality.","marker":"[63]"},{"why":"Supplies the Ising exponents nu, alpha, and omega used as inputs in the scaling analysis.","marker":"[64]"},{"why":"Provides the cluster algorithm used to obtain the accurate Ising critical energy density that yields Ec via duality.","marker":"[70]"}],"fun_headline_variants":["Quench pins gauge model's critical slowdown at z=2.61","3D Z2 gauge critical flow yields z=2.610 from quench","Out-of-equilibrium quench sharpens gauge dynamic exponent","Gauge model's relaxational flow pegged at z=2.610","Quench dynamics nails no-order-parameter criticality at z=2.61"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method assumes that the short-ranged fluctuations responsible for the analytic background of the energy density reach equilibrium much faster than the critical modes, so that at fixed $\\Theta = t/L^z > 0$ the slow $L^{-\\alpha/\\nu}$ corrections vanish and only faster corrections survive.","fun_headline_variants_meta":{"raw":{"variants":["Quench pins gauge model's critical slowdown at z=2.61","3D Z2 gauge critical flow yields z=2.610 from quench","Out-of-equilibrium quench sharpens gauge dynamic exponent","Gauge model's relaxational flow pegged at z=2.610","Quench dynamics nails no-order-parameter criticality at z=2.61"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000285,"raw_usage":{"total_tokens":1682,"prompt_tokens":951,"completion_tokens":731,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":567,"completion_tokens_details":{"reasoning_tokens":633}},"tokens_in":567,"tokens_out":731,"duration_ms":7423,"temperature":1.0,"reasoning_tokens":633,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T19:28:45.941965+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fit $t(\\Omega,\\Upsilon,L)$ at a fixed small but nonzero $\\Theta$, such as $\\Theta \\approx 0.001$, with both $L^{z-\\omega}$ and $L^{z-\\alpha/\\nu}$ correction terms included; if the $L^{-\\alpha/\\nu}$ term is statistically significant at $\\Theta > 0$, the assumed separation of time scales fails and the reported $z$ is biased. A complementary check is to measure the relaxation time of short-range plaquette fluctuations directly: if it grows like $L^z$ instead of staying short, the central assumption collapses.","supporting_citations":[{"cited_title":"Hasenbusch, The dynamic critical exponent z of the three-dimensional Ising universality class: Monte Carlo simulations of the improved Blume-Capel model, Phys","cited_arxiv_id":null,"evidence_quote":"Supplies the out-of-equilibrium energy-density scaling ansatz and the conjecture that regular backgrounds thermalize rapidly along the critical relaxational flow."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the earlier equilibrium dynamic exponent z = 2.55(6) that the new estimate improves and agrees with."},{"cited_title":"Ben-Av, D","cited_arxiv_id":null,"evidence_quote":"Provides the earlier slow-crossing estimate z = 2.70(3) that the new result is compared against."},{"cited_title":"Critical relaxational dynamics at the continuous transitions of three-dimensional spin models with ${\\mathbb Z}_2$ gauge symmetry","cited_arxiv_id":"2501.09575","evidence_quote":"Gives the 3D Ising dynamic exponent z = 2.0245(15) used to highlight that gauge and spin dynamics differ."},{"cited_title":"Panagopoulos and E","cited_arxiv_id":null,"evidence_quote":"Supplies the precise Ising critical coupling Jc from which Kc is obtained through duality."},{"cited_title":"Guida and J","cited_arxiv_id":null,"evidence_quote":"Provides the cluster algorithm used to obtain the accurate Ising critical energy density that yields Ec via duality."}],"review_version":1}