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Magnetic Field Dependence of the Spin Susceptibility on Conventional s-wave Superconductor LaRu$_4$P$_{12}$ Revealed by $^{31}$P-NMR and $^{139}$La-NMR

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read In the conventional s-wave superconductor LaRu4P12, the superconducting-state spin susceptibility is proportional to the applied magnetic field and connects smoothly to the normal-state value at the upper critical field.

desk verdict Direct NMR confirmation of H-linear spin susceptibility in an orbital-limited s-wave superconductor; clean two-site subtraction, with minor caveats about the K_dia assumption and fit range. read the letter →

arxiv 2505.20702 v1 pith:PVFL5YJD submitted 2025-05-27 cond-mat.supr-con

Riku Matsubayashi , Shiki Ogata , Taishi Ihara , Hiroyasu Matsudaira , Shunsaku Kitagawa , Kenji Ishida , Yusuke Nakai , Hitoshi Sugawara
show 1 more author
Hideyuki Sato
This is my paper · ORCID
classification cond-mat.supr-con
keywords spinsusceptibilityKnightshiftNMRs-wavesuperconductororbitalpair-breakingLaRu4P12uppercriticalfieldvortexstate
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

This paper reports direct NMR measurements of how the electron-spin part of the magnetic susceptibility of a conventional s-wave superconductor behaves as a magnetic field approaches the upper critical field. Using phosphorus-31 and lanthanum-139 Knight-shift measurements on LaRu4P12, the authors subtract the superconducting diamagnetic contribution by comparing the two lattice sites and find that the spin susceptibility in the superconducting state grows linearly with field above about 1 T, reaching the normal-state value smoothly at the upper critical field. The result matters because field-dependent spin-susceptibility data on conventional superconductors are rare, and this is a clean demonstration of the orbital pair-breaking scenario.

What carries the argument

The load-bearing object is the two-site Knight-shift difference. Because the diamagnetic shift $K_{\rm dia}$ is assumed to be identical at the $^{31}$P and $^{139}$La sites, the difference $\Delta K^{\rm La} - \Delta K^{\rm P}$ cancels it, leaving $(A_{\rm hf}^{\rm La} - A_{\rm hf}^{\rm P})\Delta\chi_{\rm spin}$. The hyperfine ratio $A_{\rm hf}^{\rm P}/A_{\rm hf}^{\rm La} = 0.60$ is read off the linear $K^{\rm P}$-versus-$K^{\rm La}$ plot from 10 K to 220 K, converting the two-site difference into the spin shift at each site. This subtraction is what lets the authors extract a field-linear spin signal from spectra that otherwise contain vortex-lattice broadening (the Redfield pattern, an asymmetric line shape from the vortex lattice) and a large diamagnetic shift at low fields.

What would settle it

A direct calculation of the local magnetic field at the phosphorus and lanthanum sites in the vortex lattice, using the reported $\kappa = 24.7$ and $H_{c2} = 3.2$ T, would show whether the diamagnetic Knight shift is site independent; if it is not, the two-site subtraction does not cancel it and the reported linear spin susceptibility would be an artifact. Comparing the extraction with a third nuclear site would test this directly.

Watch

Extended reading notes

Core claim

The central claim is that the spin susceptibility in the superconducting state of LaRu$_4$P$_{12}$ obeys $\chi_{\rm spin} = \alpha(H - H_{c2}) + \chi_{\rm normal}$, with a linear recovery in field that joins the normal-state susceptibility at $H_{c2}$. The evidence comes from the $^{31}$P and $^{139}$La Knight shifts at 1.4 K: measured against the 10 K normal-state values, the two-site difference removes the common diamagnetic shift, and the extracted $\Delta K_{\rm spin}$ increases linearly with $H$ up to $\mu_0 H_{c2} = 3.2$ T. The authors interpret this linear field dependence as the expected consequence of the quasiparticle density of states being proportional to $H/H_{c2}$ in the mixed state, realized when the upper critical field is governed by orbital pair-breaking rather than Pauli limiting.

Load-bearing premise

The analysis assumes that the magnetic-field shift caused by superconducting screening currents is exactly the same at the 31P and 139La sites, so subtracting the two-site difference removes it completely; if the vortex lattice shifts one site more than the other, the extracted spin susceptibility is systematically wrong.

Editorial extensions

If this is right

  • In LaRu$_4$P$_{12}$, the superconducting-state spin susceptibility follows $\chi_{\rm spin} = \alpha(H - H_{c2}) + \chi_{\rm normal}$, so the spin response recovers linearly with field and matches the normal-state value at $H_{c2}$.
  • The smooth recovery means no Pauli-paramagnetic anomaly or Fulde-Ferrell-Larkin-Ovchinnikov-type spin response appears near the upper critical field, consistent with orbital pair-breaking dominance.
  • The two-site subtraction recipe can be applied to other multi-site conventional superconductors to isolate the spin susceptibility from the superconducting diamagnetic background.
  • The fitted Ginzburg-Landau parameter $\kappa \approx 24.7$ and estimated lower critical field $\mu_0 H_{c1} \approx 10$ mT place LaRu$_4$P$_{12}$ in the strong type-II regime.
  • The result is consistent with the earlier $1/T_1T \propto H^2$ observation, reinforcing the orbital-pair-breaking picture from a different observable.

Reading between the lines

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

  • If the two-site cancellation of the diamagnetic shift is exact, the same linear $\chi_{\rm spin}(H)$ should be seen in other orbital-limited s-wave superconductors that have two NMR-accessible sites; the two-site subtraction could become a quick diagnostic of orbital pair-breaking.
  • The data deviate from linearity below 1 T where the diamagnetic term dominates, so the true low-field behavior remains unresolved; measurements closer to the lower critical field with oriented crystals could reveal a vortex-lattice correction hidden by the current error bars.
  • The assumed proportionality between $\chi_{\rm spin}$ and the quasiparticle density of states could be tested by comparing the NMR-derived slope $\alpha$ with the specific-heat-derived $D(E_F)$ on the same crystals; a mismatch would indicate corrections beyond the simple single-particle picture.
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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

3 major / 3 minor

Summary. The manuscript reports 31P and 139La NMR Knight-shift measurements on the conventional s-wave superconductor LaRu4P12 in the mixed state at 1.4 K. By taking the difference of the Knight shifts at the two nuclear sites, the authors argue that the superconducting diamagnetic shift Kdia cancels, allowing them to extract the spin part of the shift ΔKspin as a function of magnetic field. The extracted ΔKspin is reported to increase linearly with H and to connect smoothly to the normal-state value at Hc2, leading to the conclusion that the superconducting-state spin susceptibility follows χspin = α(H - Hc2) + χnormal, as expected when the upper critical field is governed by orbital pair breaking. The authors also estimate Kdia and fit it with a Ginzburg-Landau expression to obtain κ ≈ 24.7. The central claim is that this is a textbook demonstration of orbital-limited superconductivity with a linear field recovery of the spin susceptibility.

Significance. If the central claim holds, the paper provides a direct, two-site NMR measurement of the field-dependent spin susceptibility in a conventional s-wave superconductor, complementing earlier 1/T1T data on the same material and giving a clean experimental confirmation of the orbital pair-breaking scenario. The two-site subtraction technique is a sound and potentially reusable approach, and the hyperfine-coupling ratio used in the analysis is calibrated from independent 10–220 K data rather than fitted to the superconducting-state data. Error bars are provided for the reported Knight shifts. The main weaknesses are the reliance on an unquantified assumption that Kdia is identical at the 31P and 139La sites, the exclusion of the two lowest-field data points from the linear fit, and the absence of a finite-temperature discussion for the comparison with the T = 0 theory. These issues are addressable but require additional analysis and clarification.

major comments (3)
  1. [Eq. (4)] The central subtraction in Eq. (4) cancels Kdia only if the diamagnetic Knight shift is strictly identical at the 31P and 139La sites. The manuscript justifies this by the statement 'Since Kdia is a bulk effect, it is reasonable to assume that Kdia works at the 31P and 139La sites in the same manner' (p. 3). This assumption is load-bearing: a site-dependent residual Kdia_La - Kdia_P that varies with H would directly bias the extracted Δχspin and could produce or distort the claimed linear H dependence. No quantitative argument or experimental check is provided. In particular, the Kdia values in Fig. 6 are obtained from Eq. (2) using the same two-site subtraction, so the fit yielding κ = 24.7 is not an independent validation of the cancellation. Please add a quantitative justification for site-independent Kdia (for example, that the vortex-lattice field varies on scales λ and ξ which are orders of magnitude larger than the unit cell, so the macroscopic diamagnetic field is the same at both sites) or estimate the maximum possible field-dependent residual from the data and propagate it into the systematic uncertainty of ΔKspin.
  2. [Fig. 5] The linear fit in Fig. 5 excludes the µ0H = 0.5 T and 1.1 T points, which the text says 'deviate from this behavior with large error bar.' Because Kdia is largest at low fields (Fig. 6), this is exactly the field region where a failure of the Kdia-cancellation assumption would be most visible; excluding these points prevents the data from falsifying the cancellation. Please report the fit details (number of points, slope, intercept, reduced χ²), perform the fit with all points weighted by their errors, and show the residuals. If the low-field points are excluded, provide a quantitative justification beyond the large error bars, and state explicitly that the claimed proportionality is restricted to the fitted field range rather than the full 0 < H < Hc2 interval.
  3. [Conclusions] The linear relation χspin = α(H - Hc2) + χnormal is a zero-temperature expectation: the quasiparticle density of states in the mixed state of a type-II superconductor is proportional to H/Hc2 at T = 0. The measurements are performed at 1.4 K, and for fields close to Hc2 the superconducting transition temperature at that field is only slightly above 1.4 K (e.g., from Fig. 1(c), Tc(3 T) is only about 1.8 K), so thermal quasiparticle contributions are not negligible. Please either quantify the finite-temperature correction to the linear form in the field range used for the fit, or soften the conclusion to state that the data are consistent with the T = 0 linear form within the present experimental accuracy.
minor comments (3)
  1. [p. 2] In the sentence 'Magnetic field calibration was carried out using 63Cu and 65Cu signal arising from the NMR coil', the word 'signal' should be plural ('signals').
  2. [Fig. 5] The vertical axis label in Fig. 5 is '-ΔKspin (%)' while the text and caption refer to 'ΔKspin'. Please define the sign convention explicitly, because the description 'the value increases linearly with increasing H' is confusing if the plotted quantity is negative.
  3. [p. 3] The sentence 'If one extrapolates ΔKspin to zero magnetic field, the resulting value is nearly equal to K (10 K)' is not clear: K(10 K) is the absolute Knight shift at one site, while ΔKspin is a difference. Please restate this as an extrapolation of Kspin(1.4 K, H → 0) to zero or to a value consistent with the spin susceptibility vanishing at H = 0.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the linear field recovery of the spin susceptibility is an empirical result inferred from independently calibrated Knight-shift differences, not a consequence of the fitting or self-citation chain.

full rationale

The paper's central claim, that the superconducting-state spin susceptibility recovers linearly with magnetic field and connects smoothly to the normal-state value, is obtained from measured 31P and 139La NMR Knight shifts. The crucial two-site subtraction in Eq. (4) assumes, rather than derives, that the diamagnetic Knight shift Kdia is identical at both sites; this is an unverified assumption and a possible systematic error, but it is not circular because it does not encode the target linear H dependence. The hyperfine ratio A_P/A_La = 0.60 used in Eq. (6) is calibrated independently from the normal-state K_P versus K_La relation between 10 K and 220 K (Fig. 4), outside the superconducting state and outside the field range of the claimed effect. The linear dependence in Fig. 5 is read from the data and compared with the theoretical expectation D(E_F) ∝ H/H_c2 and with the prior 1/T1T result; it is not manufactured by the fitting parameters. The Ginzburg-Landau parameter κ = 24.7 obtained from fitting Eq. (7) to Kdia is a consistency check and is not fed back into the extraction of ΔKspin. Self-citations to refs. 34-36 and 39 supply prior spectra, Hc2 data, and a related compound comparison, but the central conclusion does not reduce to these citations. No derivation step is equivalent by construction to its own input, so there is no circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests mainly on two empirical calibrations, the hyperfine ratio and the identified diamagnetic contribution, plus standard assumptions of NMR analysis in the vortex state. No new physical entities are introduced.

free parameters (2)
  • Hyperfine coupling ratio A_P_hf / A_La_hf = 0.60
    Obtained from the slope of the linear fit of KP vs KLa between 10 K and 220 K (Fig. 4). This ratio is used in Eq. (6) to convert the two-site shift difference into the spin shift at each site, so the central result depends directly on it.
  • Ginzburg-Landau parameter κ = 24.7
    Fit of the estimated Kdia with the theoretical Brandt expression (Eq. 7). Used to estimate Hc1 and to check consistency with LaFe4P12, but not used in extracting the linear spin susceptibility.
assumptions (4)
  • domain assumption Normal-state Knight shift is independent of magnetic field.
    Explicitly assumed in the text: 'Here we assumed that normal-state Knight shift is independent of H, which is usual in conventional metals.' The difference K(10 K) - K(1.4 K) relies on this.
  • domain assumption The diamagnetic Knight shift Kdia is identical at the 31P and 139La sites.
    Explicitly assumed: 'Since Kdia is a bulk effect, it is reasonable to assume that Kdia works at the 31P and 139La sites in the same manner.' This is load-bearing for the two-site subtraction.
  • domain assumption The hyperfine coupling ratio measured between 10 K and 220 K remains valid at 1.4 K.
    Implicit. The ratio A_P/A_La = 0.60 is extracted from a high-temperature fit and applied to the low-temperature superconducting-state data without a temperature-dependence check.
  • domain assumption The peak position of the broadened vortex-state NMR spectrum is a valid measure of the average Knight shift needed for the analysis.
    Implicit in using the peak Knight shift in the SC state despite the Redfield-pattern asymmetry. The authors rely on the standard interpretation of the vortex-state spectrum.

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Cite this review

Pith. "Pith review of Magnetic Field Dependence of the Spin Susceptibility on Conventional s-wave Superconductor LaRu$_4$P$_{12}$ Revealed by $^{31}$P-NMR and $^{139}$La-NMR." pith.science (2026). https://pith.science/paper/PVFL5YJD

@misc{pith2026250520702,
  author       = {Pith},
  title        = {Pith review of: Magnetic Field Dependence of the Spin Susceptibility on Conventional s-wave Superconductor LaRu$_4$P$_12$ Revealed by $^31$P-NMR and $^139$La-NMR},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PVFL5YJD}},
  note         = {Machine review of arXiv:2505.20702}
}
abstract

The magnetic field dependence of the spin part of Knight shift, which is proportional to the superconducting-state spin susceptibility, was investigated at two nuclear sites, $^{31}$P and $^{139}$La in a conventional s-wave superconductor LaRu$_4$P$_{12}$. After the analyses, we confirmed that the superconducting-state spin susceptibility is proportional to magnetic field, and connects to the normal-state spin susceptibility smoothly. This is a textbook example, when the superconductivity is broken with the orbital pair-breaking effect.

Figures

Figures reproduced from arXiv: 2505.20702 by the authors.

Figure 1
Figure 1. (Color online) (a) Crystal structure of LaRu4P12 drawn by VESTA.40) (b) Magnetic field dependence of χAC at 1.46 K. The arrow indicates µ0Hc2 = 3.2 T. (c) H–T phase diagram of LaRu4P12. The blue points are from the previous study36) and the orange point is from the present measurement. tally, as carried out previously.38, 39) We revealed that the SC spin susceptibility above 1 T showed a H-linear de￾pendence and smo… view at source ↗
Figure 2
Figure 2. (Color online) Knight shift spectra at µ0H = 1.1 T, 2.0 T and 3.5 T. Measurements were performed at three temperatures ranging from 10 K to 1.4 K. The dashed lines represent the peak Knight shift at 10 K. The background color of spectra in the SC state is gray. At lower temperatures, the peak Knight shift of the spectrum decreases, and at the lowest temperature, the spectrum becomes broadened. upper critical field µ… view at source ↗
Figure 3
Figure 3. (Color online) H dependence of K at 1.4 K and 10 K for 31P (a) and 139La site (b). (Inset) T dependence of K at 1.1 T and 3.5 T. (c) H dependence of ∆K(= KLa − KP) at 10 K and 1.4 K. ∆χspin = χspin(10 K) − χspin(1.4 K). (5) Thus, we can get ∆Ki spin (i = 31P and 139La) as ∆Ki spin = A i hf∆χspin = Ai hf ALa hf − AP hf (∆KLa − ∆KP ). (6) Here, Ai hf is the hyperfine coupling constant at the 3 [PITH_FULL_IMAGE:figure… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: (Color online) The plot of KP against KLa above 10 K. (Inset) T dependence of K at 31P site and 139La site. 0 1 2 3 4 −0.2 −0.1 0.0 0.1 -Δ Kspin (%) μ0H (T) SC LaRu4P12 ↓ μ0Hc2 139La-NMR 31P-NMR [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: (Color online) H dependence of ∆Kspin at31P site and 139La site. The background color in the in the SC state is gray. The dashed line below µ0Hc2 represents the fitting function (fitted with a linear function). The arrow indicates µ0Hc2. i site and χspin is the spin su…

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