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Exceedingly large in-plane critical field of finite-momentum pairing state in bulk superlattices

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper reports that the bulk van der Waals superlattice Ba6Ta11S28 remains superconducting under in-plane fields exceeding eight times the Pauli paramagnetic limit, and argues that this is caused by an orbital-effect-induced…

desk verdict Credible bulk realization candidate for orbital finite-momentum pairing, but the huge Hc2 and the pairing state rest on a fitted model and an extrapolated resistive boundary. read the letter →

arxiv 2506.16039 v1 pith:WMM2RVUI submitted 2025-06-19 cond-mat.supr-con cond-mat.mtrl-sci

Junfa Lin , Ziqiao Wang , Hongyi Yan , Xiaoping Ma , Zihan Cui , Yu Zhang , Yuxuan Lei , Jie Liu
show 10 more authors
Rao Li Chuanying Xi Zengwei Zhu Huakun Zuo Yanzhao Liu Huaixin Yang Tian-Long Xia Haiwen Liu Yi Liu Jian Wang
This is my paper · ORCID
classification cond-mat.supr-concond-mat.mtrl-sci
keywords finite-momentumpairingorbitaleffectin-planeuppercriticalfieldPaulilimitIsingsuperconductivityvanderWaalssuperlatticeBa6Ta11S28Josephsonvortexlatticemelting
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 reports that the bulk van der Waals superlattice Ba6Ta11S28 stays superconducting under in-plane magnetic fields far beyond the Pauli paramagnetic limit, with an extrapolated zero-temperature in-plane upper critical field near twelve times $B_P$. The central claim is that this resilience comes from an orbital-effect-induced finite-momentum pairing state: the in-plane field shifts the Cooper-pair momentum between adjacent superconducting layers and makes the order parameter spatially modulated. A generalized Lawrence-Doniach model reproduces the pronounced low-temperature upturn in $B_{c2,\parallel}(T)$ across several samples, and transport measurements show that interlayer coherence is suppressed above a characteristic field $B^*$. If the claim is right, bulk superlattices with strong Ising spin-orbit coupling and weak interlayer coupling can host spatially modulated high-field superconductivity, a state previously reported mainly in thin flakes.

What carries the argument

The carrying object is the generalized Lawrence-Doniach free energy for a stack of superconducting layers coupled by Josephson tunneling. In layer $l$ the order parameter is written $\psi_l(x)=\Delta(x)e^{iQ_l x}$ with $Q_l = 2\pi B\sin\theta\,D l/\Phi_0$, so neighboring layers differ in momentum by $2q_0 = 2\pi B D/\Phi_0$; minimizing the free energy gives an eigenvalue equation whose largest eigenvalue sets $B_{c2,\parallel}(T)$. The spatial modulation of $\Delta(x)$ is the finite-momentum pairing state, and the characteristic field $B^*$ is identified with the melting line of the Josephson vortex solid, above which the in-plane field becomes uniform and the interlayer coherence is suppressed.

What would settle it

Measure the full in-plane resistive transition at temperatures below 1 K in a pulsed or hybrid magnet and compare the 50 percent and 90 percent thresholds with the true zero-resistance onset; if the resistivity reaches the normal-state value at a field well below the extrapolated $B_{c2,\parallel}(0)$, or if the transition shows a vortex-liquid tail, the exceedingly large critical field would not be a bulk superconducting instability. A thermodynamic probe such as specific heat or magnetization in the same field range would settle whether the transition is a genuine bulk phase boundary.

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Extended reading notes

Core claim

On the paper's own terms, Ba6Ta11S28 is a bulk superlattice made of 1H-TaS2 superconducting layers separated by Ba3TaS5 block layers, with strong Ising spin-orbit coupling and unusually weak interlayer Josephson coupling. The in-plane upper critical field shows a sharp upturn below about 2.3 K, so that $B_{c2,\parallel}(0)$ is extrapolated to roughly $12B_P$, comparable to monolayer Ising superconductors. The authors attribute the upturn to the orbital effect of the in-plane field: the field imprints an Aharonov-Bohm phase that gives adjacent layers a relative Cooper-pair momentum $2q_0 = 2eBD/\hbar$, producing a finite-momentum pairing state whose order-parameter amplitude oscillates in the intralayer direction. They support this with a generalized Lawrence-Doniach model that fits the measured phase boundary, with the sharp angular cusp near the in-plane orientation, with the marked anisotropy between interlayer and intralayer transport above $B^*$, and with the observation that moderate disorder weakens but does not destroy the upturn.

Load-bearing premise

The load-bearing premise is that the resistive thresholds used to define $B_{c2,\parallel}$ (50 percent and 90 percent of the normal-state resistivity, as used in Figs. 2c-d and 3a-b) coincide with the true superconducting instability in the high-field regime, because below 1.1 K the resistivity stays below half the normal-state value even at 41 T and the very large reported critical fields are partly extrapolated from a model fit.

Editorial extensions

If this is right

  • Bulk van der Waals superlattices with strong Ising spin-orbit coupling and weak interlayer Josephson coupling should generally be able to host finite-momentum pairing at high in-plane fields, not just monolayer or few-layer flakes.
  • The finite-momentum pairing state is robust against moderate disorder, in contrast to the conventional Zeeman-driven FFLO state, so bulk samples with natural disorder can still exhibit it.
  • Above the characteristic field $B^*$, interlayer and intralayer transport should decouple dramatically, giving a clear experimental signature in simultaneous $\rho_{ab}$ and $\rho_c$ measurements.
  • The extrapolated in-plane critical field near $12B_P$ places this bulk superlattice on par with monolayer Ising superconductors, making it a benchmark for high-field superconducting materials.
  • The fitted generalized Lawrence-Doniach model predicts how the upturn temperature and the value of $B_{c2,\parallel}(0)$ shift as the interlayer coupling, layer spacing, or disorder level is changed.

Reading between the lines

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

  • The transport data imply, but do not directly prove, that the pairing amplitude is periodically modulated; a local probe such as scanning tunneling spectroscopy or Josephson scanning microscopy should resolve spatial oscillations with period $\pi/q_0$ if the interpretation is correct.
  • The mechanism should be tunable: changing the interlayer Josephson coupling by pressure, intercalation, or layer stacking should move both $B^*$ and the upturn temperature in a predictable way, providing a sharper test than sample-to-sample disorder comparison.
  • Extending the paper's logic to other naturally layered superconductors with strong spin-orbit coupling suggests that high-field modulated superconducting states may be sought in bulk compounds beyond transition-metal dichalcogenides, for example in stripe-ordered or structurally modulated materials.
  • A clean thermodynamic measurement of the transition under in-plane field would test whether the reported $B_{c2,\parallel}$ values represent a bulk instability or an artifact of the resistive criterion.
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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 / 5 minor

Summary. The paper reports transport measurements on the bulk van der Waals superlattice Ba6Ta11S28, showing a pronounced upturn in the in-plane upper critical field Bc2,∥(T) below about 2.3 K, with reported values exceeding eight times the Pauli limit and a strong angular selectivity within roughly 0.5° of the in-plane direction. The authors interpret the upturn as evidence for an orbital-effect-induced finite-momentum pairing state, described by a generalized Lawrence-Doniach model, and they associate the characteristic field B* with melting of the Josephson vortex solid. They support this assignment with multi-sample disorder studies, simultaneous in-plane and interlayer transport, critical-current anisotropy, and exclusion of several alternative scenarios (KLB, two-band, Takahashi-Tachiki, and conventional Ising superconductivity).

Significance. If the finite-momentum assignment holds, this would be the first bulk analog of the orbital Fulde-Ferrell-Larkin-Ovchinnikov state and would establish bulk van der Waals superlattices as a tunable platform for spatially modulated superconductivity. The paper has clear experimental strengths: the large in-plane critical field and its upturn are documented in several samples and in two high-field facilities (pulsed 41 T and hybrid 44 T); the angular dependence shows a sharp cusp consistent with quasi-2D behavior; and the disorder series provides a falsifiable trend that is used to rule out KLB and related mechanisms. The main weaknesses are inferential: the high-field Bc2 values are based on resistivity thresholds that are not actually reached in the pulsed-field data, and the quantitative agreement with theory comes from a three-parameter fit based on the authors' own preprint. The significance of the paper is therefore contingent on closing the gap between the measured resistive state and the claimed superconducting phase boundary.

major comments (3)
  1. [Fig. 3a-b and the pulsed-field section of the main text] The central observable Bc2,∥ is defined by resistivity thresholds that are not reached in the field range where the main claim is made. At 1.06 K, Fig. 3a shows the resistivity still below half of the normal-state value at 41 T, and the text states that this 'yields' a Bc2,∥ beyond eight times the Pauli limit; however, the Fig. 3b caption says the critical field is determined by 90% of the normal-state resistivity, while Fig. 2d and its caption use the 50% criterion. These are different definitions, and neither boundary is actually attained at 1.06 K up to 41 T. The upturn and the >8 Bp values are therefore partly extrapolations from a fitted model rather than directly measured transitions to the normal state. If the resistive state above B* is instead a dissipative vortex liquid or a normal metal with field-induced magnetoresistance, the finite-momentum pairing assignment loses its evidentiary basis. Please provide a criterion-sensitivity analysis (e.g., 10%, 50%, and 90% thresholds, I-V characteristics, or a thermodynamic probe) and clearly separate measured boundary points from model extrapolation.
  2. [Supplementary Sec. 2 (Eqs. S1-S7) and Table S2] The quantitative agreement that anchors the finite-momentum interpretation is a three-parameter fit to the same Bc2,∥(T) data, not an independent prediction. The generalized Lawrence-Doniach model is fitted using B0, T0, and d/D, with per-sample values listed in Table S2, and the underlying theory is taken from the authors' own preprint (arXiv:2409.20336). A fit with three flexible parameters cannot by itself discriminate the orbital finite-momentum state from other mechanisms. Please provide independent estimates of B0, T0, and d/D (for example from Josephson plasma measurements, specific heat, or a predicted value of the modulation wavevector q0), or show that the fitted parameters are uniquely constrained by data outside the upturn region. Without such constraints, the 'good agreement' in Fig. 3b should be described as a consistency check rather than confirmation.
  3. [Supplementary Sec. 3 and Fig. 4a] The identification of B* with Josephson vortex-lattice melting is also based on a fitted line, with the Lindemann criterion, the in-plane penetration length λ_ab(0), and the Labusch parameter α_L used as adjustable inputs. Since the two-step resistive transition and the transport anisotropy in Figs. 3e-f could in principle be produced by a vortex-liquid regime or by thermally activated phase slips, the paper should present an independent diagnostic of the melting transition (for example a sharp feature in ρ_c, a nonlinear response signature, or a thermodynamic anomaly) before using B* as evidence for the prerequisite of the finite-momentum state. At present the melting interpretation is plausible but not uniquely established by the data shown.
minor comments (5)
  1. [Fig. 2d and Fig. 3b captions] Please reconcile the two different threshold definitions: Fig. 2d states that critical fields are determined by 50% of the normal-state resistivity, while the Fig. 3b caption states 90%. The manuscript should state which criterion was used for each figure and for Table S1.
  2. [Abstract and Fig. 4c] The abstract says the in-plane critical field 'exceeds eight times the Pauli limit', while Fig. 4c shows an extrapolated Bc2,∥(0) ≈ 12 Bp. Please label clearly which values are directly measured and which are extrapolated from the model, and use consistent wording throughout.
  3. [Fig. 3a and phase-diagram section] The characteristic field B* is introduced without a quantitative definition. Please specify the criterion used to extract B* from the two-step resistive transition, either in the main text, figure caption, or Methods.
  4. [General editorial] There are several typographical and wording issues, including 'charge particles' in the abstract and 'board temperature regime' in Supplementary Sec. 5; a careful proofreading pass would improve the manuscript.
  5. [Data and code availability] The Data Availability and Code Availability statements say that materials are available 'on reasonable request'. If possible, depositing the processed transport data and the numerical diagonalization code in a public repository would strengthen the reproducibility of the central analysis.

Circularity Check

2 steps flagged · score 5.0 of 10

Experimental upturn is independent, but the quantitative support for the finite-momentum-pairing mechanism is a three-parameter fit to the same Bc2,∥(T) data, with the mechanism itself assumed in the model.

  1. fitted input called prediction [Supplementary Note 2 (Theoretical model for the orbital effect induced finite-momentum pairing state); main-text Figs. 3b and 4a]
    "In the main text, we fit the experimental data by adjusting three key parameters (Figs. 3b and 4): the characteristic magnetic field scale B0 corresponding to the Josephson vortex solid, the characteristic temperature scale T0 corresponding to the interlayer Josephson coupling and the ratio d/D."

    The theoretical Bc2,∥(T) curve that the paper says 'is in a good agreement with the experimental phase boundary' is generated by fitting B0, T0, and d/D to those same experimental Bc2,∥(T) data. Agreement between a fit and its own input is a consistency check, not an independent validation. Moreover, the generalized LD model in the same Supplementary section already assumes the layer-dependent momentum Ql = 2πB sinθ D l/Φ0, which is precisely the finite-momentum pairing state being claimed. Hence the quantitative support for the orbital-FMP mechanism reduces, at that step, to the model's ansatz plus a fit to the curve it is supposed to explain.

  2. self citation load bearing [Main text, section 'Suppression of interlayer coherence under high in-plane fields'; Supplementary Note 3; Ref. 27]
    "According to our theoretical model, the melting of Josephson vortex solid gives rise to a uniform in-plane magnetic field, which is the prerequisite of the orbital effect induced finite-momentum pairing state."

    This load-bearing step in the interpretation of the characteristic field B* is taken from Ref. 27 (Yan et al., arXiv:2409.20336), whose authors include two co-authors of the present paper (H. Yan and H. Liu). The calculated melting line is then fitted to the same B* data, with Supplementary Fig. S8 reporting fitted values λab(0)=1.2 μm and αL=2.5×10^-2 T^2/nm^2. The B* agreement is therefore a self-consistency check with parameters adjusted to the data, not an external confirmation of the mechanism. The raw two-step resistive feature is independent, but the identification of its origin as Josephson-vortex melting imported from the authors' own preprint is load-bearing.

full rationale

The paper's raw experimental results—the in-plane Bc2 upturn, its sharp angular sensitivity, disorder dependence, and the intralayer/interlayer transport anisotropy—are independent measurements, so this is not a fully circular paper. The circularity burden sits on the quantitative theory comparison: the generalized Lawrence-Doniach curve in Fig. 3b/4a is a three-parameter fit (B0, T0, d/D) to the very Bc2,∥(T) data it is claimed to explain, so its 'good agreement' is a consistency check rather than a prediction. The same applies to the Josephson-vortex-solid melting line fitted to B*. The model itself assumes the layer-dependent momentum shift that constitutes the finite-momentum state, so the fit cannot independently prove the mechanism. The high-field Bc2 definition (50%/90% normal-state resistivity, or extrapolation when 41 T is insufficient) is a measurement-criterion concern; it weakens the strength of the claim but is not a circular derivation. Alternatives (KLB, two-band, Takahashi-Tachiki, Ising) are discussed and ruled out on grounds independent of the fitted model. Overall: partial circularity in the validation step, but a substantial independent experimental core.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The paper builds on substantial prior apparatus: the Ising-SOC suppression of Zeeman effects, the Lawrence-Doniach model for layered superconductors, and the orbital-FFLO proposal from the authors' own preprint. The new inputs are the material synthesis and transport data, plus the choice to fit B0, T0, and d/D. The fitted parameters and the measurement-criterion assumption carry much of the burden for the quantitative claim.

free parameters (6)
  • B0 (characteristic magnetic field scale) = 1.276-2.196 T by sample (Table S2)
    Sets the magnetic-field scale in the generalized Lawrence-Doniach model; fitted to each sample's Bc2,∥(T).
  • T0 (interlayer Josephson coupling temperature scale) = 0.242-0.299 K by sample (Table S2)
    Sets the interlayer coupling strength in the model; fitted to each sample's Bc2,∥(T).
  • d/D (superconducting layer effective thickness over interlayer spacing) = 0.36-0.73 by sample (Table S2)
    Controls the orbital suppression term in the model; fitted per sample and then interpreted as disorder-induced broadening.
  • Lindemann criterion parameter c_L^2 = not stated
    Used to locate the Josephson vortex melting line through the condition <u^2>/L0^2 > c_L^2; the numerical value is not given.
  • in-plane penetration length lambda_ab(0) = 1.2 micrometers
    Fitted to the B*(T) melting line in Supplementary Fig. S8; enters the vortex-lattice elastic moduli.
  • Labusch parameter alpha_L = 2.5 x 10^-2 T^2/nm^2
    Fitted to the B*(T) melting line; represents the restoring force on Josephson vortices.
assumptions (5)
  • domain assumption A Ginzburg-Landau/Lawrence-Doniach mean-field free energy with one order parameter per superconducting layer adequately describes Bc2 in this dirty layered superconductor.
    The theoretical fit in Supplementary Sec. 2 uses this free energy; strong-disorder corrections are not derived from a microscopic theory.
  • domain assumption Zeeman pair-breaking is negligible because Ising spin-orbit coupling locks spins out of plane, so the orbital effect dominates the in-plane field response.
    Invoked in the main text around Fig. 3 and Supplementary Sec. 2, based on the ARPES-reported 120 meV spin splitting from Ding et al. 2024, an external input not measured in this paper.
  • domain assumption The layer-dependent phase profile Q_l = 2*pi*B*D*l/Phi_0 is the relevant Cooper-pair momentum and the order parameter amplitude can be treated as a single function Delta(x) across the bulk.
    Follows from a gauge choice in the Lawrence-Doniach model (Supplementary Eqs. S3-S4), but identifying this gauge phase with a physical finite-momentum pairing state is the interpretation under test.
  • ad hoc to paper The 50% and 90% normal-state resistivity thresholds mark the upper critical field Bc2 even when the full normal-state resistivity is not reached.
    Fig. 3a-b: the low-temperature Bc2 values are partly extrapolated from the model because the resistivity stays below half of the normal-state value at 41 T. This is a measurement-criterion assumption specific to this paper.
  • domain assumption Josephson vortex solid melting produces an effectively uniform in-plane field and suppresses interlayer coherence, which is the prerequisite for the orbital finite-momentum pairing state.
    Main text 'Suppression of interlayer coherence' section and Supplementary Sec. 3; supported by a Lindemann-melting calculation with fitted parameters, not by direct vortex imaging.

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Pith. "Pith review of Exceedingly large in-plane critical field of finite-momentum pairing state in bulk superlattices." pith.science (2026). https://pith.science/paper/WMM2RVUI

@misc{pith2026250616039,
  author       = {Pith},
  title        = {Pith review of: Exceedingly large in-plane critical field of finite-momentum pairing state in bulk superlattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WMM2RVUI}},
  note         = {Machine review of arXiv:2506.16039}
}
read the original abstract

Magnetic flux profoundly influences the phase factor of charge particles, leading to exotic quantum phenomena. A recent example is that the orbital effect of magnetic field could induce finite-momentum pairing state in nanoflakes, which offers a new pathway to realize the spatially modulated superconductivity distinct from the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state induced by Zeeman effect. However, whether such intriguing state can exist in the bulk materials under extremely large magnetic field remains elusive. Here we report the orbital effect induced finite-momentum pairing state with exceedingly large in-plane critical field in a bulk superconducting superlattice. Remarkably, the in-plane critical field shows a pronounced upturn behavior, exceeding eight times the Pauli limit which is comparable to monolayer Ising superconductor. Under high in-plane magnetic fields, significant anisotropic transport behavior between the interlayer and intralayer directions is detected, highlighting the critical role of suppressed interlayer coherence in the orbital effect induced finite-momentum pairing state. Crucially, this finite-momentum pairing state remains robust against moderate disorder. Our findings suggest that van der Waals superlattices, with strong Ising spin-orbit coupling and tunable interlayer coherence, offer new avenues for constructing and modulating unconventional superconducting states.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Structures and proximity effects of inhomogeneous population-imbalanced Fermi gases with pairing interactions

    cond-mat.quant-gas 2026-02 conditional novelty 6.0 of 10

    In spin-imbalanced 1D Fermi gases, spatial jumps in pairing or polarization create BCS/FFLO/normal coexistence, with FFLO correlations penetrating normal regions and a buffer FFLO zone forming at BCS-normal junctions.

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