{"id":"1eececb3-239c-4559-bfa8-23c3aca009d8","arxiv_id":"2509.04098","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"In classical spin systems, the hydrodynamic tail exponent of an observable's autocorrelation function equals dm/z, where m is the order of the observable's energy-density dependence and z is the dynamical critical exponent.","lead":"This paper shows that in classical spin systems the late-time decay rate of an observable's autocorrelation function is set by two things: how energy spreads through the system, and how strongly the observable depends on energy density. This links static thermodynamic properties to dynamical transport behavior, helping predict slow relaxation in many-body systems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Hydrodynamic projection for generic observables is untested; symmetry-odd observables with zero energy overlap could break ν=dm/z.","rationale":"Reading in good faith, the paper is a plausible and well-illustrated extension of the quantum relaxation-overlap inequality to classical spin systems. The finite-size plateau scaling L^{-m} (Eq. 14) is derived carefully from ergodicity and the central limit theorem in Sec. IV B, and the numerics for the chosen observables support it. The weak point is the jump from the inequality ν≤dm/z to the equality ν=dm/z: that jump relies on the hydrodynamic projection assumption, which asserts that all non-energy modes decay exponentially and that the algebraic tail of any local observable is governed by its overlap with energy and its powers. This is precisely the assumption the reader flagged. My proposed S^y test is a clean, decisive check: a local observable with zero overlap with every power of H by symmetry should have no algebraic tail if the projection is valid. The paper does not consider such an observable, so the central claim is not yet established for generic observables. This supports the reader's CONDITIONAL verdict; I do not recommend changing it, but I suggest the concrete test as a prerequisite for upgrading to full acceptance.","tokens_in":17228,"tokens_out":9467,"duration_ms":93878,"concrete_test":"Compute C(t)=⟨S^y_j(t)S^y_j(0)⟩_{β=0} for the 1D tilted-field Ising model (Eq. 10, hx=1.1, hz=0.9, J=1) using the same Yoshida 4th-order symplectic integrator (δt=0.02) and 10^7 Monte-Carlo initial configurations as in Sec. III B, for system sizes L up to at least 200. Because the Hamiltonian and the Haar measure are invariant under S^y→−S^y, the ergodicity argument of Sec. IV B predicts the late-time plateau vanishes, and the hydrodynamic projection predicts exponential (non-power-law) decay. Plot C(t) on both log-linear and log-log scales: if a power-law tail t^{-ν} appears, the central formula Eq. (13) is contradicted for generic observables; if the decay is consistent with exponential and the plateau vanishes, the projection assumption is supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central equality ν=dm/z (Eq. 13) is not derived. Sec. IV C establishes only the inequality ν≤dm/z, under two additional assumptions: (i) the hydrodynamic projection, which states that a generic local observable's algebraic tail comes solely from its overlap with the energy density and its powers, while non-conserved modes 'give rise to subleading exponentially decaying contributions'; and (ii) monotonicity of the autocorrelator. The equality is then inferred from numerical fits for four observables chosen because they have finite overlap m with H (Appendix B). The projection assumption is never independently tested. In the tilted-field Ising model of Eq. (10), the local observable S^y_j has ⟨H^m S^y_j⟩_{β=0}=0 for every m, by the spin-reflection symmetry S^y→−S^y. Hence Sec. IV B predicts a vanishing plateau (no L^{-m} term), and the hydrodynamic projection predicts no algebraic tail: C(t) should decay exponentially, faster than any power. If S^y_j instead shows an algebraic tail, then energy projection is not the sole source of slow dynamics, and Eq. (13) fails for generic observables. The manuscript computes only observables with nonzero energy overlap, so this load-bearing premise is unchecked.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies classical spin systems with a single conserved quantity (energy) and proposes that the late-time decay exponent ν of a local observable's autocorrelator is fixed by ν = dm/z, where d is the spatial dimension, z is the energy dynamical critical exponent, and m is the order of the first nonvanishing term in the thermal expectation O(ε) around the relevant energy density. It reports extensive numerical simulations for a 1D tilted-field Ising model, a 1D long-range Ising model (α = 1.1 and 1.5), and a 2D transverse Ising model, using observables S^x_j, S^x_jS^x_{j+1}, S^x_jS^x_{j+1}S^x_{j+2}, and S^x_jS^x_{j+1}S^x_{j+2}S^x_{j+3}, with overlap orders m = 1,...,4. The finite-size plateau of the autocorrelator is shown to scale as L^{-m}, and the hydrodynamic-tail exponents are reported to be compatible with ν = dm/z. A theoretical derivation from ergodicity gives the plateau scaling and an inequality ν ≤ dm/z under additional assumptions of hydrodynamic projection and monotone decay; the equality is then inferred from the numerical fits.","tokens_in":17544,"tokens_out":8559,"duration_ms":87285,"significance":"If the central equality could be established, the paper would provide a clean classical analogue of the quantum relaxation-overlap relation, connecting thermodynamics, ergodicity, and hydrodynamics. The plateau scaling result (Eq. 14) is derived from ergodicity and is well supported by the numerics, and the analytical computation of the overlap order in Appendix B is a useful concrete ingredient. The numerical evidence is also broad: large systems (up to L = 200), two spatial dimensions, and several transport regimes. However, the headline relation is not actually proven in the text; the derivation yields only an inequality, and the key hydrodynamic-projection assumption is not tested independently. The manuscript therefore overstates the degree to which Eq. (13) is established for generic observables.","major_comments":[{"comment":"Eq. (13) is stated as an equality, ν = dm/z, and the abstract claims that the late-time tail is 'determined' by z and m. But the derivation in Sec. IV C establishes only the inequality ν ≤ dm/z, and it does so under two additional assumptions (hydrodynamic projection and monotonic decay). The text itself calls this the 'relaxation-overlap inequality'. The equality is inferred from fits, not derived. Please either provide a saturation argument or reframe the central claim as an inequality plus numerical evidence of saturation for the observables studied.","section":"Sec. III C, Eq. (13); Sec. IV C"},{"comment":"The hydrodynamic-projection assumption is load-bearing and is not tested. For a zero-overlap observable such as O = S^y_j in the model of Eq. (10), the Hamiltonian contains no S^y, so ⟨H^m S^y_j⟩_{β=0} = 0 for all m and O(ε) = 0 identically. The theory then predicts no algebraic tail and no L^{-m} plateau, only exponentially decaying non-hydrodynamic contributions. The manuscript computes only observables with nonzero overlap m. A direct numerical test of S^y_j (or a similar symmetry-odd observable) would be needed to support the claim that Eq. (13) applies to generic local observables.","section":"Sec. IV C, hydrodynamic projection"},{"comment":"The paper states that for α ≥ 1.5 energy transport is diffusive (z = 2), yet the confirmation of Eq. (13) for the α = 1.5 long-range model uses an intermediate-time exponent z = 4/3 taken from the quantum work Ref. [24], not from the energy-density scaling of this classical model. The same data also show a later tail with z = 2. It is unclear whether Eq. (13) is being tested against the asymptotic dynamical critical exponent or fit to a crossover regime with an effective exponent. The status of z = 4/3 for this classical model should be clarified, ideally by an independent measurement of the energy autocorrelator scaling.","section":"Sec. III A and III C, α = 1.5 case"}],"minor_comments":[{"comment":"The monotonicity assumption is introduced as a physically motivated hypothesis, but the raw autocorrelators in Figs. 2, 4, 6, and 8 show transient oscillations and clear plateau effects. The derivation of ν ≤ dm/z would benefit from a precise statement of how the averaging that restores monotonicity is performed.","section":"Sec. IV C"},{"comment":"The notation C(∞) for the long-time plateau in finite systems may be confused with the true t → ∞ limit in the thermodynamic limit. Consider using C_plateau(L) or explicitly defining it as the long-time average at fixed L.","section":"Sec. III C / Fig. 9"},{"comment":"The numerical section reports the integration time step but not the total integration time or the time at which the plateau is measured for each L. This information would make the fitting procedure for ν and the plateau extraction more reproducible.","section":"Sec. III B"},{"comment":"There are minor typos: 'coordin dates' should be 'coordinates', and several figure captions read 'Dashed line indicate' instead of 'Dashed lines indicate'.","section":"Sec. II B / figure captions"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely to be a useful contribution if the central claim is reframed as an inequality with numerical saturation, and if the zero-overlap case is tested. The current text's conflation of Eq. (13) with an inequality is a fixable overclaim rather than a fatal flaw. The α = 1.5 z = 4/3 issue needs to be addressed because it currently weakens the numerical test of the formula. I would not reject, but the revision must confront these points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a credible and mostly well-executed extension of the quantum relaxation-overlap inequality to classical spin systems, but the headline claim is one step ahead of what is actually derived. The paper's new content is real: the classical derivation of the plateau scaling ~L^{-m} from ergodicity plus central-limit fluctuations is clean, the analytical overlap computations in Appendix B are useful, and the numerics on larger systems than the quantum counterpart give you a genuinely new observation—two distinct hydrodynamic tails in the α=1.5 long-range model. This is not just a repackaging of Ref. [24]; the classical framework requires its own arguments, and the authors provide them for the plateau and the inequality.\n\nWhere the paper is softer: Sec. IV C derives ν ≤ dm/z, not ν = dm/z. Equality is then inferred from fits without error bars. That is a reasonable way to build a conjecture, but the abstract states the equality as a result, which is overreach. More importantly, the hydrodynamic projection assumption—that non-conserved modes contribute only exponentially decaying terms—is stated as an expectation and never tested. The stress-test suggestion is exactly right: in the tilted-field Ising model, S^y_j has zero overlap with H at infinite temperature by reflection symmetry, so the projection assumption predicts exponential decay. If S^y_j shows a power-law tail, the 'generic observable' claim fails. The authors should run that calculation; it is cheap and directly targets the load-bearing premise.\n\nThe intermediate z=4/3 tail in the α=1.5 model is also presented as an empirical match to the quantum result, without a classical justification for why that exponent should appear at intermediate times before the diffusive tail. It may be right, but it needs discussion rather than a dashed line. And the absence of error bars on the fitted exponents is a minor but real issue; a bootstrap would settle it.\n\nOverall, the paper is aimed at researchers in classical spin dynamics, hydrodynamics, and ETH-flavored statistical mechanics. It deserves a serious referee. The framework is plausible and the numerics are above the usual bar. I would send it to review with a clear request to soften the abstract to 'inequality,' test the projection assumption on a symmetry-odd observable, and add error estimates. If that work holds up, this becomes a solid contribution.","headline":"Credible classical extension of the relaxation-overlap inequality with clean numerics, but the headline equality is not derived and the 'generic observable' claim rests on an untested hydrodynamic projection assumption.","tokens_in":17974,"tokens_out":3561,"would_cite":true,"duration_ms":33876,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82C05","82C70","82B20"],"pacs":["05.20.-y","05.60.-k","75.10.Hk"],"model":"deepseek-v4-flash","headline":"For ergodic classical spin systems, the late-time autocorrelation of any local observable is determined by energy transport and the observable's thermodynamic energy-overlap order.","keywords":["classical spin systems","ergodicity","hydrodynamic tails","relaxation-overlap inequality","autocorrelation functions","dynamical critical exponent","energy transport","long-range Ising model"],"falsifier":"In a one-dimensional short-range spin chain with demonstrably diffusive energy transport (z = 2) and no other conserved charge, measure the autocorrelation of S^x_j, whose energy-overlap order is m = 1. The prediction is an algebraic tail C(t) ~ t^{-1/2} followed by a plateau ~ 1/L. A clean t^{-ν} tail with ν differing from 1/2 by more than the numerical uncertainty, or a plateau that fails to scale as 1/L, would falsify the formula.","tokens_in":17153,"feed_emoji":"🧲","tokens_out":6302,"duration_ms":63673,"temperature":0.7,"pith_summary":"This paper claims that in classical spin systems with ergodic dynamics and energy as the only conserved charge, the late-time decay of any local observable's autocorrelation function is fixed by two numbers: the dynamical critical exponent z of energy transport and the lowest order m at which the observable's thermal expectation depends on energy density. Concretely, the hydrodynamic tail decays as t^{-ν} with ν = dm/z, and the long-time finite-size plateau obeys C(∞) ~ L^{-m}. The authors verify this in one- and two-dimensional Ising-type spin models, including long-range models with anomalous transport, and show that the same relation also fixes the system-size saturation. The classical derivation replaces the eigenstate thermalization hypothesis used in the quantum case by ordinary ergodicity, so the relaxation-overlap logic applies across quantum and classical many-body physics. If the relation holds, the late-time dynamics of a generic observable can be predicted from equilibrium thermodynamics alone.","feed_headline":"Spin correlations decay as t^{-dm/z}: one universal law","feed_subtitle":"In ergodic classical spin systems, late-time tails follow from energy transport and the observable's thermodynamic overlap order.","key_machinery":"Two mechanisms carry the argument. First, ergodicity in the form that long-time averages of any observable equal its microcanonical expectation at the initial energy: applied to the autocorrelator, this turns its late-time plateau into the variance of O across the thermal ensemble, which scales as L^{-m} when O(ε) ~ ε^m. Second, hydrodynamic projection: a generic local observable is dominated in its slow dynamics by its overlap with the energy density and its powers, whose spreading follows the scaling ⟨h(x,t)h(0,0)⟩ ~ t^{-1/z}F(x/t^{1/z}). Since energy spreads over a region of size t^{1/z}, finite-size saturation occurs at t ~ V^{z/d}; monotone decay then forces ν ≤ dm/z, and the numerical","core_discovery":"On the paper's own terms: for a chaotic, ergodic classical spin system at high temperature with energy as the only conserved charge, define m such that the thermal expectation O(ε) grows as ε^m near the relevant energy density. Then the equilibrium autocorrelator exhibits a hydrodynamic tail ⟨O(t)O⟩_c ~ t^{-dm/z}, where z is the dynamical critical exponent of energy spreading, and its infinite-time plateau decays as L^{-m}. The plateau scaling is derived from ergodicity through a saddle-point expansion of the microcanonical variance; the connection between plateau and tail follows from hydrodynamic projection and monotone decay of the autocorrelator, yielding the relaxation-overlap inequalit","pith_inferences":["The same logic should apply charge by charge: in systems with additional conserved quantities, the natural replacement is the observable's overlap with that charge's density and its powers, yielding a family of ν = dm/z relations indexed by each conservation law.","The formula is the saturation of an inequality; near-integrable, scarred, or slowly relaxing regimes could show ν strictly smaller than dm/z, offering an independent test of the hydrodynamic-projection assumption.","Because the plateau amplitude at fixed L is controlled by equilibrium thermodynamic derivatives, C(∞) scaling could be used as a diagnostic: measure the plateau at a few sizes, extract m, and compare it with the first nonvanishing energy derivative of the observable's thermal expectation.","For the α = 1.5 long-range model, the crossover between the z = 4/3 and z = 2 tails should occur on a timescale set by L^{z/d}; this is a quantitative prediction one could verify with time-resolved data at larger sizes."],"forward_implications":["The hydrodynamic tail of a local observable is not universal: in a fixed diffusive system, observables with energy-overlap orders m = 1, 2, 3, 4 decay as t^{-d/z}, t^{-2d/z}, t^{-3d/z}, t^{-4d/z} respectively.","The finite-size plateau of the autocorrelator scales as L^{-m}, giving a direct dynamical readout of the observable's thermodynamic overlap order.","In the long-range model with α = 1.5, the theory predicts an intermediate superdiffusive tail with z = 4/3 followed by a late-time diffusive tail with z = 2; the large-system numerics confirm this two-stage decay.","In the long-range model with α = 1.1, the dynamical exponent z = 2α - 1 persists without a crossover to diffusion, so the tail exponent is m/(2α - 1) throughout.","In two-dimensional diffusive systems, the same observables show tails t^{-m}, matching ν = dm/z with d = 2, z = 2."],"supporting_citations":[{"why":"Introduces the relaxation-overlap inequality in quantum ETH systems; this paper extends it to classical ergodic systems.","marker":"[24]"},{"why":"Supplies the hydrodynamic scaling form and the definition of the dynamical critical exponent z used in Eq. (29).","marker":"[11]"},{"why":"Provides the energy-transport exponent z = 2α - 1 for the long-range Ising model and the predicted tails compared here.","marker":"[39]"},{"why":"Formulates shell-ergodicity, the precise ergodicity notion behind Eq. (16).","marker":"[33]"},{"why":"Develops hydrodynamic projections, the mechanism used to project generic local observables onto energy density and its powers.","marker":"[22]"},{"why":"Establishes clustering and extensivity of cumulants, underpinning the saddle-point and central-limit estimates for the plateau.","marker":"[38]"}],"fun_headline_variants":["Spin tails obey t^{-dm/z} from ergodicity","Classical spins: universal decay exponent from ergodicity","One law rules spin correlation tails: t^{-dm/z}","Ergodicity dictates late-time spin decay","Quantum to classical: same hydrodynamic tail law"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The result rests on the hydrodynamic projection assumption: at late times a generic local observable's autocorrelation is governed solely by its projection onto the energy density and its powers, with non-conserved modes contributing only exponentially small corrections; if another conserved quantity or a slow non-hydrodynamic mode dominates the observable, ν = dm/z can fail.","fun_headline_variants_meta":{"raw":{"variants":["Spin tails obey t^{-dm/z} from ergodicity","Classical spins: universal decay exponent from ergodicity","One law rules spin correlation tails: t^{-dm/z}","Ergodicity dictates late-time spin decay","Quantum to classical: same hydrodynamic tail law"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000174,"raw_usage":{"total_tokens":1060,"prompt_tokens":628,"completion_tokens":432,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":372,"completion_tokens_details":{"reasoning_tokens":355}},"tokens_in":372,"tokens_out":432,"duration_ms":5288,"temperature":1.0,"reasoning_tokens":355,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T10:23:23.451523+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In a one-dimensional short-range spin chain with demonstrably diffusive energy transport (z = 2) and no other conserved charge, measure the autocorrelation of S^x_j, whose energy-overlap order is m = 1. The prediction is an algebraic tail C(t) ~ t^{-1/2} followed by a plateau ~ 1/L. A clean t^{-ν} tail with ν differing from 1/2 by more than the numerical uncertainty, or a plateau that fails to scale as 1/L, would falsify the formula.","supporting_citations":[{"cited_title":"Capizzi, J","cited_arxiv_id":null,"evidence_quote":"Introduces the relaxation-overlap inequality in quantum ETH systems; this paper extends it to classical ergodic systems."},{"cited_title":"Spohn, Large scale dynamics of interacting particles (Springer Science & Business Media, 2012)","cited_arxiv_id":null,"evidence_quote":"Supplies the hydrodynamic scaling form and the definition of the dynamical critical exponent z used in Eq. (29)."},{"cited_title":"Nishikawa and K","cited_arxiv_id":null,"evidence_quote":"Provides the energy-transport exponent z = 2α - 1 for the long-range Ising model and the predicted tails compared here."},{"cited_title":"Ergodicity, eigenstate thermalization, and the foundations of statistical mechanics in quantum and classical systems","cited_arxiv_id":"1904.02336","evidence_quote":"Formulates shell-ergodicity, the precise ergodicity notion behind Eq. (16)."},{"cited_title":"Doyon, Diffusion and superdiffusion from hydro- dynamic projections, Journal of Statistical Physics186, 25 (2022)","cited_arxiv_id":null,"evidence_quote":"Develops hydrodynamic projections, the mechanism used to project generic local observables onto energy density and its powers."},{"cited_title":"Friedli and Y","cited_arxiv_id":null,"evidence_quote":"Establishes clustering and extensivity of cumulants, underpinning the saddle-point and central-limit estimates for the plateau."}],"review_version":1}