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REVIEW 2 major objections 5 minor 47 references

$b$-axis and $c$-axis Knight shift measurements in the superconducting state on ultraclean UTe$_2$ with $T_c$ = 2.1 K

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper establishes that the spin susceptibility of UTe2 drops along all three crystallographic axes in the superconducting state, not just the a axis, by cancelling the superconducting diamagnetic background with a difference of…

desk verdict Good b-axis evidence for a spin-susceptibility drop in clean UTe2; the c-axis leg needs error bars before the all-three-axes d-vector conclusion is secure. read the letter →

arxiv 2505.19615 v1 pith:TB665WII submitted 2025-05-26 cond-mat.supr-con cond-mat.str-el

Hiroki Matsumura , Yuki Takahashi , Katsuki Kinjo , Shunsaku Kitagawa , Kenji Ishida , Yo Tokunaga , Hironori Sakai , Shinsaku Kambe
show 7 more authors
Ai Nakamura Yusei Shimizu Yoshiya Homma Dexin Li Fuminori Honda Atsushi Miyake Dai Aoki
This is my paper · ORCID
classification cond-mat.supr-concond-mat.str-el
keywords UTe2spin-tripletsuperconductivityKnightshift125TeNMRd-vectorspinsusceptibilityheavy-fermionsuperconductor
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 aims to settle whether the tiny drops in the b- and c-axis Knight shifts of the spin-triplet superconductor UTe2 below its 2.1 K transition come from a real loss of spin susceptibility or merely from the superconducting diamagnetic shielding. The authors measure 125Te NMR at two crystallographically distinct Te sites and subtract one site's shift from the other, which cancels any site-independent diamagnetic background. They find that the difference Kb,II − Kb,I and Kc,I − Kc,II clearly drops at Tc, proving the spin susceptibility falls along both axes. Combined with the previously reported large a-axis drop, this establishes that the d-vector has finite components along all three crystal axes, an important constraint on the pairing state.

What carries the argument

The central object is the difference of 125Te Knight shifts measured at the two crystallographically distinct Te sites, Te(I) and Te(II), for each field direction. Because the orbital shift is temperature-independent and the superconducting diamagnetic shift is assumed to be site-independent, the subtraction Kα,II − Kα,I cancels the diamagnetic term and leaves only the spin contribution proportional to (Aα,II − Aα,I)χspin(T). A supplementary Γ-ratio analysis, using the linear relation between the two site shifts below 20 K, estimates the diamagnetic shift K_dia for the b axis and confirms that the spin part is consistent with the direct site-difference result.

What would settle it

Repeat the b- and c-axis Knight-shift measurements at a second applied field well above Hc1 (for example 1.5 T) and check whether Kb,II − Kb,I and Kc,I − Kc,II show the same drop below Tc; a field-dependent drop would mean the diamagnetic shielding affects the two Te sites differently, and the spin-susceptibility conclusion would collapse.

Watch

Extended reading notes

Core claim

This work reports 125Te Knight-shift measurements along the b and c axes of an ultraclean UTe2 single crystal with Tc = 2.1 K, down to 70 mK. In the superconducting state, the raw b- and c-axis Knight shifts decrease by only about 3% of their normal-state values, comparable to the estimated diamagnetic shielding, so the paper cannot rely on those raw changes alone. The decisive step is to take the difference of the Knight shifts measured at the two crystallographically distinct Te sites; because the diamagnetic shift is assumed identical at both sites, it cancels in the difference, leaving only the spin part. The difference drops below Tc for both field directions, showing that the spin susceptibility along the b and c axes genuinely decreases. Together with the large a-axis reduction reported earlier, the paper concludes that the d-vector has components along all three crystal axes, consistent with an Au odd-parity pairing state or an f-wave state, and that the superconducting gap is likely anisotropic full-gap or point-node, not line-node.

Load-bearing premise

The load-bearing assumption is that the superconducting diamagnetic shift is exactly the same at the two Te sites, so that subtracting one Knight shift from the other removes it entirely; if the mixed-state field distribution at 0.8 T shifts the two sites differently, the observed difference drop could be diamagnetic rather than spin.

Editorial extensions

If this is right

  • The spin susceptibility of UTe2 decreases along the a, b, and c axes in the superconducting state, so the d-vector has nonzero components along all three axes.
  • The single-component B3u scenario, which has no a-axis spin reduction, is ruled out for the ultraclean 2.1 K sample.
  • The gap is not line-nodal; the low-temperature spin susceptibility is compatible with a full gap or point nodes only if 2Δ(0)/kBTc exceeds the BCS value of 3.5.
  • The absence of Pauli depairing near the estimated Pauli limit, despite the large a-axis spin drop, supports spin-triplet pairing with field-aligned spins.
  • The difference-of-sites method provides a way to extract spin susceptibility changes even when the raw diamagnetic shift is of the same size as the spin effect.

Reading between the lines

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

  • If the site-independence of the diamagnetic shift is generic, the same two-site subtraction could be applied to other multi-site superconductors with small Knight-shift changes, converting the diamagnetic background into a built-in null reference.
  • The c-axis analysis is the least certain because the two Te peaks overlap and Γc is close to 1; a measurement at higher field or on a differently oriented sample that resolves the two peaks could tighten the c-axis spin-drop claim.
  • The nearly temperature-independent spin susceptibility below 1 K leaves open a finite residual density of states; low-temperature 1/T1 measurements would test whether the gap is truly full.
  • If the d-vector truly has all three components, rotating the field in the bc and ab planes should reveal field-induced d-vector reorientation, connecting these results to the field-reinforced superconducting phases of UTe2.
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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

2 major / 5 minor

Summary. This paper reports 125Te NMR Knight-shift measurements on an ultraclean UTe2 single crystal with Tc = 2.1 K, for H applied along the b and c axes, resolved at the two crystallographically distinct Te sites. The authors observe small decreases of Kb and Kc below Tc that are comparable in size to the expected superconducting diamagnetic shift. To remove the diamagnetic contribution, they use the difference between the two Te-site Knight shifts, arguing that the diamagnetic shift is site-independent. They report a decrease of this difference for both H || b and H || c, which, combined with the previously reported large a-axis Knight-shift decrease, leads them to conclude that the spin susceptibility decreases along all three axes and that the d-vector has components along all three crystal axes. They also analyze the b-axis spin shift using full-gap, point-node, and line-node gap models, finding that the line-node model can be ruled out while full-gap and point-node scenarios cannot be distinguished.

Significance. If the c-axis result is quantitatively secure, this is an important experimental constraint on the UTe2 superconducting order parameter: it would imply a spin-triplet d-vector with components along all three axes, favoring an Au-type state or f-wave pairing and distinguishing the ultraclean 2.1 K sample from earlier lower-Tc samples. The b-axis part of the paper is convincing: the two Te resonances are well separated, the site-difference method cancels the common-mode diamagnetic shift, and the authors are appropriately cautious about auxiliary model fits. The paper also has the strength of using a disorder-free sample and of transparently acknowledging the limitations of the c-axis fit and of the Kdia_c determination. However, the central all-axes conclusion depends on a c-axis difference whose statistical and systematic uncertainties are not quantified, and the wording 'unambiguously decreases' is stronger than the presented evidence.

major comments (2)
  1. [Section III, Fig. 5] The conclusion that the spin susceptibility decreases along the c axis rests entirely on the temperature dependence of Kc,I − Kc,II. In the same section the authors state that the two peaks are 'partially overlapped,' that the separation between the two peaks becomes 'unclear' in the superconducting state, and that 'the decrease in Kc,I may have been overestimated due to this fitting uncertainty.' No error bars are reported for the fitted peak positions or for the differences plotted in Fig. 5. Because a downward bias in the fitted Kc,I directly produces a spurious decrease in Kc,I − Kc,II, the observed drop cannot currently be distinguished from an artifact of the two-peak Lorentzian fit. To support the central d-vector claim, the authors should provide error propagation from the fits, show representative fitted spectra with residuals at several temperatures, and test the robustness of the decrease under alternative fitting choices (for example, fixed linewidths, different line shapes, or constraints from the normal-state peak separation). If such an analysis is not possible, the c-axis conclusion should be moderated.
  2. [Section III, Eq. (1) and Fig. 6] The extraction of ΔK_spin_b,II depends on Γb, on K_dia_b(0), and on the assumed temperature form K_dia_b(T) = K_dia_b(0)[1 − (T/Tc)^2]. The manuscript reports Γb = 1.32 ± 0.01 but does not propagate the uncertainty of the linear fit or of K_dia_b(0) into ΔK_spin_b,II or into the gap-model comparison. The stated agreement between ΔK_spin_b,II and the b-axis site difference is not an independent check, since both quantities are derived from the same raw Knight-shift data. In addition, the 1 − (T/Tc)^2 form is introduced without justification for a type-II superconductor in the mixed state at 0.8 T. This analysis does not affect the qualitative difference-based conclusion, but it does affect the quantitative claims about the magnitude and temperature dependence of the spin susceptibility and the subsequent gap-structure discussion; those claims need uncertainty estimates and a justification of the temperature form.
minor comments (5)
  1. [Section III, text near Fig. 5] The sentence 'Figure 5 shows Kb,II − Kb,I for H ∥ b and Kc,I − Kb,II for H ∥ c' contains a typo: the second quantity should be Kc,I − Kc,II.
  2. [Fig. 1 caption and Section III] The word 'Lorentian' appears instead of 'Lorentzian' in the description of the two-peak fit.
  3. [Section III, near the end] The phrase 'the large reduction in the quantized a axis' is unclear; it should likely read 'the large reduction in the a-axis Knight shift' or similar.
  4. [Fig. 3 caption] The caption says 'in (a) H ∥ b and H ∥ c below 4 K'; it should specify '(a) H ∥ b and (b) H ∥ c' to match the two panels.
  5. [Abstract and Conclusion] Given the acknowledged c-axis fitting uncertainty, the word 'unambiguously' in the abstract and conclusion is too strong unless quantitative error analysis is added; 'is consistent with' would be more precise at the present level of evidence.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central spin-susceptibility decrease is a measured interval, and the diamagnetic-subtraction assumption is a systematic-risk assumption, not a derivation from the conclusion.

full rationale

The central derivation chain is an experimental subtraction, not a theory-derived prediction. In Sec. III the decrease of Kb,II − Kb,I (and Kc,I − Kc,II) below Tc is read directly from measured spectra (Fig. 5); the only input is the assumption that the diamagnetic shift K_dia is site-independent, so that it cancels in the difference. That assumption is an empirical/systematic hypothesis and a potential error source, not a definition of the conclusion; it cannot make the result circular. The separate K_dia_b estimate via eq. (1) uses the normal-state Γ_b = Kb,II/Kb,I and is then subtracted to obtain ΔK_spin_b,II; the agreement between this quantity and Kb,II − Kb,I is algebraic (both are proportional to the same measured interval under the assumed linear relation), and the paper uses it as a self-consistency check, not as an independent prediction. The c-axis conclusion rests on the measured interval Kc,I − Kc,II; the authors explicitly flag possible overestimation of Kc,I from the two-peak fit, which is a disclosed systematic uncertainty, not a circular step. The prior same-group result for Ka [23] is load-bearing for the three-axis d-vector statement, but it is an independent published experimental measurement, not an unverified premise imported to force the conclusion. No fitted parameter is renamed as a prediction and no equation is defined in terms of the result it is supposed to establish.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard NMR analysis assumptions: site-independent diamagnetic shift, constant orbital shift, and temperature-independent hyperfine couplings. The main fitted parameters are the normal-state ratio Γ_b and the diamagnetic shift amplitude, neither of which defines the existence of the spin decrease. No new entities are introduced.

free parameters (3)
  • Gamma_b (ratio of Kb,II to Kb,I) = 1.32 ± 0.01
    Fitted from the linear Kb,II vs Kb,I plot below 20 K, used in Eq. (1) to separate the diamagnetic shift from the spin shift.
  • K_dia_b(0) (zero-temperature diamagnetic shift along b) = -0.10%
    Fitted to the evaluated K_dia_b(T) using the form [1-(T/Tc)^2]; used to extract the spin part of the Knight shift.
  • 2Δ(0)/kBTc (superconducting gap ratio) = Resulting values shown for each gap model
    Free parameter when fitting the calculated spin susceptibility temperature dependence to the data for full-gap, point-node, and line-node models.
assumptions (5)
  • domain assumption The superconducting diamagnetic shift K_dia is identical for the two Te sites.
    Required for the subtraction Kα,II - Kα,I to cancel the diamagnetic contribution; stated in Sec. III.
  • domain assumption The hyperfine coupling constants A_α,i and their ratio Γ are temperature-independent, and the linear K_II vs K_I relation holds in the SC state.
    Used to derive Eq. (1) and extrapolate the normal-state relation below Tc.
  • standard math The orbital Knight shift K_orb is constant at low temperatures.
    Standard assumption in NMR analysis; the orbital shift does not change in the SC state at these temperatures.
  • domain assumption A two-peak Lorentzian fit reliably determines the peak positions of the overlapping c-axis NMR spectra in the SC state.
    The authors note the spectra are partially overlapped and the separation becomes unclear; the fit is used to extract Kc,I and Kc,II.
  • domain assumption The diamagnetic shift follows K_dia(T) = K_dia(0)[1-(T/Tc)^2].
    Phenomenological form used to fit the evaluated K_dia_b(T); standard for the temperature dependence of the screening current density.

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Cite this review

Pith. "Pith review of $b$-axis and $c$-axis Knight shift measurements in the superconducting state on ultraclean UTe$_2$ with $T_c$ = 2.1 K." pith.science (2026). https://pith.science/paper/TB665WII

@misc{pith2026250519615,
  author       = {Pith},
  title        = {Pith review of: $b$-axis and $c$-axis Knight shift measurements in the superconducting state on ultraclean UTe$_2$ with $T_c$ = 2.1 K},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TB665WII}},
  note         = {Machine review of arXiv:2505.19615}
}
abstract

Knight shifts along the $b$ and $c$ axes ($K_b$ and $K_c$) at two crystallographically distinct Te sites were measured down to 70 mK using $^{125}$Te nuclear magnetic resonance (NMR) on an ultraclean UTe$_2$ single crystal with a superconducting (SC) transition temperature $T_{\mathrm{c}}$ = 2.1 K. This was carried out to determine the $\boldsymbol{d}$-vector components, which are the order parameter in the spin-triplet pairing. Although the decrease in $K_b$ and $K_c$ is comparable to the theoretical estimation of the SC diamagnetic shielding effect, it is confirmed, by taking the difference between two Knight shifts at the distinct Te sites, that the spin susceptibility along the $b$ and $c$ axes decreases in the SC state. Taking into account the large decrease in $K_a$ in the SC state, we conclude that the $\boldsymbol{d}$ vector has components along all three crystal axes.

Figures

Figures reproduced from arXiv: 2505.19615 by the authors.

Figure 1
Figure 1. FIG. 1. (Color online) (a) Crystal structure of UTe [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (Color online) (a) Te(I) and (b) Te(II)-NMR peaks [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. (Color online) (a) Temperature dependence of [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: FIG. 5. (Color online) Temperature dependence of [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]
Figure 6
Figure 6. Figure 6: (c). Although the line-node model can be ruled out, the current data cannot distinguish between the full-gap or point-node scenarios if an SC gap larger than the standard BCS value [2∆(0)/kBTc = 3.5] is assumed. This result does not contradict with both the full-gap be…
Figure 7
Figure 7. Figure 7: To compare with the experimental data, we fit [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

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