{"id":"2cf95aec-ffd6-474f-b2f5-354d3f223500","arxiv_id":"2506.03509","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Field-dependent muon-spin and upper-critical-field data are best described by a two-band model, indicating multigap superconductivity in NbRuSi and TaRuSi.","lead":"This paper finds that two superconducting materials, NbRuSi and TaRuSi, have more than one set of electronic bands participating in superconductivity. The authors use field-dependent muon-spin measurements, resistivity, specific heat, and band calculations to show that a two-band model matches the data better than a one-band model.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The multiband evidence from sigma_sc(H) rests on a fixed-weight two-band fit with no model selection: w=0.7 is forced for both compounds although DFT indicates ~85% for TaRuSi, so the multiband claim is conditional until w is varied and fit statistics are reported.","rationale":"In good faith, the paper does several things right: it presents multiple independent datasets, notes the prior failure of temperature-dependent TF-muSR to distinguish gaps, reports positive curvature in Hc2(T), and connects the multiband scenario to DFT bands crossing the Fermi level. These are real, non-circular supports for plausibility. However, the central assertion of 'solid evidence for multiband SC' is load-bearing on the sigma_sc(H) fit, and that fit is currently presented in a way that cannot be independently checked: the weight w is fixed, no model-selection statistics are given, and the underlying two-component TF-muSR decomposition is not documented. The reader's weakest assumption identifies exactly this point, and I agree with it. The concern is not that the authors are wrong; it is that the manuscript does not yet allow the reader to distinguish a genuine second band from a flexible three-parameter fit. A focused refitting exercise with w free and with the DFT-derived weights, plus AIC/BIC comparison, would settle the matter. If the free-w fit remains clearly superior and stable, ACCEPT would be justified; if not, the claim should be softened. Since the reader already returned CONDITIONAL with a request for model-selection analysis and/or data release, my read does not change the verdict.","tokens_in":17031,"tokens_out":6542,"duration_ms":79859,"concrete_test":"Refit the field-dependent sigma_sc data for NbRuSi and TaRuSi with Eq. (3) in three variants: (i) w free; (ii) w fixed to the DFT DOS weights (0.7 for NbRuSi, 0.85 for TaRuSi); (iii) single-band Eq. (2). For each, report chi-squared, AICc, and the 95% confidence intervals of lambda_0, xi_1, xi_2, and w. If the free-w fit gives Delta AICc < 10 over the single-band model, or if its w interval includes 0 or 1, the multiband evidence from sigma_sc(H) should be downgraded; if w for TaRuSi drifts substantially from 0.7, the reported xi values are not robust. The same procedure should be applied to Hc2(T) with the Gurevich model.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive evidence is Fig. 5: the two-band modified-London fit of sigma_sc(H) at 0.3 K (Eq. 3) is described as 'clearly superior' to the single-band model (Eq. 2). Three things make this claim fragile. (1) No statistical model comparison is reported: no chi-squared, AIC/BIC, or residual analysis, so 'superior' is visual. Since Eq. (3) has three fitted parameters (lambda_0, xi_1, xi_2) plus a fixed weight w, versus Eq. (2)'s two, some improvement is expected even if the second band is spurious. (2) w is fixed to 0.7 for both compounds, but the DOS analysis in Fig. 7(d) and (h) gives ~70% for NbRuSi and >85% for TaRuSi; the same w is not derived from a fit or from the band weights. With w fixed, the quoted xi_1 and xi_2 uncertainties (1-2 nm) are conditional and do not reflect the dominant systematic uncertainty. (3) The sigma_sc values themselves are obtained from a two-oscillation TF-muSR fit (Eq. 1) whose individual amplitudes, frequencies, Gaussian widths, and field evolution are not reported; the effective second moment can be sensitive to this decomposition. If the two-oscillation fit is unstable, the field dependence could be an artifact. The independent support from Hc2(T) positive curvature is suggestive but also comes from a multi-parameter two-band model with no model-selection metric. Thus the strongest claim, 'solid evidence for multiband SC', is not yet supported at that strength; at best it is conditional on the fixed-w two-band description.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper reports a combined muon-spin rotation, electrical-resistivity, and specific-heat study of the noncentrosymmetric superconductors NbRuSi and TaRuSi, together with DFT band-structure calculations. The central claim is that both compounds exhibit multiband (multigap) superconductivity. The evidence presented is: (i) a positive curvature in Hc2(T) and a better description of Hc2(T) by a two-band model than by single-band WHH or GL fits; (ii) a field-dependent muon-spin relaxation rate σsc(H) at 0.3 K that is better described by a two-band modified-London model than by a single-band model; (iii) DFT results showing multiple bands at the Fermi level, with two dominant bands contributing most of the DOS. The authors conclude that the data provide solid evidence for multigap superconductivity in both compounds.","tokens_in":17403,"tokens_out":4444,"duration_ms":43979,"significance":"If the multiband scenario is confirmed, the paper would be a valuable addition to the study of unconventional superconductivity in noncentrosymmetric, topologically nontrivial materials, since it would identify a concrete system in which multigap pairing coexists with time-reversal-symmetry breaking. The experimental work is careful and combines several independent techniques: field-dependent TF-µSR, Hc2(T) from transport and specific heat, and DFT with spin-orbit coupling. The observation that temperature-dependent superfluid density alone cannot distinguish one-gap from two-gap behavior is correctly presented, and the paper honestly notes the limitation of that previous dataset. The DFT analysis is explicit about orbital contributions and DOS weights. The main weakness is that the decisive two-band fits are not subjected to quantitative model-selection tests, and the fixed weight w=0.7 is not justified from the data, so the strength of the central claim currently exceeds what the analysis supports.","major_comments":[{"comment":"The assertion that the two-band model is 'clearly superior' to the single-band model for σsc(H) is based only on visual inspection. No goodness-of-fit metric (χ², reduced χ², AIC/BIC, or residual analysis) is reported, and the two-band model has three fitted parameters (λ0, ξ1, ξ2) plus a fixed weight w, versus two parameters (λ0, Hc2) for the single-band model. An improvement in fit is therefore expected even if the underlying physics is single-band. Please report quantitative fit statistics and residuals over the full field range, and specify how many degrees of freedom each fit actually has.","section":"§III, Fig. 5 and Eq. (3)"},{"comment":"The weight w is fixed to 0.7 for both NbRuSi and TaRuSi, but the DOS analysis in Fig. 7(d) and (h) gives approximately 70% for NbRuSi and more than 85% for TaRuSi. The quoted uncertainties of 1–2 nm on ξ1, ξ2, and λ0 are therefore conditional on an externally chosen parameter and do not reflect the dominant systematic uncertainty. Please show how the fitted parameters and the resulting Hc2 values change when w is varied over a physically reasonable range (e.g., 0.5–0.9), and state whether the multiband conclusion is robust to this variation.","section":"§III, Fig. 5 and Table I"},{"comment":"The σsc values are extracted from a two-oscillation TF-µSR fit, but the individual amplitudes A1, A2, frequencies B1, B2, Gaussian widths σ1, σ2, and their field dependence are not reported. The effective second moment defined after Eq. (1) is sensitive to how the two oscillating components are separated, particularly if one component is weak or if there is background contamination. Please provide the full set of fit parameters as a function of field, and justify the two-oscillation model against plausible alternatives (e.g., one sample component plus a broad distribution) so that the field dependence of σsc is not an artifact of the decomposition.","section":"§III, Eq. (1) and Fig. 4"},{"comment":"The claim that the two-band model is 'clearly superior' to WHH and GL for Hc2(T) also lacks statistical support. Please report the fit quality (residuals or χ²) for each model, with the number of free parameters and any constraints, so that the comparison is quantitative rather than visual. In addition, the consistency check between Hc2 derived from ξ2 and the bulk Hc2 values is loose for NbRuSi: the values 1.9(3) T and 1.40(5) T differ by about 0.5 T, which is not trivial compared with the quoted errors. Please clarify whether this difference is expected from the temperature dependence between 0.3 K and T=0, or whether it indicates a systematic discrepancy.","section":"§III, Fig. 3 and Hc2(T)"}],"minor_comments":[{"comment":"There is a duplicated word: 'time-reversal, and and parity variants' should be 'time-reversal, and parity variants'.","section":"Introduction"},{"comment":"There is a typo in 'we used both a singe-band and a two-band model' (should be 'single-band').","section":"§III, after Eq. (1)"},{"comment":"The definition of the reciprocal lattice vectors is difficult to read: 'q = 4π/√3 a (m√3/2, n + m/2)' should be written explicitly with m and n integers and with the two vector components clearly separated. Please also define the summation range or truncation used in the numerical evaluation.","section":"Eq. (3)"},{"comment":"The two-band coupling constants λ11, λ22, and λ12 are quoted without uncertainties. If they come from a fit, please provide errors; if they are derived or estimated, say so explicitly.","section":"Table I"},{"comment":"The inset contour plots showing positive curvature are small and hard to read. Please enlarge them or add explicit guides to the eye, since they are used as visual evidence for the multiband scenario.","section":"Fig. 3"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a relevant question and the experimental data appear to be of good quality, but the central claim needs to be backed by quantitative model comparison and a robustness check of the fixed weight w. These are standard and feasible analyses, so I believe a major revision is appropriate rather than rejection. The 'solid evidence' wording in the abstract and conclusion should be softened until those analyses are provided."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Good to see this follow-up on NbRuSi and TaRuSi. The new content is the field-dependent muSR and Hc2(T) data, and that is genuinely useful. The previous paper's T-dependent superfluid density could not separate one-gap from two-gap; the field dependence is a smarter probe. The measurements look careful: resistivity and specific heat under field give consistent Tc values, and the Hc2(T) curves show positive curvature, which is a standard multiband signature. The single-band fit to sigma_sc(H) clearly fails above 200 mT, and the two-band modified-London model tracks the data over the full range. The DFT also shows multiple bands crossing EF, with two bands dominating the DOS. That is a coherent picture.\n\nThe soft spot is the statistical support for the central claim. The paper says the two-band model is 'clearly superior' to single-band, but there is no quantitative model comparison: no chi-square, no AIC/BIC, no residual analysis. With three fitted parameters plus a fixed weight w, versus two in the single-band model, some improvement is guaranteed even if the second band is spurious. The weight w is set to 0.7 for both compounds, but the DFT DOS weights are ~70% for NbRuSi and >85% for TaRuSi. That inconsistency is never addressed. If w were allowed to vary, the quoted xi_1 and xi_2 uncertainties would almost certainly grow. The two-oscillation TF-muSR decomposition that yields sigma_sc is also not documented in detail, so the effective second moment could be sensitive to the fitting assumptions.\n\nNone of this makes the multiband conclusion wrong. The positive curvature in Hc2 and the DFT band count are independent strands pointing the same way, and the authors themselves note that low-temperature specific heat remains crucial. But the phrase 'solid evidence' overstates what is currently testable from the manuscript. The evidence is suggestive and worth taking seriously, not yet decisive.\n\nWhere does that leave it? I'd send it to review, but I would insist on model-selection statistics, a free weight or a sensitivity analysis, and ideally raw or processed sigma_sc(H) data so the fits can be checked. If the authors can show that the two-band model wins on an information criterion and that the parameters are stable across a reasonable range of w, the paper becomes convincing. As it stands, it is a good experimental report with an analysis that needs to catch up to the data.","headline":"Useful field-dependent muSR data, but the 'solid evidence' for multiband superconductivity is not yet supported without proper model selection and a free two-band weight.","tokens_in":18022,"tokens_out":2556,"would_cite":false,"duration_ms":24521,"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":"NbRuSi and TaRuSi superconduct in two bands, not one: field-dependent muon-spin relaxation and upper critical field data both require a two-band description where single-band models fail.","keywords":["multiband superconductivity","multigap superconductivity","muon spin relaxation","upper critical field","Kramers nodal-line semimetal","noncentrosymmetric superconductor","NbRuSi","TaRuSi"],"falsifier":"Measure $\\sigma_{\\rm sc}(H)$ below 10 mT at base temperature and check whether the relaxation drops as the field approaches $H_{c1}$, as the two-band fit implies; or re-analyze the published $\\sigma_{\\rm sc}(H)$ and $H_{c2}(T)$ datasets with an explicit model-selection criterion (for instance Akaike or Bayesian information) comparing the two-band model against a disorder-broadened single-band model and a three-band model; or measure the specific heat down to about 0.3 K to look for the second gap's distinct thermodynamic signature. If a single-band model with disorder or a three-band model fits at least as well, or if the sub-10 mT drop is absent, the multiband conclusion would be overturned.","tokens_in":16773,"feed_emoji":"🧲","tokens_out":12209,"duration_ms":112769,"temperature":0.7,"pith_summary":"The paper claims that the noncentrosymmetric superconductors NbRuSi and TaRuSi, both predicted to be three-dimensional Kramers nodal-line semimetals, have multiband (multigap) superconductivity, a possibility that earlier temperature-dependent muon measurements could not distinguish from a single gap. The evidence is twofold: the upper critical field $H_{c2}(T)$ is reproduced over the full field range only by a two-band model, and the field-dependent muon-spin relaxation rate $\\sigma_{\\rm sc}(H)$ in the vortex state departs strongly from the single-band prediction above roughly 200 mT while the two-band modified London model fits the whole range. Band-structure calculations reinforce the picture, since two of the several Fermi-level bands contribute 70 to 85 percent of the density of states. If the claim holds, NbRuSi and TaRuSi join the short list of superconductors where multigap pairing, time-reversal-symmetry breaking, and topological band geometry appear in the same material.","feed_headline":"Two bands, not one, carry superconductivity in NbRuSi and TaRuSi","feed_subtitle":"Field-dependent muon and critical-field data rule out a single gap in both compounds.","key_machinery":"The load-bearing object is the two-band modified London model for the second moment of the vortex-lattice field distribution (Eq. 3), in which the muon relaxation rate $\\sigma_{\\rm sc}$ is a sum over two bands, each with its own coherence length $\\xi_1$, $\\xi_2$ and relative weight $w$, sharing one penetration depth $\\lambda_0$. Fitting $\\sigma_{\\rm sc}(H)$ at fixed $w = 0.7$ recovers the bulk upper critical field from the shorter coherence length $\\xi_2$ and yields a lower virtual critical field $H^*_{c2}$ for the smaller gap from $\\xi_1$. Its partner is the two-band theory of $H_{c2}(T)$, whose positive curvature reflects the field suppression of the weaker band, in contrast to the single-band WHH and GL forms that underestimate $H_{c2}(0)$.","core_discovery":"The paper establishes that the superconducting state of NbRuSi and TaRuSi involves at least two active bands with comparable weights. Temperature-dependent resistivity and specific heat under fields up to 5 T give upper critical fields $\\mu_0H_{c2}(0) = 1.40(5)$ T and $3.20(5)$ T, respectively; the $H_{c2}(T)$ curves show a positive curvature that the single-band Werthamer-Helfand-Hohenberg and Ginzburg-Landau models cannot capture at high fields, whereas a two-band model fits the full range. Transverse-field muon-spin relaxation at 0.3 K in fields from 10 to 800 mT yields a superconducting relaxation rate $\\sigma_{\\rm sc}(H)$ that falls far more steeply with field than the single-band vortex-lattice formula predicts; the two-band modified London model with fixed weight $w = 0.7$ describes the data with penetration depths $\\lambda_0 = 338(2)$ nm (NbRuSi) and $216(2)$ nm (TaRuSi) and two distinct coherence lengths, the shorter one recovering the measured $H_{c2}$ and the longer one defining a virtual critical field $\\mu_0H^*_{c2} \\approx 0.8$ to $0.9$ T that suppresses the smaller gap. Density-functional calculations show several bands crossing the Fermi level, with two dominant bands carrying most of the density of states. The paper concludes that the positive curvature of $H_{c2}(T)$, the distinct field response of $\\sigma_{\\rm sc}(H)$, and the band structure together constitute solid evidence for multiband superconductivity in both compounds.","pith_inferences":["A decisive experiment the paper leaves open is the sub-10 mT regime: the two-band fit predicts a downturn in $\\sigma_{\\rm sc}(H)$ as the field approaches $H_{c1}$, and the authors note they did not investigate it.","If the two-gap picture is correct, specific-heat measurements down to about 0.3 K should reveal the thermodynamic fingerprint of the second gap; the paper itself flags such measurements as crucial.","The analysis fixes the band weight and never applies a formal model-selection criterion, so re-fitting the same data with a three-band model or with disorder-broadened single-band line shapes would test whether the two-band model wins on merit rather than by having more parameters.","Should the multiband claim survive, the combination of multigap pairing, time-reversal-symmetry breaking, and Kramers nodal-line topology makes NbRuSi and TaRuSi natural platforms for studying how superconducting gap structure and topological band geometry influence each other; the paper does not itself claim topological superconductivity."],"forward_implications":["Future analyses of superfluid density, vortex dynamics, or surface states in NbRuSi and TaRuSi must treat both compounds as at least two-band superconductors.","Temperature-dependent superfluid density alone cannot resolve the two gaps because they are close in size ($\\Delta_{0,1}/\\Delta_{0,2} \\approx 0.80$ to $0.85$) with weight $w = 0.7$; field-dependent probes are the discriminating tool for this system.","The two-band penetration depths, $\\lambda_0 = 338(2)$ nm and $216(2)$ nm, agree with the values from earlier temperature-dependent muon measurements, so the two analyses are mutually consistent.","Because the intra-band couplings ($\\lambda_{11} \\approx \\lambda_{22} = 0.22$ to $0.29$) are roughly five times the inter-band couplings ($\\lambda_{12} = 0.043$ to $0.06$), the positive curvature in $H_{c2}(T)$ is more pronounced than in the related compound NbReSi.","Both upper critical fields lie below the Pauli limit, indicating that orbital pair breaking, not spin effects, controls the destruction of superconductivity by a magnetic field."],"supporting_citations":[{"why":"The prior study of NbRuSi and TaRuSi that established time-reversal-symmetry breaking and (s+ip) pairing, and supplied the temperature-dependent superfluid-density data reanalyzed here.","marker":"[16]"},{"why":"The ZrNiAl-type NbReSi superconductor whose comparable inter- and intra-band couplings contrast with the weaker inter-band coupling found in the present compounds.","marker":"[49]"},{"why":"The single-band WHH theory of the upper critical field that fails to fit the high-field data, serving as the baseline the two-band model must beat.","marker":"[57]"},{"why":"The two-band model for Hc2(T) used to fit the full field range and to extract the intra- and inter-band couplings.","marker":"[58]"},{"why":"The earlier rhenium-boron multigap muon-spin study that established the distinct field response used as the reference for single-gap versus two-gap behavior.","marker":"[62]"},{"why":"Provides the method for extracting the effective Gaussian relaxation rate sigma_eff from vortex-state muon line shapes used in Eq. (1).","marker":"[63]"},{"why":"Source of the single-band sigma_sc(H) relation (Eq. 2), the failing baseline for the field-dependent relaxation data.","marker":"[64]"},{"why":"The ideal Ginzburg-Landau vortex-lattice theory that underlies the single-band sigma_sc(H) relation of Eq. (2).","marker":"[65]"},{"why":"The two-gap flux-lattice field-distribution framework on which the modified London model of Eq. (3) is built.","marker":"[66]"}],"fun_headline_variants":["Two bands, not one, drive superconductivity in NbRuSi and TaRuSi","Multiband superconductivity confirmed in topological semimetals","Two-band superconductivity in NbRuSi and TaRuSi","NbRuSi and TaRuSi superconduct with two bands"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The conclusion that two superconducting bands exist rests on assuming that the two-band London model, with its band weight fixed at 0.7 and three adjustable parameters, is the true source of the field dependence in the muon-spin data, rather than a statistically better-looking version of a single-band fit with extra freedom, or a field dependence produced by vortex-lattice disorder or anisotropy.","fun_headline_variants_meta":{"raw":{"variants":["Two bands, not one, drive superconductivity in NbRuSi and TaRuSi","Multiband superconductivity confirmed in topological semimetals","Two-band superconductivity in NbRuSi and TaRuSi","NbRuSi and TaRuSi superconduct with two bands"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000971,"raw_usage":{"total_tokens":4200,"prompt_tokens":1086,"completion_tokens":3114,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":702,"completion_tokens_details":{"reasoning_tokens":3040}},"tokens_in":702,"tokens_out":3114,"duration_ms":27562,"temperature":1.0,"reasoning_tokens":3040,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:01:47.609325+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure $\\sigma_{\\rm sc}(H)$ below 10 mT at base temperature and check whether the relaxation drops as the field approaches $H_{c1}$, as the two-band fit implies; or re-analyze the published $\\sigma_{\\rm sc}(H)$ and $H_{c2}(T)$ datasets with an explicit model-selection criterion (for instance Akaike or Bayesian information) comparing the two-band model against a disorder-broadened single-band model and a three-band model; or measure the specific heat down to about 0.3 K to look for the second gap's distinct thermodynamic signature. If a single-band model with disorder or a three-band model fits at least as well, or if the sub-10 mT drop is absent, the multiband conclusion would be overturned.","supporting_citations":[{"cited_title":"Shang, J","cited_arxiv_id":null,"evidence_quote":"The prior study of NbRuSi and TaRuSi that established time-reversal-symmetry breaking and (s+ip) pairing, and supplied the temperature-dependent superfluid-density data reanalyzed here."},{"cited_title":"Shang, D","cited_arxiv_id":null,"evidence_quote":"The ZrNiAl-type NbReSi superconductor whose comparable inter- and intra-band couplings contrast with the weaker inter-band coupling found in the present compounds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The single-band WHH theory of the upper critical field that fails to fit the high-field data, serving as the baseline the two-band model must beat."},{"cited_title":"Gurevich, Iron-based superconductors at high magnetic fields, Rep","cited_arxiv_id":null,"evidence_quote":"The two-band model for Hc2(T) used to fit the full field range and to extract the intra- and inter-band couplings."},{"cited_title":"Shang, W","cited_arxiv_id":null,"evidence_quote":"The earlier rhenium-boron multigap muon-spin study that established the distinct field response used as the reference for single-gap versus two-gap behavior."},{"cited_title":"Maisuradze, R","cited_arxiv_id":null,"evidence_quote":"Provides the method for extracting the effective Gaussian relaxation rate sigma_eff from vortex-state muon line shapes used in Eq. (1)."},{"cited_title":"Barford and J","cited_arxiv_id":null,"evidence_quote":"Source of the single-band sigma_sc(H) relation (Eq. 2), the failing baseline for the field-dependent relaxation data."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The ideal Ginzburg-Landau vortex-lattice theory that underlies the single-band sigma_sc(H) relation of Eq. (2)."},{"cited_title":"Serventi, G","cited_arxiv_id":null,"evidence_quote":"The two-gap flux-lattice field-distribution framework on which the modified London model of Eq. (3) is built."}],"review_version":1}