{"id":"5c101b05-3b24-4fa3-ae28-34bf96407133","arxiv_id":"2607.11066","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"When driven from an extraordinary-log boundary critical state into the 3D Heisenberg surface special transition, the surface order parameter obeys M_s^2 ∝ R^{(1+η_s)/r_s} [log(LR^{1/r_s})]^{-q}, a new log-corrected finite-time scaling relation.","lead":"By simulating a 3D Heisenberg magnet with a tuned surface, this paper finds that quenches starting from a special 'extraordinary-log' boundary state create a new finite-time scaling law with an extra logarithmic factor. It shows how the memory of a logarithmically ordered surface changes Kibble-Zurek-style predictions, a result that may sharpen future simulations and experiments on boundary criticality.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (5)'s log correction is fixed by hand; the concrete test is an independent derivation or a refit with q free.","rationale":"The reader's weakest_assumption is exactly the load-bearing step of the paper: the transfer of the equilibrium initial-state log decay into the FTS scaling function with the same q and argument. My independent reading of the Theoretical Analysis section confirms this is asserted rather than derived. I considered whether this is merely a standard FTS practice — initial-state power laws are inserted as U(m0^2,b) → m0^2 b^{...} in Eq. (3) — but for a log decay there is no analogous scaling dimension, so the deduction is not a standard move and is exactly where the claim could fail. I also considered whether the data collapse could independently validate the form; it can, but only if q is not fixed by hand. The paper does not state whether q is fitted or fixed in the collapse, and no fit metrics are given. I do not see a basis to reject: Eq. (5) is plausible, the four other protocols are consistent with BFTS, and the collapse across L=40-128 is encouraging. But the central new claim rests on one undefended step, so CONDITIONAL is right. The concrete test I propose would settle it: a free-q refit, which is computationally trivial using the same data and would directly test whether the log correction is real or an artifact of fixing q≈2.1. I also note a minor typo: the abstract writes [log(LR^{1/η_s})]^{-q} while the main text uses [log(LR^{1/r_s})]^{-q}; the latter is clearly intended, but it underscores the need for care around this relation.","tokens_in":9735,"tokens_out":3448,"duration_ms":27537,"concrete_test":"Refit the Fig. 2(c) data with M_s^2 = A R^α [log(B L R^{1/r_s})]^{-q_dyn} with α, q_dyn, A, B free (or at least q_dyn free), using the same L and R range. If q_dyn is consistent with q≈2.1 within error and the fit is stable under dropping the smallest R and smallest L points, the claim survives; if q_dyn drifts or a fixed A/B is needed to force q≈2.1, Eq. (5)'s specific logarithmic correction is not established. As a complement, independently re-derive Eq. (5) by applying the RG transformation (3) to a boundary two-point function known to decay as [log(L)]^{-q} and check that the log-argument LR^{1/r_s} and q indeed emerge without extra constants.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Eq. (5): M_s^2 ∝ R^{(1+η_s)/r_s} [log(L R^{1/r_s})]^{-q}, for driving from the extraordinary-log boundary critical state into the special point. The paper's derivation is a one-line 'we deduce' (text near Eq. (4)-(5)): because M_s^2 ∝ [log L]^{-q} initially, and M1(x) is dimensionless, M1(x) ∝ [log(L R^{1/r_s})]^{-q} with the same q≈2.1. This is an assertion of a specific functional form, not a derivation. Nothing in the FTS RG framework Eq. (3) guarantees that the initial-state memory enters solely through M_s^2 ∝ [log L]^{-q} times a pure power of R, nor that q does not renormalize (it likely is the same q as the equilibrium extraordinary-log exponent, but that must be shown). The agreement with data in Fig. 2(d) is a collapse using exactly the assumed form and exponent q≈2.1, so it is partially circular as a test. If q in the driven system differs from the equilibrium q, or if the log acquires an additive constant or a different argument, Eq. (5) is not the correct large-rate form. The reader flagged this exact assumption; it is the load-bearing step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies finite-time scaling (FTS) of the special surface transition in a three-dimensional classical Heisenberg model on an L×L×L lattice with open boundaries. It performs Monte Carlo simulations for four driving protocols: temperature heating and cooling through the special transition, and surface-coupling ramps from the ordinary and extraordinary-log boundary critical states into the special point. The temperature protocols are shown to obey the boundary FTS form in Eq. (2). For ramps from the ordinary state, the surface order parameter M_s^2 is dominated by a non-singular L^-2 term, so the two-point correlation C(L/2) is used to extract the singular contribution. The central claim is Eq. (5): for ramps from the extraordinary-log state, in the large-rate limit M_s^2 ∝ R^{(1+η_s)/r_s} [log(L R^{1/r_s})]^{-q} with q≈2.1, supported by data collapse for L=40–128. The paper argues that non-power-law initial-state memory changes the standard large-rate scaling.","tokens_in":10153,"tokens_out":6449,"duration_ms":61011,"significance":"If Eq. (5) is correct, the paper provides a genuinely new extension of FTS/KZM: logarithmic initial-state correlations are remembered in the driven large-rate limit and produce a logarithmic correction to the usual power-law scaling. This goes beyond existing boundary FTS for Ising systems and is relevant to the recently discovered extraordinary-log boundary universality class. The numerical study is broad, covering four protocols, a wide range of system sizes and driving rates, and it uses independently determined equilibrium exponents. However, the key theoretical step is an assertion rather than a derivation, and the central collapse is a consistency test of the assumed scaling form rather than an independent quantitative confirmation.","major_comments":[{"comment":"The step from Eq. (4) to Eq. (5) is not derived. The sentence 'Since M1(x) is a dimensionless scaling function, we deduce...' assumes that the initial-state memory enters M1(x) exactly as [log x]^{-q}, with the same q as the equilibrium extraordinary-log decay and with no additive constant or renormalization. The RG transformation U(m0^2,b) in Eq. (3) could in principle produce a more general function of x=LR^{1/r_s}. This is the load-bearing step. Please provide a derivation of the scale transformation for a logarithmically correlated initial state, or at least an independent test: fit the large-R data with q as a free parameter (and, if useful, with log(x+c)) and report the best-fit q and the collapse residual. The reported effective exponent R^{0.149(3)} cannot by itself validate Eq. (5).","section":"Theoretical analysis, Eqs. (3)–(5)"},{"comment":"The central numerical evidence is visual. No collapse metric, error bar, or fit range is given for the rescaling in Fig. 2(d), and it is not stated whether q≈2.1 is fixed to the equilibrium value from Ref. [61] or fitted to the driven data. Since the rescaling uses Eq. (5) as the ansatz, the collapse is a self-consistency check, not a falsifiable test. Please add a quantitative collapse measure (e.g., scatter of the collapsed curves) and a sensitivity analysis with respect to η_s, r_s, and q. Also define the crossover between the low-rate regime (inset, Eq. (2)) and the large-rate regime (main panel, Eq. (5)).","section":"Fig. 2(d) and data-collapse quality"},{"comment":"The scaling dimension r_s = z + 1/ν_s is computed using the bulk dynamic exponent z≈2.033. The paper does not justify or test that the surface special transition is governed by the same z; if a distinct surface dynamic exponent exists, the prefactor exponent in Eq. (5) changes. Please either cite evidence that z is the correct dynamic exponent for Model-A dynamics at the special transition, or fit the collapse exponent independently and compare with z + 1/ν_s.","section":"Models and method; definition of r_s"}],"minor_comments":[{"comment":"The abstract (and the line near the end of the full text) writes [log(LR^{1/η_s})]^{-q}; this should be [log(LR^{1/r_s})]^{-q}. The exponent 1/η_s is dimensionally inconsistent and appears to be a typo.","section":"Abstract and Eq. (5)"},{"comment":"For C(L/2), the fit gives R^{-0.83(3)} while the predicted exponent is R^{-0.772}; the text says the theory 'can well describe' the results without discussing the discrepancy. Also, near the end of the ordinary-state paragraph, 'Fig. 1(f)' should be 'Fig. 2(f)', and the panel references in the Fig. 2 caption are inconsistent.","section":"Ordinary-state analysis and Fig. 2"},{"comment":"Several typos: 'extraordianry-log', 'the the', 'KibbleZurek' (missing hyphen), and duplicated words in the Fig. 2 caption. Please copy-edit carefully.","section":"Throughout"},{"comment":"The constraint term \\sum_i [S_i^2 + λ(S_i^2 - 1)^2] is unusual for a unit-length Heisenberg model; a one-sentence explanation of the soft-spin representation and why λ=5.2 suppresses finite-size effects would improve clarity.","section":"Hamiltonian, Eq. (1)"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely a useful contribution, and the data are suggestive. My main concern is the derivation of Eq. (5): the log correction is essentially an ansatz read off the initial state, so the authors should not claim a derivation without a more careful RG treatment or a free-q refit. If that step is strengthened, I would support publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Start with what is new. The paper extends boundary finite-time scaling to an initial state with log-correlations, the extraordinary-log boundary state of the 3D Heisenberg model. The claimed result, Eq. (5), is a power law times a log factor, and it replaces the pure power-law exponent R^0.221 with numerical R^0.149(3). That is a real, nontrivial deviation, and the data collapse across L=40–128 and a wide range of rates is visually convincing. The temperature-driven protocols and the ordinary-critical-state ramp are also consistent with the existing BFTS, which is a useful sanity check.\n\nThe central soft spot is exactly where the stress test points. The step from Eq. (4) to Eq. (5) is a single sentence: because the initial state has M_s^2 ∝ [log L]^{-q} and M1(x) is dimensionless, the authors 'deduce' M1(x) ∝ [log(LR^{1/r_s})]^{-q}. That is an assertion, not a derivation. The RG framework does not force the memory of the initial state to factorize that way, and q could renormalize under the drive. The collapse is built on the assumed form, so it is a consistency check, not a test of the assumption. The fix is straightforward: fit q from the driven data, or derive the log factor from an RG calculation. This is the load-bearing step, and the paper would be much stronger with either.\n\nMinor issues: the abstract writes LR^{1/η_s} instead of LR^{1/r_s}; the text refers to Fig. 1(f) for C(L/2) which is actually in Fig. 2(f); and the simulation plots have no visible error bars or collapse metrics. None of these change the physics, but they make it harder to evaluate the quality of the collapse.\n\nMy overall take: this is an honest extension of FTS to a non-power-law initial state. The result is plausible and the numerical evidence is real, but Eq. (5) is not yet established because the key mapping is assumed. I would send it to peer review and ask the referee to focus on the derivation and on a free-q fit. It is a useful paper for the FTS and boundary-criticality subfield, and it does not deserve a desk reject.","headline":"A log-corrected FTS form for extraordinary-log initial states, supported by a decent collapse but with the key step asserted rather than derived.","tokens_in":10585,"tokens_out":3554,"would_cite":false,"duration_ms":32109,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82B27","82B20","82C80"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper shows that driving a 3D Heisenberg magnet from the extraordinary-log boundary critical state into the surface special transition makes the surface order parameter follow M_s^2 ∝ R^{(1+η_s)/r_s} [log(LR^{1/r_s})]^{-q}, a logarithm","keywords":["finite-time scaling","boundary criticality","special surface transition","extraordinary-log phase","3D Heisenberg model","Kibble-Zurek mechanism","Monte Carlo simulations","logarithmic corrections"],"falsifier":"Measure M_s^2 at a fixed large R for several system sizes L, and plot y = M_s^2 R^{-(1+η_s)/r_s} [log(LR^{1/r_s})]^q against x = R^{1/r_s} L. If y is not flat at large x, or if the best-fit q changes appreciably when the R window is varied, the assumed logarithmic argument and exponent are wrong.","tokens_in":9623,"feed_emoji":"🧲","tokens_out":8682,"duration_ms":71601,"temperature":0.7,"pith_summary":"The paper studies what happens when a 3D classical Heisenberg magnet with open boundaries is driven quickly to its special surface transition point. For temperature quenches and for ramps starting from the ordinary boundary critical state, the surface order parameter follows the usual boundary finite-time scaling. The central claim concerns ramps that start from the extraordinary-log boundary critical state, where correlations decay logarithmically: in that case the fast-driving scaling acquires a logarithmic factor and becomes M_s^2 ∝ R^{(1+η_s)/r_s} [log(LR^{1/r_s})]^{-q}. The authors derive this form from a generalized finite-time scaling ansatz that keeps a memory of the logarithmic initial state, and they verify it by data collapse over system sizes L = 40 to 128. A sympathetic reader would care because it shows that the Kibble–Zurek-style scaling of boundary critical dynamics is not universal in initial-condition class: a non-power-law initial state leaves a detectable trace in the driven regime.","feed_headline":"Logarithmic memory changes fast-quench scaling at surface transitions","feed_subtitle":"Fast quenches from a logarithmic boundary state obey a new scaling law, confirmed by Monte Carlo collapse.","key_machinery":"The carrying mechanism is the generalized finite-time scaling relation M_s^2(R,L,m_0^2) = b^{-(1+η_s)} M_s^2(R b^{r_s}, L b^{-1}, U(m_0^2,b)), with b the rescaling factor and U the scale transformation of the initial-state squared magnetization. Choosing b = R^{-1/r_s} puts the order parameter in the form R^{(1+η_s)/r_s} M_1(LR^{1/r_s}, U(m_0^2,R^{-1/r_s})). The paper then lets the logarithmic boundary initial state enter through the memory term U, deducing that the dimensionless function M_1(x) behaves as [log x]^{-q} at large x. This single hypothesis—logarithmic initial-state memory—converts the standard power-law finite-time scaling into Eq. (5).","core_discovery":"The central discovery is a new fast-ramp scaling law for the surface order parameter: when the 3D Heisenberg model is driven from the extraordinary-log boundary critical state to the special surface transition, M_s^2 ∝ R^{(1+η_s)/r_s}[log(LR^{1/r_s})]^{-q}, with η_s≈−0.473, r_s≈2.393, and q≈2.1. The paper derives this from generalized finite-time scaling by assuming the dimensionless scaling function inherits the logarithmic form [log x]^{-q} of the initial state. Monte Carlo data collapse onto one curve for L=40–128 when plotted as [M_s^2 R^{-(1+η_s)/r_s}]^{-1/q} against log(R^{1/r_s}L), showing non-power-law initial-state memory persists into the driven regime.","pith_inferences":["Inference: Eq. (5) implies the effective frozen scale in the large-rate limit still depends on the system size through log(LR^{1/r_s}); a direct prediction is that at fixed large R, M_s^2 continues to drift with L in a logarithmic way, which could be checked by varying L alone.","Inference: If the exponent q is truly unchanged between equilibrium and driven settings, the same q must collapse data at multiple R windows; a drift in the best-fit q would signal a dynamic renormalization of the log exponent, a possibility the paper does not address.","Inference: The same log-memory mechanism should apply to other continuous-symmetry O(N) models and to the extraordinary-log phase in other geometries, although the paper only demonstrates it for the 3D Heisenberg special transition."],"forward_implications":["Temperature heating and cooling across the special transition, and surface-coupling ramps from the ordinary critical state, remain described by the standard boundary finite-time scaling; the new log term appears only for extraordinary-log initial states.","For fast ramps from the extraordinary-log state, the naive power-law prediction M_s^2 ∝ R^{(1+η_s)/r_s} ≈ R^{0.221} fails; the measured exponent 0.149(3) is explained only after including the log factor.","The log-corrected scaling holds over L = 40–128 and a wide range of driving rates, indicating it is a genuine scaling feature rather than a finite-size artifact.","The derivation generalizes the boundary finite-time scaling framework beyond power-law initial conditions, showing that initial-state universality classes matter for the Kibble–Zurek scaling at boundaries."],"fun_headline_variants":["Logarithmic corrections drive new boundary scaling","Fast quench from log state yields new scaling law","New quench law for log-memory surface states","Log-memory initial states alter critical scaling","Boundary log-memory reshapes fast-quench scaling"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is the paper's assertion that in the large-rate limit the dimensionless scaling function takes exactly the same logarithmic form as the equilibrium initial state, [log(LR^{1/r_s})]^{-q} with q ≈ 2.1; this step is stated (the paper says 'we deduce') rather than derived, and if the log argument or exponent is not precisely the initial-state one, Eq. (5) is incomplete.","fun_headline_variants_meta":{"raw":{"variants":["Logarithmic corrections drive new boundary scaling","Fast quench from log state yields new scaling law","New quench law for log-memory surface states","Log-memory initial states alter critical scaling","Boundary log-memory reshapes fast-quench scaling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000589,"raw_usage":{"total_tokens":2654,"prompt_tokens":848,"completion_tokens":1806,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":592,"completion_tokens_details":{"reasoning_tokens":1733}},"tokens_in":592,"tokens_out":1806,"duration_ms":13072,"temperature":1.0,"reasoning_tokens":1733,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T07:02:01.497879+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure M_s^2 at a fixed large R for several system sizes L, and plot y = M_s^2 R^{-(1+η_s)/r_s} [log(LR^{1/r_s})]^q against x = R^{1/r_s} L. If y is not flat at large x, or if the best-fit q changes appreciably when the R window is varied, the assumed logarithmic argument and exponent are wrong.","supporting_citations":[],"review_version":2}