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REVIEW 4 major objections 5 minor 70 references

Multiband superconductivity in the topological Kramers nodal-line semimetals

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2506.03509 v1 pith:HAZXDGLH submitted 2025-06-04 cond-mat.supr-con cond-mat.str-el

Tian Shang , Jianzhou Zhao , Keqi Xia , Lun-Hui Hu , Yang Xu , Qingfeng Zhan , Dariusz Jakub Gawryluk , Toni Shiroka This is my paper
classification cond-mat.supr-concond-mat.str-el
keywords multibandsuperconductivitymultigapmuonspinrelaxationuppercriticalfieldKramersnodal-linesemimetalnoncentrosymmetricsuperconductorNbRuSiTa
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $H_{c2}(0)$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

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.

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 (4)
  1. [§III, Fig. 5 and Eq. (3)] 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.
  2. [§III, Fig. 5 and Table I] 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.
  3. [§III, Eq. (1) and Fig. 4] 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.
  4. [§III, Fig. 3 and Hc2(T)] 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.
minor comments (5)
  1. [Introduction] There is a duplicated word: 'time-reversal, and and parity variants' should be 'time-reversal, and parity variants'.
  2. [§III, after Eq. (1)] There is a typo in 'we used both a singe-band and a two-band model' (should be 'single-band').
  3. [Eq. (3)] 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.
  4. [Table I] 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.
  5. [Fig. 3] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the multiband claim rests on new field-dependent muon data, Hc2(T), and DFT band-structure calculations, not on a self-citation chain or parameters that are inputs by construction.

full rationale

The paper's central claim of multiband superconductivity is supported by new experimental data (field-dependent sigma_sc(H) at 0.3 K, rho(T,H), C(T,H)) and by density-functional-theory band-structure calculations, rather than by a definitional reduction or a load-bearing self-citation. The two-band modified-London model of Eq. (3) is fitted to the measured sigma_sc(H) data and compared with the single-band model of Eq. (2); the fitted parameters (lambda_0, xi_1, xi_2) are not preset to the bulk Hc2 values. The Hc2 values quoted from xi_2 via Phi_0/(2*pi*xi_2^2) are derived from the fitted coherence lengths and are used as internal consistency checks, not as independent predictions that define the model inputs. Similarly, the Hc2(T) two-band fit is performed on the measured upper-critical-field data. The authors' prior work Ref. [16] supplies the lambda_eff^-2(T) data, the TRS-breaking context, and the earlier failure to distinguish one-gap from two-gap behavior, but the multiband conclusion also relies on new measurements and on independent DFT results. No uniqueness theorem or ansatz is imported from the authors' own prior work to force the two-band choice; the two-band model for sigma_sc(H) is attributed to external literature (Refs. [63,66,67]). The main weaknesses are statistical rather than circular: the 'clearly superior' two-band fit is assessed visually with no information criterion or residual analysis, and the weight w is fixed to 0.7 rather than fitted or derived from the DFT weights. These are robustness and model-selection concerns, not cases where a prediction is equivalent to its input by construction. The paper also honestly notes that the one-gap and two-gap fits to lambda_eff^-2(T) are practically indistinguishable, which further shows that the multiband evidence is not being manufactured from that fit.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

No new particles, fields, symmetries, or conserved quantities are postulated. The two electronic bands are derived from DFT, not invented. The free parameters are the fitted or hand-selected quantities of the two-band and two-gap models used to support the multiband claim.

free parameters (5)
  • Two-band weight w = 0.7 (fixed/selected)
    Used in both the sigma_sc(H) fit and the lambda_eff^-2(T) two-gap fit; motivated by DFT DOS (two bands contribute ~70% and ~85%), but no uncertainty is given and the same value is imposed on both compounds.
  • Magnetic penetration depth lambda_0 (two-band sigma_sc fit) = 338(2) nm (NbRuSi), 216(2) nm (TaRuSi)
    Fitted to the field-dependent relaxation data in Fig. 5 using Eq. (3).
  • Coherence lengths xi_1, xi_2 (two-band sigma_sc fit) = NbRuSi: 19(1)/13(1) nm; TaRuSi: 18(1)/10.5(5) nm
    Fitted to the same sigma_sc(H) dataset; Hc2 values quoted in the text are then computed from xi_2.
  • Two-band coupling constants lambda_11, lambda_22, lambda_12 = NbRuSi: 0.22/0.22/0.043; TaRuSi: 0.29/0.29/0.06
    Fitted to Hc2(T) in Fig. 3 using the Gurevich two-band model; used to argue that intra-band coupling dominates.
  • Two-gap magnitudes Delta_0,1 and Delta_0,2 = NbRuSi: 0.44(2)/0.55(2) meV; TaRuSi: 0.58(3)/0.68(3) meV
    Obtained from the two-gap s-wave fit to lambda_eff^-2(T) with w fixed at 0.7; the paper itself notes this fit is nearly indistinguishable from the one-gap fit.
assumptions (5)
  • standard math Modified London model for a hexagonal flux-line lattice (Eq. 3) describes the measured second moment of the field distribution.
    Invoked in the analysis of sigma_sc(H) in Section III; taken from Refs. [66,67]. Assumes an ideal vortex lattice and isotropic penetration depth.
  • standard math Ginzburg-Landau, WHH, and two-band Gurevich models for Hc2(T) are valid for these polycrystalline superconductors.
    Used to fit the Tc(H) data in Fig. 3; the two-band model in particular assumes two coupled bands with intra- and inter-band coupling constants.
  • domain assumption The electronic structure is well described by PBE-DFT with spin-orbit coupling; the two dominant bands contribute ~70% (NbRuSi) and >85% (TaRuSi) of the DOS at EF.
    Used to motivate the fixed weight w=0.7 and to argue that at least two bands are active; not directly verified by angle-resolved photoemission in this work.
  • domain assumption Isotropic spherical Fermi surface and s-wave two-gap model are adequate for analyzing lambda_eff^-2(T).
    Stated explicitly in Section III: 'we ignore the influence of the Fermi-surface shape... assume an isotropic spherical Fermi surface'. The authors argue that an (s+ip) model would give similar results based on Ref. [16].
  • domain assumption The normal-state nuclear relaxation rate sigma_n is constant and can be subtracted in quadrature to obtain sigma_sc.
    Used in Section III to extract the superconducting contribution; assumes no field or temperature dependence of the nuclear contribution.

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Cite this review

Pith. "Pith review of Multiband superconductivity in the topological Kramers nodal-line semimetals." pith.science (2026). https://pith.science/paper/HAZXDGLH

@misc{pith2026250603509,
  author       = {Pith},
  title        = {Pith review of: Multiband superconductivity in the topological Kramers nodal-line semimetals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HAZXDGLH}},
  note         = {Machine review of arXiv:2506.03509}
}
abstract

Recent band-structure calculations predict that the ruthenium-based ternary silicides are three-dimensional Kramers nodal line semimetals. Among them, NbRuSi and TaRuSi show bulk superconductivity (SC) below $T_c \sim 3$ K and 4 K, as well as spontaneous magnetic fields. The latter indicates the breaking of time-reversal symmetry and, thus, unconventional SC in both compounds. Previous temperature-dependent muon-spin spectroscopy studies failed to distinguish whether such compounds exhibit single-gap or multi-gap SC. Here, we report on systematic measurements of the field-dependent muon-spin relaxation rates in the superconducting state and on temperature-dependent electrical resistivity and specific heat under applied magnetic fields. Both the upper critical field and the field-dependent superconducting relaxation are well described by a two-band model. By combining our experimental results with numerical band-structure calculations, we provide solid evidence for multiband SC in NbRuSi and TaRuSi, and thus offer further insight into the unconventional- and topological nature of their superconductivity.

Figures

Figures reproduced from arXiv: 2506.03509 by the authors.

Figure 1
Figure 1. FIG. 1. Crystal structure (unit cell) of orthorhombic TiFeSi-type [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. NbRuSi temperature-dependent electrical resistivity [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Superconducting transition temperature [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. TF- [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Field-dependent superconducting Gaussian relaxation rate [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Temperature-dependent inverse square of the effective [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Electronic band structures of NbRuSi calculated by considering (a) and by ignoring (b) the spin-orbit coupling. The bands that [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Representative Fermi surfaces for NbRuSi (a) and TaRuSi [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]

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