Pith. sign in

REVIEW 3 major objections 4 minor 64 references

Lorentz-violating QED inspired superconductivity

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Adding a CPT-odd Carroll-Field-Jackiw term to the Ginzburg-Landau free energy produces, for couplings k\lambda_L > 0.39, non-Meissner magnetic phases with in-plane fields and a central anti-vortex.

desk verdict CFJ-in-GL is a natural extension, but a sign error in the current equation means the modified London equation and the vortex phase diagram are unsupported as written. read the letter →

arxiv 2505.16930 v1 pith:VTAENGCJ submitted 2025-05-22 cond-mat.supr-con hep-th

A.F.Morais , M.C.Araújo , T.T.Saraiva , J. Furtado This is my paper · ORCID
classification cond-mat.supr-conhep-th
keywords superconductivityGinzburg-LandautheoryLondonequationLorentzviolationCarroll-Field-JackiwtermMeissnereffectvorticesUTe2
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 argues that a single Lorentz-violating, CPT-odd term added to the standard Ginzburg-Landau free energy changes the London equation so that the magnetic field couples to its own curl. In the London limit the field obeys \tilde{\nabla}^2 B = B + 2(k\lambda_L)\tilde{\nabla}\times B, controlled by one parameter k\lambda_L. Numerical solution on circular and square cross-sections shows ordinary Meissner screening for k\lambda_L \lesssim 0.39, and above that a breakdown of the Meissner effect: in-plane magnetic field components become dominant and a central anti-vortex appears at larger values. The authors propose this as a possible signature of stratified superconductors such as UTe2.

What carries the argument

The load-bearing object is the Carroll-Field-Jackiw term k \mathbf{A}\cdot(\nabla\times\mathbf{A}) added to the Ginzburg-Landau free energy, with k identified as the temporal component of the CPT-odd vector (k_{AF})^\mu in the static limit. It produces the modified London equation \tilde{\nabla}^2 B = B + 2(k\lambda_L)\tilde{\nabla}\times B, whose dimensionless coupling k\lambda_L regulates the transfer of field into transverse components. The numerical machinery is a finite-difference solution of the coupled Ginzburg-Landau equations on two-dimensional meshes (circular and square cross-sections) for a long cylinder with an axial magnetic field.

What would settle it

Solve the full static Ginzburg-Landau equations retaining all four components of (k_{AF})^\mu for the same square cross-section: if the threshold k\lambda_L \approx 0.39 and the central anti-vortex do not survive when spatial components are included, the static-limit reduction is the fragile step; alternatively, scanning Hall probe or SQUID microscopy on a stratified superconductor such as UTe2 should reveal a central flux spot opposite to the applied field—its absence would falsify the predicted anti-vortex phase.

Watch

Extended reading notes

Core claim

The central discovery is that the Carroll-Field-Jackiw term, which couples the vector potential to its own curl, does not merely renormalize the penetration length but qualitatively changes the magnetic response of a superconductor. After the static-limit reduction and the London approximation, the magnetic field obeys \tilde{\nabla}^2 B = B + 2(k\lambda_L)\tilde{\nabla}\times B, where the extra term transfers flux from the applied direction into orthogonal components. For k\lambda_L > 0.39 the transverse component becomes the total field on a loop around the center; near k\lambda_L \approx 0.8 the usual Meissner effect breaks down. The numerical solutions also reveal strong in-plane currents and, at k\lambda_L = 1, an anti-vortex at the center encircled by four vortices with opposite current circulation.

Load-bearing premise

The derivation rests on assuming that in the static limit only the temporal component of the Lorentz-violating vector contributes, so the extra free-energy term is exactly k \mathbf{A}\cdot(\nabla\times\mathbf{A}); if spatial components survive, the modified London equation and all predicted phases change.

Editorial extensions

If this is right

  • For k\lambda_L \lesssim 0.39 the model reproduces standard Meissner screening, so weak Lorentz violation would be hard to detect in the magnetic response.
  • Above k\lambda_L \approx 0.39, significant in-plane magnetic field components appear inside the superconductor; above k\lambda_L \approx 0.8 the Meissner effect breaks down.
  • At k\lambda_L \approx 1 the field configuration contains a central anti-vortex with flux opposite to the applied field, surrounded by four vortices with opposite circulation.
  • The mean magnetic field and the effective penetration length rise sharply for k\lambda_L \gtrsim 0.1 and saturate for k\lambda_L \gtrsim 1.
  • If reproduced experimentally, such field and current patterns could serve as a hallmark of unconventional superconducting states in stratified materials such as UTe2.

Reading between the lines

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

  • If the static-limit reduction is relaxed and spatial components of (k_{AF})^\mu are kept, the modified London equation would acquire extra terms coupling to A_0 and to gradients of k; the threshold k\lambda_L \approx 0.39 and the anti-vortex phase could shift, so the threshold should be read as a sensitivity estimate within the stated approximation.
  • The London approximation assumes a nearly uniform order parameter; in type-I superconductors or near T_c, where the order parameter varies, the vortex and anti-vortex phases may be substantially modified, and mapping the Ginzburg-Landau parameter range is a natural next step.
  • Reversing the sign of k\lambda_L should reverse the handedness of the vortex current circulation while preserving the threshold, a concrete and testable prediction of the model.
  • The model implies that in-plane magnetic field components inside a superconductor are a generic signature of Lorentz violation, so scanning Hall probe or SQUID microscopy on a candidate material could directly look for this effect.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a Lorentz-violating extension of Ginzburg-Landau theory by adding a Carroll-Field-Jackiw (CFJ) term to the free energy, reduces the resulting equations in the London limit to a modified London equation for the magnetic field, and numerically studies the magnetic response of cylindrical and square superconducting samples as a function of the dimensionless parameter kλL. The authors report a transition near kλL ≈ 0.39 from a conventional Meissner response to phases with strong in-plane fields, anomalous vortex configurations, and a central anti-vortex, and they discuss possible connections to stratified superconductors such as UTe2.

Significance. If the central derivation were correct, the paper would provide a concrete, falsifiable scenario in which a Lorentz-violating term in the gauge sector changes the Meissner response and produces vortex structures not present in standard London theory. The reduction to a single dimensionless parameter kλL is elegant, and the predicted threshold is a crisp, testable claim. The paper also builds its Ginzburg-Landau coefficients from prior BCS/Gor'kov results rather than fitting them to the target phase diagram. However, the main London equation contains a sign error that invalidates the derivation of Eq. (14), and the numerical phase diagram is presented without convergence checks or stated boundary conditions. Because the central physical claim depends on the sign in Eq. (14), the paper in its current form does not establish its stated results.

major comments (3)
  1. [Eqs. (7)-(10) and (13)-(14)] The supercurrent term in the current equation has the wrong sign. Minimizing the free energy (3) with respect to A for constant Ψ gives (1/4π)∇×∇×A + 2K(2e/ħc)^2|Ψ|^2 A = 0, so Eq. (10) should have a minus sign on the RHS supercurrent term. A direct check is the standard Meissner solution for k=0: B = B0 e^{-x/λ} ẑ and A = -λ B0 e^{-x/λ} ŷ gives ∇×∇×A = -(1/λ^2)A, opposite to the positive coefficient in Eq. (10). Taking the curl of the corrected equation gives ∇^2 B = -(1/λ_L^2)B - 2k∇×B, not the positive-sign Eq. (13) or Eq. (14). Since Eq. (14) is the basis for the reported threshold kλL ≈ 0.39 and the anti-vortex phases, the central claim is not supported by the derivation as written.
  2. [Paragraph before Eq. (2)] The reduction of the CFJ term to k A·(∇×A) assumes that only the temporal component (k_AF)^0 contributes and that A_0 can be set to zero. The text states this without justification. If spatial components of k_AF are present or if A_0 is not dropped, additional couplings such as A_0 B_i appear in the free energy and modify the London equation. The authors should either justify the A_0=0 choice within the London approximation or show that spatial components do not affect the qualitative conclusions.
  3. [Numerical results, Figs. 2-3] The numerical results are presented without convergence checks, error bars, or a stated grid-size dependence, and the boundary conditions are not specified in the main text ('details about the numerical methods are shown in the Supplemental Material'). This is especially problematic because the physical interpretation depends on the sign of the CFJ coupling; with the corrected sign from Eq. (10), the mode structure changes qualitatively. The threshold kλL ≈ 0.39 and the phase diagram in Fig. 3 therefore cannot be assessed as reliable quantitative predictions.
minor comments (4)
  1. [Text after Eq. (6)] The temperature scaling k ∝ τ^{1/2} is introduced to make all free-energy terms scale as τ^2, but the next sentence says 'we do not consider temperature dependence of this term and rather express all quantities in temperature reduced dimensionless units.' This is confusing and should be clarified: is k treated as temperature dependent in the derivation or not?
  2. [Fig. 2 caption and Sec. III] The text describes a long cylindrical superconductor with circular cross-section in Fig. 1, but Fig. 2 is described as a wire with a square cross-section; the transition between geometries is not explained in the main text.
  3. [Conclusion] The first sentence of the Conclusion contains a typo: 'In summa,' should be 'In summary,'.
  4. [References [9] and [10]] References [9] and [10] are assigned the same URL; the bibliographic details should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation chain is self-contained and kλL thresholds are numerical outputs, not fitted or self-referential inputs.

full rationale

The claimed derivation proceeds from the Lagrangian (1), through the static-limit reduction (2), the GL free energy (3), the current equation (7)/(10), and the London-limit equation (14). Each step uses the preceding equation as input, and the phase thresholds (kλL ≈ 0.39 and the anti-vortex configurations) are obtained by numerically solving Eq. (14) over a scanned range of kλL, not by fitting parameters to those target features. The GL coefficients a, b, K are taken from standard Gor'kov/BCS results (ref. [53]) and are not tuned to the predicted phase diagram. The speculative UTe2 discussion appears only as an interpretive afterthought, not as an input to the model. Self-citations (refs. [18], [19], [34], [35], [67]) are background or peripheral and are not load-bearing for the London equation or the phase diagram. The in-scope caveats are genuine limitations but not circularities: the static-limit choice that only the temporal CFJ component contributes (paragraph before Eq. (2)) is an unproven assumption, the numerical boundary conditions are deferred to the Supplemental Material, and the temperature dependence of k is set aside; these are completeness/correctness concerns. The sign error noted by the skeptic in Eq. (7)/(10), if present, would make Eq. (14) not a consequence of the free energy, but an incorrect derivation is not a case of a prediction reducing by construction to its input. No circular step satisfying the quoted-evidence standard was found.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The model rests on standard GL coefficients taken from BCS/Gor'kov and one new dimensionless parameter kλL scanned over many orders of magnitude. No parameter is fitted to experimental data; the phase transition thresholds are outputs of the numerical solutions. Axioms include the static reduction (only k0) and the London limit.

free parameters (1)
  • k (CFJ coefficient, equivalent dimensionless kλL) = scanned 10^-6 to 10^2; threshold near 0.39
    The Lorentz-violating coupling is the new input to the free energy; its dimensionless combination with the London penetration depth controls all numerical results.
assumptions (4)
  • ad hoc to paper Only the temporal component (k_AF)^0 of the CFJ vector contributes in the static limit, reducing the term to k A·(∇×A).
    Stated immediately before Eq. (2); spatial components can couple to A_0 B unless A_0=0, which is not established.
  • domain assumption London limit: the order parameter is uniform (∇Ψ≈0, |Ψ|=Ψ_u) and λ_L is the relevant length scale.
    Used to derive Eqs. (9)-(14); standard for weak fields but restricts validity.
  • ad hoc to paper Temperature scaling k ∝ τ^(1/2) ensures all free energy terms scale as τ^2; afterward k is treated as temperature independent in dimensionless units.
    Introduced after Eq. (6); affects how kλL is interpreted across temperature.
  • domain assumption GL coefficients a, b, K are taken from BCS/Gor'kov with standard values.
    Inputs from prior literature, not derived here.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Lorentz-violating QED inspired superconductivity." pith.science (2026). https://pith.science/paper/VTAENGCJ

@misc{pith2026250516930,
  author       = {Pith},
  title        = {Pith review of: Lorentz-violating QED inspired superconductivity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VTAENGCJ}},
  note         = {Machine review of arXiv:2505.16930}
}
abstract

We studied a Lorentz-violating inspired Ginzburg-Landau model for superconductivity where we considered a CPT-odd contribution given by $(k_{AF})^{\mu}$, also known as the Carroll-Field-Jackiw term. In the static limit of the equations, we could find a pair of modified Ginzburg-Landau equations. Furthermore, these equations were reduced to the London equation for the magnetic field when assumed that the characteristic length of the order parameter is much smaller than the characteristic length of the magnetic field, i.e. the London penetration length. Our numerical solutions showed a simple Meissner state when this new term is small compared to $\lambda_L$ and a phase transition into phases with strong in-plane currents and anomalous vortices for large contributions. This model becomes useful in exemplifying the changes in the phenomenology of superconductors when the setup of the system shows an important breakdown of Lorentz invariance. Based on these results, we discuss how such models might be the hallmark of unusual superconducting states where there is a direction where the system shows stratification, as in anapole superconductors UTe$_2$.

Figures

Figures reproduced from arXiv: 2505.16930 by the authors.

Figure 1
Figure 1. (a) Magnetic field intensity, B, for kλL ≈ 0.25. The ar￾rows indicate the magnetic field component in the xy-direction (B∥). The external field is applied solely in the perpendicular direction, with Bz = 1 Oe. (b) Corresponding current density distribution per￾pendicular to the circular cross-section of the circular sample. The London penetration depth is λL = 400 nm, and the cross-section of the long cylinder is ci… view at source ↗
Figure 2
Figure 2. The first column (I) presents a color map of the magnetic field intensity, [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Dependence of the mean magnetic field (red curve, left scale) and the e [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

64 extracted references · 51 canonical work pages

  1. [1]

    V .Barger and K.L.Whisnant, The physics of neu- trinos, v

    D. V .Barger and K.L.Whisnant, The physics of neu- trinos, v. 5 (Princeton University Press, 2012), ISBN 9780080570464, URL https://books.google.ru/books? id=VzgJN-XPTRsC

    work page 2012

  2. [2]

    Arkani–Hamed, S

    N. Arkani–Hamed, S. Dimopoulos, and G. Dvali, Physics Letters B 429, 263 (1998), ISSN 0370-2693, URL https://www.sciencedirect.com/science/article/ pii/S0370269398004663

    work page 1998

  3. [3]

    Colladay and V

    D. Colladay and V . A. Kostelecký, Phys. Rev. D 55, 6760 (1997), URL https://link.aps.org/doi/10.1103/ PhysRevD.55.6760

    work page 1997

  4. [4]

    Colladay and V

    D. Colladay and V . A. Kostelecký, Phys. Rev. D 58, 116002 (1998), URL https://link.aps.org/doi/10. 1103/PhysRevD.58.116002

    work page 1998

  5. [5]

    V . A. Kostelecký and S. Samuel, Phys. Rev. D 39, 683 (1989), URL https://link.aps.org/doi/10.1103/ PhysRevD.39.683

    work page 1989

  6. [6]

    J. Link, M. Reyes, P. Yager, J. Anjos, I. Bediaga, C. Gö- bel, J. Magnin, A. Massa fferri, J. de Miranda, I. Pepe, et al., Physics Letters B 556, 7 (2003), ISSN 0370- 2693, URL https://www.sciencedirect.com/science/ article/pii/S0370269303001035

    work page 2003

  7. [7]

    Aubert, M

    B. Aubert, M. Bona, D. Boutigny, Y . Karyotakis, J. P. Lees, V . Poireau, X. Prudent, V . Tisserand, A. Zghiche, J. Garra Tico, et al. (BABAR Collaboration), Phys. Rev. Lett. 100, 131802 (2008), URL https://link.aps.org/doi/10. 1103/PhysRevLett.100.131802

    work page 2008

  8. [8]

    Babusci, I

    D. Babusci, I. Balwierz-Pytko, G. Bencivenni, C. Bloise, F. Bossi, P. Branchini, A. Budano, L. Caldeira Balkeståhl, G. Capon, F. Ceradini, et al., Physics Letters B 730, 89 (2014), ISSN 0370-2693, URL https://www.sciencedirect.com/ science/article/pii/S0370269314000380

    work page 2014

Show all 64 references
  1. [9]

    Babusci, I

    D. Babusci, I. Balwierz-Pytko, G. Bencivenni, C. Bloise, F. Bossi, P. Branchini, A. Budano, L. Caldeira Balkeståhl, G. Capon, F. Ceradini, et al., Found. Phys. 40, 852 (2010), ISSN 0370-2693, URL https://www.sciencedirect.com/ science/article/pii/S0370269314000380

    2010

  2. [10]

    V . M. Abazov, B. Abbott, B. S. Acharya, M. Adams, T. Adams, J. P. Agnew, G. D. Alexeev, G. Alkhazov, A. Al- ton, A. Askew, et al. (D0 Collaboration), Phys. Rev. Lett. 115, 161601 (2015), URL https://link.aps.org/doi/10. 1103/PhysRevLett.115.161601

    2015

  3. [11]

    R. Aaij, C. Abellán Beteta, B. Adeva, M. Adinolfi, Z. Ajal- touni, S. Akar, J. Albrecht, F. Alessio, M. Alexander, S. Ali, et al. (LHCb Collaboration), Phys. Rev. Lett. 116, 241601 (2016), URL https://link.aps.org/doi/10. 1103/PhysRevLett.116.241601

    2016

  4. [12]

    Adamson, C

    P. Adamson, C. Andreopoulos, K. E. Arms, R. Armstrong, D. J. Auty, D. S. Ayres, B. Baller, G. Barr, W. L. Barrett, B. R. Becker, et al. (MINOS Collaboration), Phys. Rev. Lett. 101, 151601 (2008), URL https://link.aps.org/doi/10. 1103/PhysRevLett.101.151601

    2008

  5. [13]

    Adamson, D

    P. Adamson, D. J. Auty, D. S. Ayres, C. Backhouse, G. Barr, W. L. Barrett, M. Bishai, A. Blake, G. J. Bock, D. J. Boehnlein, et al. (MINOS Collaboration), Phys. Rev. Lett. 105, 151601 (2010), URL https://link.aps.org/doi/10. 1103/PhysRevLett.105.151601

    2010

  6. [14]

    KATORI, Modern Physics Letters A 27, 1230024 (2012), https://doi.org/10.1142/S0217732312300248, URL https:// doi.org/10.1142/S0217732312300248

    T. KATORI, Modern Physics Letters A 27, 1230024 (2012), https://doi.org/10.1142/S0217732312300248, URL https:// doi.org/10.1142/S0217732312300248

    2012 doi

  7. [15]

    Aguilar-Arevalo, C

    A. Aguilar-Arevalo, C. Anderson, A. Bazarko, S. Brice, B. Brown, L. Bugel, J. Cao, L. Coney, J. Conrad, D. Cox, et al., Physics Letters B 718, 1303 (2013), ISSN 0370- 2693, URL https://www.sciencedirect.com/science/ article/pii/S0370269312012592

    2013

  8. [16]

    Jackiw and V

    R. Jackiw and V . A. Kostelecký, Phys. Rev. Lett. 82, 3572 (1999), URL https://link.aps.org/doi/10.1103/ PhysRevLett.82.3572

    1999

  9. [17]

    V . A. Kostelecký, C. D. Lane, and A. G. M. Pickering, Phys. Rev. D 65, 056006 (2002), URL https://link.aps.org/ doi/10.1103/PhysRevD.65.056006

    2002 doi

  10. [18]

    A. F. Ferrari, J. Furtado, J. F. Assunção, T. Mariz, and A. Y . Petrov, Europhysics Letters 136, 21002 (2022), URL https: //dx.doi.org/10.1209/0295-5075/ac419a

    2022 doi

  11. [20]

    Lehum, T

    A. Lehum, T. Mariz, J. Nascimento, and A. Petrov, Physics Letters B 790, 129 (2019), ISSN 0370-2693, URL https://www.sciencedirect.com/science/article/ pii/S0370269319300309

    2019

  12. [22]

    Belich, L

    H. Belich, L. Bernald, P. Gaete, and et al., Eur. Phys. J. C 291, 31001 (2015), URLhttps://doi.org/10.1140/epjc/ s10052-015-3447-0

    2015 doi

  13. [23]

    Colladay and P

    D. Colladay and P. McDonald, Phys. Rev. D83, 025021 (2011), URL https://link.aps.org/doi/10.1103/PhysRevD. 83.025021

    2011 doi

  14. [24]

    S. M. Carroll, J. A. Harvey, V . A. Kostelecký, C. D. Lane, and T. Okamoto, Phys. Rev. Lett. 87, 141601 (2001), URL https://link.aps.org/doi/10.1103/PhysRevLett.87. 141601

    2001 doi

  15. [25]

    Hayakawa, Physics Letters B 478, 394 (2000), ISSN 0370- 2693, URL https://www.sciencedirect.com/science/ article/pii/S0370269300002422

    M. Hayakawa, Physics Letters B 478, 394 (2000), ISSN 0370- 2693, URL https://www.sciencedirect.com/science/ article/pii/S0370269300002422

    2000

  16. [26]

    J. F. Assunção and T. Mariz, Europhysics Letters 110, 41002 (2015), URL https://dx.doi.org/10.1209/0295-5075/ 110/41002

    2015 doi

  17. [28]

    Goswami, G

    P. Goswami, G. Sharma, and S. Tewari, Phys. Rev. B 92, 161110 (2015), URL https://link.aps.org/doi/10. 1103/PhysRevB.92.161110

    2015

  18. [29]

    Sharma, P

    G. Sharma, P. Goswami, and S. Tewari, Phys. Rev. B 96, 045112 (2017), URL https://link.aps.org/doi/10. 1103/PhysRevB.96.045112

    2017

  19. [30]

    Katsnelson and G

    M. Katsnelson and G. V olovik, Phys. Rev. B 95, 411 (2012), URL https://link.aps.org/doi/10.1103/PhysRevB. 96.045112

    2012 doi

  20. [31]

    G. Baym, D. H. Beck, J. P. Filippini, C. J. Pethick, and J. Shel- ton, Phys. Rev. D 102, 035014 (2020), URL https://link. aps.org/doi/10.1103/PhysRevD.102.035014

    2020 doi

  21. [32]

    E. R. Pereira and F. Moraes, Central Eur. J. Phys. 9, 1100 (2011), URL https://doi.org/10.2478/ s11534-010-0109-6

    2011

  22. [33]

    Bazeia, F

    D. Bazeia, F. Brito, and J. Mota-Silva, Physics Let- ters B 762, 327 (2016), ISSN 0370-2693, URL https://www.sciencedirect.com/science/article/ pii/S0370269316305470

    2016

  23. [34]

    Furtado, R

    J. Furtado, R. M. M. C. Filho, A. F. Morais, and I. C. Jardim, Europhysics Letters 136, 51001 (2022), URL https://dx. doi.org/10.1209/0295-5075/ac36f0. 7

    2022 doi

  24. [35]

    Araújo, I

    M. Araújo, I. Jardim, D. Veras, and J. Furtado, Annals of Physics 468, 169726 (2024), ISSN 0003-4916, URL https://www.sciencedirect.com/science/article/ pii/S0003491624001349

    2024

  25. [36]

    T. Zhou, Y . Gao, and Z. D. Wang, Phys. Rev. B 93, 094517 (2016), URL https://link.aps.org/doi/10. 1103/PhysRevB.93.094517

    2016

  26. [37]

    Bednik, A

    G. Bednik, A. A. Zyuzin, and A. A. Burkov, Phys. Rev. B 92, 035153 (2015), URL https://link.aps.org/doi/10. 1103/PhysRevB.92.035153

    2015

  27. [38]

    Wei, S.-P

    H. Wei, S.-P. Chao, and V . Aji, Phys. Rev. B 89, 014506 (2014), URL https://link.aps.org/doi/10. 1103/PhysRevB.89.014506

    2014

  28. [39]

    Roy and V

    B. Roy and V . Juri ˇci´c, Phys. Rev. B 90, 041413 (2014), URL https://link.aps.org/doi/10.1103/PhysRevB.90. 041413

    2014 doi

  29. [40]

    Cohnitz, A

    L. Cohnitz, A. De Martino, W. Häusler, and R. Egger, Phys. Rev. B 96, 140506 (2017), URL https://link.aps.org/ doi/10.1103/PhysRevB.96.140506

    2017 doi

  30. [41]

    Madsen, Phys

    J. Madsen, Phys. Rev. Lett. 85, 10 (2000), URL https:// link.aps.org/doi/10.1103/PhysRevLett.85.10

    2000 doi

  31. [42]

    Bonanno and A

    L. Bonanno and A. Sedrakian, A&A 539, A16 (2012), URL https://doi.org/10.1051/0004-6361/201117832

    2012 doi

  32. [43]

    Ebert, A

    D. Ebert, A. V . Tyukov, and V . C. Zhukovsky, Phys. Rev. D 76, 064029 (2007), URL https://link.aps.org/doi/10. 1103/PhysRevD.76.064029

    2007

  33. [44]

    X. Gao, A. García-García, H. Zeng, and et al, J. High Energ. Phys. 19, 2014 (2014), URL https://doi.org/10.1007/ JHEP06(2014)019

    2014

  34. [45]

    C. P. Herzog, Journal of Physics A: Mathematical and Theo- retical 42, 343001 (2009), URL https://dx.doi.org/10. 1088/1751-8113/42/34/343001

    2009

  35. [46]

    Ghoroku, K

    K. Ghoroku, K. Kashiwa, Y . Nakano, M. Tachibana, and F. Toy- oda, Phys. Rev. D 99, 106011 (2019), URL https://link. aps.org/doi/10.1103/PhysRevD.99.106011

    2019 doi

  36. [47]

    Tinkham, Introduction to Superconductivity (McGraw-Hill, 1996), 2nd ed

    M. Tinkham, Introduction to Superconductivity (McGraw-Hill, 1996), 2nd ed

    1996

  37. [48]

    De Gennes, Superconductivity Of Metals And Alloys , Advanced Books Classics Series (Westview Press, 1999), ISBN 9780813345840, URL https://books.google.com

    P. De Gennes, Superconductivity Of Metals And Alloys , Advanced Books Classics Series (Westview Press, 1999), ISBN 9780813345840, URL https://books.google.com. br/books?id=ZMf_ticFcWYC

    1999

  38. [49]

    Fu, Phys

    L. Fu, Phys. Rev. Lett. 115, 026401 (2015), URL https:// link.aps.org/doi/10.1103/PhysRevLett.115.026401

    2015 doi

  39. [50]

    Kozii and L

    V . Kozii and L. Fu, Phys. Rev. Lett. 115, 207002 (2015), URL https://link.aps.org/doi/10.1103/PhysRevLett. 115.207002

    2015 doi

  40. [51]

    Schumann, L

    T. Schumann, L. Galletti, H. Jeong, K. Ahadi, W. M. Strick- land, S. Salmani-Rezaie, and S. Stemmer, Phys. Rev. B 101, 100503 (2020), URL https://link.aps.org/doi/10. 1103/PhysRevB.101.100503

    2020

  41. [52]

    Kanasugi and Y

    S. Kanasugi and Y . Yanase, Communications Physics 5, 39 (2022), ISSN 2399-3650, URL https://doi.org/10.1038/ s42005-022-00804-7

    2022

  42. [53]

    L. P. Gor’kov, J. Exptl. Theoret. Phys. 34(7) (1958)

    1958

  43. [54]

    V . L. Ginzburg and L. D. Landau, On the Theory of Super- conductivity (Springer Berlin Heidelberg, Berlin, Heidelberg, 2009), pp. 113–137, ISBN 978-3-540-68008-6, URL https: //doi.org/10.1007/978-3-540-68008-6_4

    2009 doi

  44. [55]

    Burkhardt and D

    H. Burkhardt and D. Rainer, An- nalen der Physik 506, 181 (1994), https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.19945060305, URL https://onlinelibrary.wiley.com/doi/abs/10. 1002/andp.19945060305

    1994 doi

  45. [56]

    E. Berg, E. Fradkin, S. A. Kivelson, and J. M. Tranquada, New Journal of Physics 11, 115004 (2009), URL https:// dx.doi.org/10.1088/1367-2630/11/11/115004

    2009 doi

  46. [57]

    A. A. Abrikosov, J. Exptl. Theoret. Phys. 32, 1174 (1957)

    1957

  47. [58]

    Aoki, J.-P

    D. Aoki, J.-P. Brison, J. Flouquet, K. Ishida, G. Knebel, Y . Tokunaga, and Y . Yanase, Journal of Physics: Condensed Matter 34, 243002 (2022), URL https://dx.doi.org/10. 1088/1361-648X/ac5863

    2022

  48. [59]

    S. Ran, C. Eckberg, Q.-P. Ding, Y . Furukawa, T. Metz, S. R. Saha, I.-L. Liu, M. Zic, H. Kim, J. Paglione, et al., Science 365, 684 (2019), https://www.science.org/doi/pdf/10.1126/science.aav8645, URL https://www.science.org/doi/abs/10.1126/ science.aav8645

    2019 doi

  49. [60]

    L. Jiao, S. Howard, S. Ran, Z. Wang, J. O. Rodriguez, M. Sigrist, Z. Wang, N. P. Butch, and V . Madhavan, Nature 579, 523 (2020), ISSN 1476-4687, URL https://doi.org/ 10.1038/s41586-020-2122-2

    2020 doi

  50. [61]

    Knafo, G

    W. Knafo, G. Knebel, P. Ste ffens, K. Kaneko, A. Rosuel, J.- P. Brison, J. Flouquet, D. Aoki, G. Lapertot, and S. Raymond, Phys. Rev. B104, L100409 (2021), URLhttps://link.aps. org/doi/10.1103/PhysRevB.104.L100409

    2021 doi

  51. [62]

    Gardner and X

    S. Gardner and X. Yan, Phys. Rev. D 93, 096008 (2016), URL https://link.aps.org/doi/10.1103/PhysRevD.93. 096008

    2016 doi

  52. [63]

    Kayser and A

    B. Kayser and A. S. Goldhaber, Phys. Rev. D 28, 2341 (1983), URL https://link.aps.org/doi/10.1103/ PhysRevD.28.2341

    1983

  53. [64]

    Kayser, Phys

    B. Kayser, Phys. Rev. D 30, 1023 (1984), URL https:// link.aps.org/doi/10.1103/PhysRevD.30.1023

    1984 doi

  54. [65]

    Feinberg and S

    G. Feinberg and S. Weinberg, Il Nuovo Cimento (1955-1965) 14, 571 (1959), ISSN 1827-6121, URL https://doi.org/ 10.1007/BF02726388

    1959 doi

  55. [66]

    Vagov, A

    A. Vagov, A. A. Shanenko, M. V . Miloševi ´c, V . M. Axt, V . M. Vinokur, J. A. Aguiar, and F. M. Peeters, Phys. Rev. B 93, 174503 (2016), URL https://link.aps.org/doi/10. 1103/PhysRevB.93.174503

    2016

  56. [67]

    T. T. Saraiva, A. Vagov, V . M. Axt, J. A. Aguiar, and A. A. Shanenko, Phys. Rev. B 99, 024515 (2019), URL https:// link.aps.org/doi/10.1103/PhysRevB.99.024515. 8 Figure 3: Dependence of the mean magnetic field (red curve, left scale) and the e ffective penetration length norm...

    2019 doi

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.