{"id":"e432cbe4-0000-4cc7-a3a8-e215e7ad6084","arxiv_id":"2501.14876","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Reanalysis of old and new muon-spin data on Sr2RuO4 shows no real difference between them and favors a T-squared low-temperature penetration depth, while the fitted square vortex lattice field profile imposes a new order-parameter constraint.","lead":"This paper re-examines older and newer muon-spin measurements of the superconductor Sr2RuO4 and finds that the low-temperature penetration depth follows a T-squared, not T-linear, trend. The work also argues that current models cannot extract a reliable absolute penetration depth, and that an unusual square vortex-lattice field pattern is a new clue to the superconducting pairing symmetry.","discovery_kind":"replication","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The new order-parameter constraint rests on a fit to the Eu model the authors reject; without a model-independent check of the field-minimum location, the claim is not established.","rationale":"The reader's conditional verdict is reasonable; I focus on the most load-bearing positive claim, the field-minimum constraint. This claim is the main new result in the abstract and conclusion, and it is the part most likely to be cited. It is also the least secure because the field profile is not measured directly but reconstructed through a GL model whose order-parameter symmetry is explicitly disfavored by current evidence. The paper provides no model-independent check of the minimum's location. My concrete test would settle whether the minimum is imposed by the data or by the Eu parametrization. I do not fully agree that Eq. (1)'s validity is the single weakest point: even if Eq. (1) distorts absolute values and some temperature dependence, the paper's conclusion that the scatter prevents distinguishing T-linear from T^2 is largely robust. The reader's weakest_assumption does also mention the Eu-model dependence, so my concern is partially shared, but I would elevate it to the primary issue. Since the concern weakens the novel contribution but does not invalidate the comparative data analysis, the conditional verdict stands unchanged.","tokens_in":10857,"tokens_out":7580,"duration_ms":70426,"concrete_test":"Reanalyze the T=0.02 K, B=15 mT TF-μSR spectrum of Ref. [20] (requesting the raw asymmetry data from the authors if not in the paper) with a model-independent square-VL Fourier expansion B(r)=Σ_G[c_G cos(G·r)+d_G sin(G·r)] truncated at ~10 shells, retaining the noise and broadening terms of Eq. (3), and locate the minimum of the reconstructed field. If the best-fit minimum is at the unit-cell center, or if the fit is insensitive to the minimum position, the Sec. III claim that the nearest-neighbor-midpoint minimum is a real constraint on the order parameter is unsupported. As a cross-check, fit the same spectrum to a d_{x^2-y^2} order-parameter model and compare the resulting field profile and chi-squared with the Eu-model result.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's novel positive claim (Sec. III, Fig. 3(c)) is that the vortex-lattice field minimum lies midway between nearest-neighbor vortices and that this 'must be a manifestation of the true superconducting order parameter.' But this B(r) was obtained only by fitting the TF-μSR spectra to the two-component Eu GL model of Agterberg (Sec. II.D). The authors themselves note (Introduction, Sec. II.D) that revised NMR results make Eu pairing unlikely. A fit to a model whose order parameter is believed to be wrong can imprint the model's topology on the extracted field profile; the low-field shoulder and left peak in the Fourier transform (Fig. 3) may indicate an anisotropic field distribution without uniquely fixing the location of the global minimum. Table I shows reduced chi-squared 1.281 for the Eu model versus 1.077 for a simple second-moment fit, but no parameter-count-adjusted comparison of nested models is given, and only three models are tested. Therefore the statement 'no valid model' and the new order-parameter constraint are stronger than the evidence. The T^2 comparison, which uses Eq. (1) outside the regime the authors themselves quote (Sec. II.A), is a separate but related weakness.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript reanalyzes transverse-field muon spin rotation data from an earlier study of Sr2RuO4 (Luke et al., Ref. [20]) and compares it with a recent study (Khasanov et al., Ref. [19]) under a common analysis. Using a multi-Gaussian decomposition and the second-moment Ginzburg-Landau formula, the authors conclude that the two datasets agree and that the low-temperature penetration depth is compatible with a T^2 rather than a T-linear dependence. They further argue that no existing model reliably determines the absolute value of lambda_ab from TF-muSR, and they identify, from fits to a two-component Eu Ginzburg-Landau model, a square vortex lattice whose field minimum occurs midway between nearest-neighbor vortices, which they propose as a new constraint on the superconducting order parameter.","tokens_in":11089,"tokens_out":6309,"duration_ms":55705,"significance":"The paper addresses a real discrepancy in the muSR literature on Sr2RuO4 and has the virtue of applying the same analysis workflow to both datasets, explicitly reporting reduced chi-squared values for three field-profile models (Table I), and flagging the limited validity of Eq. (1). If the T^2 interpretation survives a quantitatively robust comparison, it would reconcile vortex-state muSR with Meissner-state measurements; if the field-minimum feature is confirmed model-independently, it would be a genuinely new order-parameter constraint. At present, however, the two main positive claims rest on an approximate formula used outside its stated range and on a fit to a pairing model the authors themselves consider disfavored, so the significance is conditional rather than established.","major_comments":[{"comment":"The central comparison of 1/lambda_ab^2 and the T^2 fit in the inset of Fig. 2 are computed from Eq. (1), which the manuscript itself states is strictly valid for kappa >= 5 and restricted b. For the parameters quoted here (xi_ab ~ 663 Å, lambda_ab ~ 1,240-1,740 Å), kappa is only about 1.9-2.6, so the data lie outside the stated validity range. The paper needs either a quantitative estimate of how much the T-vs-T^2 discrimination is distorted by this extrapolation (for example, by repeating the analysis with a low-kappa GL or quasi-classical field profile), or a clear statement that the conclusion is only a consistency check within the same approximate analysis. As it stands, the T^2 claim does not have a firm quantitative basis.","section":"Section II.A, Eq. (1), Fig. 2"},{"comment":"The preference for T^2 over T-linear is not supported by any quantitative model comparison. The inset shows linear fits of lambda_ab versus (T/T_c)^2, but no slopes, intercepts, uncertainties, reduced chi-squared values, or residual analysis are given, and the text concedes that the scatter in the data prevents reliably distinguishing the two forms. The authors should provide a formal comparison (for example, an F-test on nested fits, information criteria, or bootstrap confidence intervals on the exponent) before concluding that both datasets are compatible with T^2 rather than T-linear.","section":"Section II.A, Fig. 2 inset"},{"comment":"The conclusion that there is at the present time no valid model for determining the absolute value of lambda_ab in Sr2RuO4 from TF-muSR measurements is categorical, but the evidence is limited to three models applied to one dataset and one field/temperature point. The reduced chi-squared values in Table I (1.936, 2.576, 3.336, 1.281) show relative performance, but no parameter-count-adjusted model selection is given, and no alternative order-parameter models (e.g., the d-wave model cited in Ref. [33]) are fitted. The claim should be restricted to the models tested, or supported by fits to a broader set of candidate models.","section":"Section III, Table I"},{"comment":"The new order-parameter constraint, that the field minimum lies midway between nearest-neighbor vortices, is obtained exclusively from a fit to the two-component Eu model, which the authors state is likely not the true pairing symmetry. Because the assumed order parameter can imprint its symmetry on B(r), this inference needs a model-independent check. A direct reconstruction of the field distribution from the time spectrum (e.g., maximum-entropy methods), or a fit using the d-wave square-vortex-lattice profiles of Ref. [33], would show whether the midway minimum is robust or an artifact of the Eu model. Without such a check, the statement that this feature must be a manifestation of the true superconducting order parameter is stronger than the evidence.","section":"Section II.D, Fig. 3(c)"}],"minor_comments":[{"comment":"The caption reports xi_ab = 2.65(7) Å for the two-component Eu fit, but Table I and the text identify this number as kappa; the symbol and the units should be corrected.","section":"Section II.D, Fig. 3(c) caption"},{"comment":"There are several typographical errors, including reliablly in the abstract, addtion in Section II.A, indentifiable in Section II.A, and likley in Section II.A.","section":"Section II.A"},{"comment":"Kappa is dimensionless, but the caption lists fit values such as kappa = 0.74(1) Å and kappa = 6.4(7) Å; the Å unit should be removed from kappa.","section":"Fig. 3 caption"},{"comment":"The depolarization rate sigma_dis is described as proportional to 1/lambda_ab^2, but the proportionality constant and the procedure for fixing the nuclear dipole broadening sigma_n from data above T_c are not specified; providing these details would make the fits reproducible.","section":"Eq. (3)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the journal's scope and the reanalysis is useful, but the categorical statements exceed the evidence. I would encourage the editor to request a revision that either adds the missing quantitative analyses or softens the claims; in particular, the no-valid-model sentence and the must-be-a-manifestation sentence should be conditional unless new model-independent evidence is added. The paper reanalyzes data from one of its own earlier publications, so a data/code availability statement would also be beneficial."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know about this paper. First, it makes a solid negative point: the T-linear penetration depth reported by Khasanov et al. doesn't survive reanalysis of the same data alongside the older Luke et al. data. The scatter is too large to distinguish T from T^2, and the authors correctly note that the second-moment formula used in both studies is strictly valid for kappa >= 5, while Sr2RuO4 has kappa around 2.6. That alone makes the paper worth reading. Second, the paper's positive claim—that the vortex-lattice field minimum lying midway between nearest-neighbor vortices is a new order-parameter constraint—is much weaker. It comes from a fit to the two-component Eu GL model, a model the authors themselves believe is wrong for Sr2RuO4 (the revised NMR results indicate even-parity singlet pairing). A fit to a model you reject can imprint that model's topology on the extracted B(r). The Fourier transforms clearly show a low-field shoulder, but that doesn't uniquely fix where the global minimum sits.\n\nThe common-analysis comparison is genuinely new, and the authors are honest about limitations. Table I, with reduced chi-squared for four fits, is useful even though the model comparison isn't nested. They also flag the inconsistency in Ref. [19]'s use of Eq. (1) outside its stated regime, which is a fair and important observation.\n\nThe soft spots are real but not fatal. The T^2 conclusion itself rests on applying Eq. (1) outside that same regime; the authors don't give error bars or goodness-of-fit for the T^2 lines in the inset of Fig. 2. The \"no valid model\" claim is categorical but tested against only three models—I'd soften it to \"none of the models we tried.\" And the field-minimum constraint is a model-dependent inference, not a model-independent observable, so saying it \"must be\" a manifestation of the true order parameter overreaches.\n\nThis deserves peer review. The negative claim about T-linear is important and should be refereed. A serious referee should push for quantifiable T^2 fits, error propagation, and a model-independent check of the field minimum location. I'd send it to PRB or a similar venue, with the expectation of moderate revision. The paper is honest and useful, even if its main positive claim needs a firmer foundation.","headline":"The T-linear muSR claim is convincingly undermined by the common reanalysis, but the new vortex-lattice 'constraint' is built on the very model the authors reject and needs independent support.","tokens_in":11686,"tokens_out":2283,"would_cite":true,"duration_ms":22275,"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":"Reanalysis of muon data says Sr2RuO4's penetration depth follows T2, not T-linear.","keywords":["muon spin rotation","Sr2RuO4","magnetic penetration depth","vortex lattice","Ginzburg-Landau theory","unconventional superconductivity","order parameter","transverse-field muSR"],"falsifier":"A clean test would be a low-temperature measurement of $\\lambda_{ab}(T)$ by a technique sensitive to absolute values in the Meissner state, such as microwave surface impedance or a tunnel-diode resonator: a genuine $\\lambda_{ab} \\propto T$ term persisting below $T/T_c \\approx 0.2$ would falsify the paper's $T^2$ conclusion. A second falsifier is direct imaging of the vortex-lattice field profile, for example by small-angle neutron scattering or scanning superconducting quantum interference device microscopy, showing the field minimum at the center of the square unit cell rather than midway between nearest-neighbor vortices.","tokens_in":10561,"feed_emoji":"🧲","tokens_out":6789,"duration_ms":56516,"temperature":0.7,"pith_summary":"This paper re-examines two transverse-field muon spin rotation (μSR) studies of the unconventional superconductor Sr2RuO4 and argues that the apparent disagreement between them is an artifact of different data-analysis methods. When the same multi-Gaussian fit and Ginzburg-Landau second-moment conversion are applied to both data sets, the temperature dependence of the in-plane penetration depth $\\lambda_{ab}$ is compatible with a limiting $\\lambda_{ab} \\propto T^2$ law and cannot support the claimed $\\lambda_{ab} \\propto T$ behavior at low temperatures. The paper further claims that no existing theoretical model reliably gives the absolute value of $\\lambda_{ab}$ in this material from μSR, because the standard formula is only valid for Ginzburg-Landau parameter $\\kappa \\geq 5$, while Sr2RuO4 has $\\kappa \\approx 2.6$. Instead, the paper identifies an unusual square vortex lattice whose magnetic-field minimum lies midway between nearest-neighbor vortices, and argues this feature must be a manifestation of the true superconducting order parameter.","feed_headline":"Reanalysis: Sr2RuO4 muon data fit T2, not T-linear","feed_subtitle":"Old and new μSR spectra agree once analyzed alike; the square vortex lattice adds a new order-parameter constraint.","key_machinery":"The argument is carried by three objects. Equation (1) is the Ginzburg-Landau second-moment formula relating the square-root variance of the vortex-lattice field distribution to $\\lambda_{ab}^{-2}$, with prefactor $A = 5.07$ for a square lattice; comparing the two data sets through this common formula is what dissolves the reported T-linear dependence. The three candidate spatial field profiles $B(r)$ — the nonlocal London model, the iterative GL solution, and the two-component Eu-state GL model — are fitted directly to the μSR time spectra; only the Eu model reproduces the low-field shoulder in the Fourier spectrum. Within that fitted profile, the location of the minimum of $B(r)$, midway between nearest-neighbor vortices, is the new constraint that any candidate order parameter would have to produce.","core_discovery":"The central claim is that the widely discussed T-linear low-temperature dependence of $\\lambda_{ab}$ in Sr2RuO4 reported in the recent μSR study is not present in the data when the earlier study is analyzed the same way; both samples' data, converted via the same formula, agree below $T \\approx 0.4T_c$ and scatter too much to distinguish T-linear from $T^2$. The paper concludes the limiting low-T behavior is consistent with the $T^2$ dependence of $\\Delta\\lambda_{ab}(T)$ measured by other probes in the Meissner state. The second and more forward-looking claim is that the square vortex lattice in Sr2RuO4 has a highly atypical field profile: in the only model that fits the spectrum (a two-component Eu-order-parameter GL model), $B(r)$ is minimized midway between nearest-neighbor vortices rather than at the center of the square unit cell. Since the same model also assumes a pairing symmetry that the authors believe is likely wrong, they interpret this unusual minimum location as a real signature of the true order parameter, and note that a nodal $d_{x^2-y^2}$ state with nonlocal electrodynamics naturally produces such a profile.","pith_inferences":["Editorial inference: the location of the vortex-lattice field minimum could serve as a generic diagnostic of fourfold order-parameter anisotropy in other low-$\\kappa$ superconductors, not only Sr2RuO4.","Editorial inference: if the field-minimum position is truly fixed by the order parameter, then the failure of isotropic single-component GL models to fit the spectrum at any $\\kappa$ is independent evidence against simple s-wave pairing in the vortex state.","Editorial inference: the paper's negative result on absolute $\\lambda_{ab}$ implies published μSR penetration depths for Sr2RuO4 should be treated as relative until a low-$\\kappa$-valid model is derived, which would recalibrate several prior comparisons.","Editorial inference: a field-dependent study of the same fits would test whether the minimum stays between nearest neighbors as $b$ increases; competition between order-parameter symmetries could move it toward the unit-cell center."],"forward_implications":["The T-linear low-temperature dependence of $\\lambda_{ab}$ reported in Ref. [19] is an artifact of comparing data analyzed by different methods; both data sets are compatible with $\\lambda_{ab} \\propto T^2$.","Absolute values of $\\lambda_{ab}$ in Sr2RuO4 quoted from transverse-field μSR should not be relied on until a model valid at $\\kappa \\approx 2.6$ is available.","Any candidate superconducting order parameter for Sr2RuO4 must produce a square vortex lattice whose field minimum sits between nearest-neighbor vortices, not at the unit-cell center.","A nodal $d_{x^2-y^2}$ order parameter with nonlocal electrodynamics remains a viable explanation, since it both gives the observed $T^2$ dependence and suppresses the field at the saddle point between nearest neighbors.","Future μSR analyses of low-$\\kappa$ superconductors should report fits to field-profile models rather than second moments alone."],"supporting_citations":[{"why":"supplies the recent TF-μSR data set whose claimed T-linear $\\lambda_{ab}$ dependence is re-analyzed and disputed.","marker":"[19]"},{"why":"supplies the earlier TF-μSR data set and the original two-component Eu state global-fit analysis used for comparison.","marker":"[20]"},{"why":"derives the second-moment Ginzburg-Landau formula (Eq. 1) and its strict validity condition $\\kappa \\geq 5$, which the paper argues is violated.","marker":"[21]"},{"why":"provides the iterative GL method tested as one of the three $B(r)$ models for the vortex lattice.","marker":"[30]"},{"why":"gives the two-component Eu state GL model for the square vortex lattice whose fitted $B(r)$ shows the atypical field minimum.","marker":"[32]"},{"why":"explains the expected vortex profile for a $d_{x^2-y^2}$ order parameter, which the paper argues matches the observed field-minimum location.","marker":"[33]"},{"why":"reports the square vortex lattice in Sr2RuO4 by small-angle neutron scattering, establishing the VL symmetry assumed in the fits.","marker":"[23]"},{"why":"predicts the $T^2$ dependence of $\\Delta\\lambda_{ab}$ from nonlocal electrodynamics that the paper says is consistent with both μSR data sets.","marker":"[17]"},{"why":"applies nonlocal electrodynamics specifically to Sr2RuO4, supporting the $T^2$ interpretation of Meissner-state measurements.","marker":"[18]"},{"why":"provides the standard method for fitting theoretical vortex-lattice field profiles directly to TF-μSR time spectra.","marker":"[25]"}],"fun_headline_variants":["μSR reanalysis: Sr2RuO4 λ is T², not T-linear","Square vortex lattice in Sr2RuO4 ups order-parameter constraints","T-linear claim for Sr2RuO4 fails reanalysis; T² wins","No T-linear λ for Sr2RuO4; square lattice hints at odd pairing"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the Ginzburg-Landau second-moment formula used to compare the two data sets remains accurate enough at $\\kappa \\approx 2.6$ to preserve the temperature dependence, even though the paper itself notes it is strictly valid only for $\\kappa \\geq 5$; if that formula distorts the T-dependence outside its stated range, the conclusion that both data sets follow $T^2$ rather than T-linear loses its quantitative basis.","fun_headline_variants_meta":{"raw":{"variants":["μSR reanalysis: Sr2RuO4 λ is T², not T-linear","Square vortex lattice in Sr2RuO4 ups order-parameter constraints","T-linear claim for Sr2RuO4 fails reanalysis; T² wins","No T-linear λ for Sr2RuO4; square lattice hints at odd pairing"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000746,"raw_usage":{"total_tokens":3371,"prompt_tokens":1039,"completion_tokens":2332,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":655,"completion_tokens_details":{"reasoning_tokens":2256}},"tokens_in":655,"tokens_out":2332,"duration_ms":15518,"temperature":1.0,"reasoning_tokens":2256,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:50:20.994143+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A clean test would be a low-temperature measurement of $\\lambda_{ab}(T)$ by a technique sensitive to absolute values in the Meissner state, such as microwave surface impedance or a tunnel-diode resonator: a genuine $\\lambda_{ab} \\propto T$ term persisting below $T/T_c \\approx 0.2$ would falsify the paper's $T^2$ conclusion. A second falsifier is direct imaging of the vortex-lattice field profile, for example by small-angle neutron scattering or scanning superconducting quantum interference device microscopy, showing the field minimum at the center of the square unit cell rather than midway between nearest-neighbor vortices.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the recent TF-μSR data set whose claimed T-linear $\\lambda_{ab}$ dependence is re-analyzed and disputed."},{"cited_title":"Bonalde, B","cited_arxiv_id":null,"evidence_quote":"supplies the earlier TF-μSR data set and the original two-component Eu state global-fit analysis used for comparison."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"derives the second-moment Ginzburg-Landau formula (Eq. 1) and its strict validity condition $\\kappa \\geq 5$, which the paper argues is violated."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the iterative GL method tested as one of the three $B(r)$ models for the vortex lattice."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the two-component Eu state GL model for the square vortex lattice whose fitted $B(r)$ shows the atypical field minimum."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"explains the expected vortex profile for a $d_{x^2-y^2}$ order parameter, which the paper argues matches the observed field-minimum location."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"reports the square vortex lattice in Sr2RuO4 by small-angle neutron scattering, establishing the VL symmetry assumed in the fits."},{"cited_title":"Kashiwaya, K","cited_arxiv_id":null,"evidence_quote":"predicts the $T^2$ dependence of $\\Delta\\lambda_{ab}$ from nonlocal electrodynamics that the paper says is consistent with both μSR data sets."},{"cited_title":"Willa, M","cited_arxiv_id":null,"evidence_quote":"applies nonlocal electrodynamics specifically to Sr2RuO4, supporting the $T^2$ interpretation of Meissner-state measurements."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the standard method for fitting theoretical vortex-lattice field profiles directly to TF-μSR time spectra."}],"review_version":1}