Atypical vortex lattice and the magnetic penetration depth in superconducting Sr$_2$RuO$_4$ deduced by $\mu$SR

J. E. Sonier; M. Yakovlev; Z. Kartsonas

REVIEW 4 major objections 4 minor 40 references

Atypical vortex lattice and the magnetic penetration depth in superconducting Sr$_2$RuO$_4$ deduced by $\mu$SR

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Reanalysis of muon data says Sr2RuO4's penetration depth follows T2, not T-linear.

desk verdict 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. read the letter →

arxiv 2501.14876 v1 pith:UHHN2EQH submitted 2025-01-24 cond-mat.supr-con

M. Yakovlev , Z. Kartsonas , J. E. Sonier This is my paper · ORCID

classification cond-mat.supr-con

keywords muonspinrotationSr2RuO4magneticpenetrationdepthvortexlatticeGinzburg-Landautheoryunconventionalsuperconductivityorderparametertransverse-fieldmuSR

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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

The 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.

The 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.

This 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.

Referee Report

4 major / 4 minor

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.

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 (4)

[Section II.A, Eq. (1), Fig. 2] 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 II.A, Fig. 2 inset] 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 III, Table I] 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 II.D, Fig. 3(c)] 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.

minor comments (4)

[Section II.D, Fig. 3(c) caption] 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 II.A] 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.
[Fig. 3 caption] 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.
[Eq. (3)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: external benchmarks anchor the T^2 comparison; the field-minimum inference is model-dependent, not construction-equivalent.

full rationale

The paper's derivation chain is self-contained rather than circular. The central temperature-dependence comparison (Sec. II.A, Figs. 1-2) re-derives lambda_ab from the same published TF-muSR spectra using Eq. (1) and compares it against the external T^2 dependence of Delta_lambda_ab(T) measured in the Meissner state by other techniques [13-16]; this is an external benchmark, not a fitted input. The observation that Eq. (1) is applied outside its stated validity regime (kappa >= 5) is a correctness or applicability concern, not a circularity. The field-minimum claim (Sec. II.D, Fig. 3(c)) is obtained by fitting the two-component Eu model of Agterberg to the measured TF-muSR spectrum; the model is adopted by citation, but the fit is to the data and the authors explicitly acknowledge that the Eu order parameter is itself now considered unlikely, so the resulting inference is a model-dependent physical interpretation rather than a definitional equivalence or a fitted parameter renamed as a prediction. The 'no valid model' conclusion is a negative assessment supported by the demonstrated failures of several models, not a derivation from the target result. Self-citations such as Refs. [20], [25], [26], [29], and [31] are original data and analysis-methodology references; they are not used as unverified load-bearing justifications. No step reduces by construction to its own input.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central T-dependence comparison rests on Eq. 1 applied outside its stated validity regime and on fitted Gaussian components; the vortex-lattice constraint rests on a fitted two-component Eu model whose underlying order parameter the authors themselves doubt. No new physical entities are introduced.

free parameters (4)

Gaussian component amplitudes, depolarization rates, and fields (a_i, sigma_i, B_i, i=1..3) in Eq. 2 fits = not tabulated
The second-moment lambda_ab(T,B) values and their T-dependence are calculated from these fitted components; no values are listed.
GL parameter kappa in the two-component Eu model = 2.65(7) (Table I)
Fitted to the same TF-muSR spectrum and used to generate the B(r) contour whose field-minimum location is the paper's new order-parameter constraint.
Fermi surface anisotropy nu in the two-component Eu model = 0.31(1) (Table I)
Common temperature-independent fit parameter in the Eu model; affects the vortex-lattice field profile shape.
Vortex-lattice disorder depolarization sigma_dis = not tabulated
Assumed proportional to 1/lambda_ab^2(T) in Eq. 3; the proportionality is not specified and would need fitting.

assumptions (5)

domain assumption The sample TF-muSR asymmetry can be represented as a sum of Gaussian-damped cosine functions (Eq. 2), with n=3 for the early Sr2RuO4 data.
Used to extract the second moment; the extra component is justified by a visual Fourier shoulder, not by an independent microscopic model.
ad hoc to paper Eq. (1) computes lambda_ab from the second moment even though the paper states Eq. (1) is only strictly valid for kappa >= 5 and a restricted b range.
The whole T-dependence comparison of Section II.A relies on Eq. (1); its validity for Sr2RuO4 (kappa about 2.6, b about 0.25) is simultaneously denied for absolute values but assumed for T-dependence.
domain assumption The low-frequency shoulder in the early TF-muSR spectrum comes from the vortex-lattice field distribution, not from muons stopping elsewhere or from a spurious phase.
The vortex-lattice model fits and the field-minimum constraint assume this attribution; no control experiment is presented.
domain assumption Nuclear dipole broadening sigma_n is temperature independent and is fixed from spectra above Tc (Section II.A, Eq. 3).
Standard muSR assumption, unverified for this specific sample but needed for the Eq. (3) fits.
standard math GL theory, including the iterative GL method and the two-component Eu GL model of Ref. 32, applies to the low-kappa vortex state of Sr2RuO4 at b about 0.25.
The field profiles B(r) and the resulting constraints are computed with these published GL frameworks, which are imported rather than derived here.

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Cite this review

Pith. "Pith review of Atypical vortex lattice and the magnetic penetration depth in superconducting Sr$_2$RuO$_4$ deduced by $\mu$SR." pith.science (2026). https://pith.science/paper/UHHN2EQH

@misc{pith2026250114876,
  author       = {Pith},
  title        = {Pith review of: Atypical vortex lattice and the magnetic penetration depth in superconducting Sr$_2$RuO$_4$ deduced by $\mu$SR},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UHHN2EQH}},
  note         = {Machine review of arXiv:2501.14876}
}

abstract

The muon spin rotation ($\mu$SR) technique has been applied to determine the behavior of the in-plane magnetic penetration depth ($\lambda_{ab}$) in the vortex state of the unconventional superconductor Sr$_2$RuO$_4$ as a means of gaining insight into its still unknown superconducting order parameter. A recent $\mu$SR study of Sr$_2$RuO$_4$ reported a $T$-linear temperature dependence for $\lambda_{ab}$ at low temperatures that was not identified in an earlier $\mu$SR study. Here we show that there is no significant difference between the data in the early and recent $\mu$SR studies and both are compatible with the limiting low-temperature $\lambda_{ab} \sim T^2$ dependence expected from measurements of the change in $\lambda_{ab}(T)$ in the Meissner state by other techniques. However, we argue that at this time there is no valid theoretical model for reliably determining the absolute value of $\lambda_{ab}$ in Sr$_2$RuO$_4$ from $\mu$SR measurements. Instead, we identify the formation of an unusual square vortex lattice that introduces a new constraint on candidate superconducting order parameters for Sr$_2$RuO$_4$.

Figures

Figures reproduced from arXiv: 2501.14876 by the authors.

**Figure 2.** in Ref. [20], so the sample-component frequencies remain distinct. The absence of the third lowerfrequency sample component in the single crystals investigated in Ref. [19] indicates a significant difference in the VL magnetic field profile B(r) and suggests an hexagonal rather than square arrangement of the vortices in this sample. Regardless, the calculated values of 1/λ2 ab(0, b) from Eq. (1) for A = 4.83 and… view at source ↗

**Figure 3.** (b)]. Like the modified nonlocal London model, the iterative GL model fails to describe the low-field shoulder and the left peak of the Fourier transformed TF-µSR signal. D. Two-component Eu state model The temperature dependence of λab in Sr2RuO4 for b ∼ 0.25 was determined in Ref. [20] from a global fit of the TF-µSR spectra to a model of B(r) derived from GL theory for a two-component complex order parameter bel… view at source ↗

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Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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B. M. Andersen, A. Kreisel, and P. J. Hirschfeld, Spon- taneous time-reversal symmetry breaking by disorder in superconductors, Front. Phys. 12, 1353425 (2024)

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[4] [1]

is an approximation that is strictly valid for κ = λ ab/ξ ab ≥ 5 and 0 . 25/κ 1. 3 ≲ b ≪ 1, where ξab is the in-plane GL coherence length [21]. For the Sr 2RuO4 sin- gle crystals studied in Ref. [19], B‖ c2 ∼ 75 mT, which cor- responds to ξab ∼ 663 ˚ A calculated from the GL relation B‖ c2 = Φ 0/ 2πξ 2 ab. This implies that Eq. (

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[5] [2]

Yet this is far greater than the zero- temperature values of 1240 ˚ A ≲ λ ab(0, b ) ≲ 1740 ˚ A for 0 ≤ b ≤ 0

is valid pro- vided λ ab > 3315 ˚ A. Yet this is far greater than the zero- temperature values of 1240 ˚ A ≲ λ ab(0, b ) ≲ 1740 ˚ A for 0 ≤ b ≤ 0. 6 reported in Ref. [19]. In addtion to this in- consistency, Eq. (

work page

[6] [3]

Before discussing the analysis method applied in the earlier TF-µ SR study of [20], we present results obtained from applying the same analysis method of Ref

is derived from a spatial ﬁeld proﬁle B(r) that does not account for the ﬁnite size of the vortex cores and assumes the individual vortices have a circular cross section, despite the assumption of a square VL. Before discussing the analysis method applied in the earlier TF-µ SR study of [20], we present results obtained from applying the same analysis met...

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[7] [4]

with A = 5 . 07. We note that the Fourier transform of the TF- µ SR asymmetry spectrum provides a visual approximation of n(B) associated with the VL and the distribution of ﬁeld sensed by muons that missed the sample. /s48 /s49 /s50 /s51 /s52 /s53 /s54 /s48/s46/s50 /s48/s46/s49 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s65 /s40/s116/s41 /s116 /s32/s40 /s10...

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[8] [5]

83 and A = 5

for A = 4 . 83 and A = 5 . 07 diﬀer by only about 5 %. The T -linear temperature dependence of λ ab reported in Ref. [19] was inferred from a linear ﬁt of λ ab(T, B ) extending up to T ∼ 0. 57Tc, which is far above temper- atures at which quasiparticle excitations are conﬁned to nodal regions on the Fermi surface. The inset of Fig. 2 shows that the variat...

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