Comment on the paper by D. Efremov and Yu.N. Ovchinnikov "Singular ground state of multiband inhomogeneous superconductors", Phys. Rev. B 99, 224508 (2019)

Egor Babaev; Mihail Silaev; Thomas Winyard

REVIEW 3 major objections 4 minor 20 references

Comment on the paper by D. Efremov and Yu.N. Ovchinnikov "Singular ground state of multiband inhomogeneous superconductors", Phys. Rev. B 99, 224508 (2019)

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The zero-field ground state proposed for broken-time-reversal multiband superconductors is not a solution of the Ginzburg-Landau equations.

desk verdict The comment raises a legitimate structural objection to Efremov-Ovchinnikov, but the linearized proof has a load-bearing sign error in Eq. (14) and is not reliable as written. read the letter →

arxiv 1908.08459 v1 pith:U34YAXVJ submitted 2019-08-22 cond-mat.supr-con

Mihail Silaev , Thomas Winyard , Egor Babaev This is my paper · ORCID

classification cond-mat.supr-con

keywords multibandsuperconductivitytime-reversalsymmetrybreakings+isstates+idGinzburg-Landautheoryspontaneousmagneticfieldszero-currentconstraintanisotropicderivativecoupling

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 comment targets a recent claim that inhomogeneous multiband superconductors with broken time-reversal symmetry cannot host spontaneous magnetic fields. The comment argues that the zero-field state proposed in that work is not a stationary solution of the two-component Ginzburg-Landau free energy for an s+id superconductor. The earlier work imposed a zero-current restriction but dropped the Euler-Lagrange equation for the phase difference between the two condensate components, creating an overdetermined system. When all equations are kept and the fields are linearized around the uniform broken-time-reversal ground state, the only consistent solution for a radial impurity has no fluctuations at all, so the assumed defect and its zero-field state cannot coexist. If the argument holds, spontaneous magnetic fields near impurities, domain walls, and fluctuations are generic in s+is, s+id, and p+ip superconductors.

What carries the argument

The load-bearing object is the full set of Ginzburg-Landau stationarity equations, in particular the Euler-Lagrange equation for the relative phase, which the commented paper omitted and replaced with the zero-current condition. The argument also relies on the anisotropy tensor $K_\alpha$ with $K_x=-K_y=1$ in the derivative coupling, which couples the two condensate gradient modes and produces the angular harmonics that make the linearized problem inconsistent. In the linearized system the zero-current restriction splits into two constraints, and with the phase-difference equation the three equations for the two fluctuating fields are overdetermined; elimination and angular-harmonic analysis force the fluctuations to zero.

What would settle it

Solve the full nonlinear Ginzburg-Landau equations, without linearization, for the same s+id model with a small radially symmetric impurity and search numerically for a stationary, finite-energy solution with zero supercurrent throughout. If such a solution exists for any nonzero impurity strength, the blanket claim that the zero-field state does not exist would be falsified.

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Extended reading notes

Core claim

In the model defined by the two-component Ginzburg-Landau free energy, a valid equilibrium must satisfy all variational equations: the two amplitude equations, current conservation, and the Euler-Lagrange equation for the phase difference. The commented paper's zero-field state satisfies the amplitude equations and the current-conservation equation but not the phase-difference equation, which was replaced by the extra restriction of zero current. At linear order the zero-current restriction splits into two constraints, and together with the phase-difference equation these form three independent equations for only two fluctuating fields. For a radially symmetric impurity the angular structure of the equations forces the phase fluctuation to vanish, and then forces all fluctuations to vanish unless the impurity coefficients themselves vanish, contradicting the assumed presence of a defect. The comment concludes that the proposed zero-field state does not exist and that the earlier no-spontaneous-field conclusion is incorrect.

Load-bearing premise

The proof assumes the linearized equations capture the full content of the stationarity conditions; for a large impurity or with nonlinear terms retained, the three equations could in principle have a common solution, so the no-zero-field conclusion is rigorously established only in the weak-perturbation regime.

Editorial extensions

If this is right

If the argument is correct, any inhomogeneity in a broken-time-reversal multiband superconductor, whether an impurity, a domain wall, or a thermal fluctuation, generically produces a magnetic response.
The zero-field ground state reported in the commented paper cannot serve as a basis for interpreting experiments, so observations of spontaneous magnetic fields in candidate materials remain consistent with time-reversal symmetry breaking.
Zero-field configurations, if any exist, require fine-tuned impurity profiles and occupy a zero-measure set of parameter space.
The full set of Euler-Lagrange equations, including the phase-difference equation, must be checked before a constraint-based ansatz is identified as a ground state.

Reading between the lines

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

The same overdetermination mechanism should appear for non-radial impurities and for s+is and p+ip analogues, since the angular harmonic coupling that kills the radial solution is not special to a circular defect; a direct extension would be to repeat the proof for an impurity with elliptical symmetry.
This comment implicitly predicts that accurate numerical minimization of the same free energy for a single small impurity will find an equilibrium with nonzero supercurrent and magnetic field, and that any numerical work claiming a zero-field ground state should be checked against the phase-difference equation residual.
More generally, the episode illustrates a methodological caution: replacing an Euler-Lagrange equation with an extra physical constraint before deriving the stationarity conditions can manufacture spurious stationary points.

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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

Quick take: the structural criticism of EO is worth taking seriously, but the proof as written has a sign error in the key relation, so the comment does not currently establish its conclusion.

The new thing here is the observation that EO imposed j=0 and dropped δF/δφ12=0. That is a real and important point; if correct, it would restore spontaneous fields in BTRS multiband systems and undercut a published counterexample. The authors also give a concrete axial-symmetric demonstration, which is the right way to make the contradiction tangible.

I checked the algebra, and the stress-test note is right. From their own (13) you get p2 = -(u1^2 m2)/(u2^2 m1) p1. Since ∇φ12 = p1 - p2, Eq. (14) should read (1 + u1^2 m2/(u2^2 m1)) p1, not with a minus. The minus propagates into the denominators of (15)-(17), so the overdetermined system as printed is not correctly derived. The subsequent angular argument therefore does not go through as written. This is load-bearing, not a typo: the whole proof of inconsistency hangs on that relation.

I also have a question about the step from j=0 to (12)-(13). Linearly, j=0 gives a = -K b (with a from the p-terms and b from the amplitude gradients), not a=0 and b=0. So (12) and (13) look like extra restrictions unless there is a further argument not stated. If they are extra restrictions, the proof only shows inconsistency of a restricted class, not of the EO state.

What is good: the paper is short, the notation is mostly clear, and the identification of the neglected GL equation is a genuine structural criticism. The citations are fine; self-citations are to prior work on spontaneous fields, which is appropriate context.

Bottom line: this is for people working on multiband superconductivity and spontaneous magnetic fields. The idea deserves a careful referee, but the manuscript needs substantial correction before it can serve as a reliable refutation of EO. If I were the editor I would send it out, with the expectation that the authors redo the linearization carefully.

Referee Report

3 major / 4 minor

Summary. The manuscript is a Comment on Efremov and Ovchinnikov (EO), Phys. Rev. B 99, 224508 (2019). It argues that EO's conclusion—that inhomogeneous multiband time-reversal-symmetry-breaking superconductors have no spontaneous magnetic fields—is incorrect, because EO imposed the zero-current condition j=0 while neglecting the independent relative-phase Ginzburg-Landau equation δF/δϕ12=0. The authors linearize the current and GL equations about a uniform s+id ground state, obtain three equations, Eqs. (15), (16) and (17), for two unknown fields, and exhibit a contradiction for an axially symmetric weak impurity. They conclude that the EO state is not a solution of the GL equations and that fluctuations, defects, or domain walls necessarily generate a magnetic response.

Significance. If the proof could be repaired, the Comment would address a real controversy by showing, within the target paper's own model and notation, that the zero-field EO state is overconstrained. The argument is transparent, has no fitted parameters, and is explicitly falsifiable through the linearized equations. The main strength is that the inconsistency is demonstrated inside the same two-band GL framework used by EO, rather than by invoking a different model. However, the current manuscript contains a central algebraic error in Eq. (14) that invalidates the derivation of Eqs. (15)–(18), so the significance is conditional on a corrected proof.

major comments (3)

[Section III, Eq. (14)] Equation (14) contains a sign error that invalidates the subsequent linearized proof. From p1−p2=∇φ̃12 and the linear zero-current condition (13), one obtains (u1²/m1+u2²/m2)p1=(u2²/m2)∇φ̃12, hence ∇φ̃12=(1+u1²m2/(u2²m1))p1. The printed minus sign is not merely a cosmetic variant: substituting ∇φ̃12=(1−u1²m2/(u2²m1))p1 back into Eq. (13) leaves 2(u1²/m1)p1=0, so the printed relation can hold only for p1=0. Since Eqs. (15)–(17) are derived from Eq. (14), the overdetermined system and the axial-symmetry contradiction are not established as written.
[Section III, Eqs. (15)–(18)] Because the erroneous relation in Eq. (14) is used to eliminate p1 and p2, the error propagates into the coefficients of the Kα∇²αφ̃12 terms in the reduced amplitude equations. With the corrected plus sign, the two amplitude equations do not have the same K-term coefficient, so subtracting them does not remove the φ̃12 dependence and the radial form of Ψ̃1 in Eq. (18) does not follow. The axial-impurity argument in Eqs. (18)–(22) is therefore not a consequence of the stated linearized equations. A revised proof must redo the reduction with the correct relation and re-examine whether the overdetermination persists.
[Section III and Conclusions] The proof is explicitly perturbative: all fields are expanded to first order in δαi and in the fluctuations, and the inconsistency is demonstrated only for weak inhomogeneities. The abstract and conclusions state the stronger claim that no zero-current EO state exists in general. Please restrict the claims to the weak-perturbation regime, or supply a nonlinear argument that excludes the possibility of a solution at finite δαi.

minor comments (4)

[Section III, Eq. (19)] Equation (19) contains 'cos(4x)' where the angular variable should be θ; this is presumably a typo but should be corrected.
[Section III, Eq. (17)] In Eq. (17) the denominator 'm1u2² − m2u2²' appears to have an index error: the second term should probably involve u1² rather than u2²; please check the derivation.
[References] Reference 1 lists the target paper with author order 'Y.N. Ovchinnikov and D. Efremov,' while the title and text use 'D. Efremov and Yu.N. Ovchinnikov'; please verify consistency with the published record.
[Title page] The PACS numbers field is empty and should either be completed or removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the comment's inconsistency proof is derived from the target paper's own Ginzburg–Landau functional and contains no fit-renamed-as-prediction or load-bearing self-citation.

full rationale

This Comment does not fit a parameter, rename a known result, or import a uniqueness theorem from the authors' prior work. Its central claim—that the Efremov–Ovchinnikov zero-current, zero-field state is not a solution of the full Ginzburg–Landau equations—is established by writing down the complete set of variational equations (9), (10) and (11) for the same functional used in Ref. 1, linearizing around the uniform ground state, and showing that the zero-current restriction (13) together with the phase equation (11) overdetermines the fluctuations. The steps are explicit algebraic deductions from the model, not inputs recycled as outputs: Eq. (14) is derived within the paper from the linearized current conditions, and Eqs. (15)–(17) are the corresponding reduced GL equations. No quantity is fitted to a subset of data and then 'predicted'; no claim rests on a self-citation. The authors do cite their own earlier papers in the introduction to motivate the existence of spontaneous magnetic fields, but those citations are contextual and the explicit inconsistency proof in Section III is self-contained and would stand or fall on its own algebra. A possible algebraic objection to Eq. (14) would be a correctness concern, not circularity: even if the sign were wrong, the argument would be invalid rather than equivalent to its assumptions. Since the derivation chain is not circular, the appropriate score is 0.

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

No fitted parameters are introduced; the proof operates with the model parameters and ground-state values inherited from the paper under comment. The axioms are physical modeling assumptions and the weak-inhomogeneity linearization. No new entities are postulated.

assumptions (3)

domain assumption A physical equilibrium is a critical point of the Ginzburg-Landau free energy with respect to |Psi_k| and phi_12, so both the amplitude equations and the phase-difference equation must hold.
Used in Section II.B to define the full GL equations (9), (10) and (11) that any ground state must satisfy.
domain assumption The impurity potential is weak, with |delta_alpha_i| much smaller than |alpha_i^(0)|, allowing linearization around the uniform ground state.
Invoked at the start of Section III to expand all fields and neglect quadratic and higher-order terms.
domain assumption The uniform ground state has |Psi_k| = u_k, phi_12 = pi/2 and p_k = 0.
This is the natural uniform s+id ground state used to linearize around, stated in Section III before the fluctuation expansion.

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Cite this review

Pith. "Pith review of Comment on the paper by D. Efremov and Yu.N. Ovchinnikov "Singular ground state of multiband inhomogeneous superconductors", Phys. Rev. B 99, 224508 (2019)." pith.science (2026). https://pith.science/paper/U34YAXVJ

@misc{pith2026190808459,
  author       = {Pith},
  title        = {Pith review of: Comment on the paper by D. Efremov and Yu.N. Ovchinnikov "Singular ground state of multiband inhomogeneous superconductors", Phys. Rev. B 99, 224508 (2019)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U34YAXVJ}},
  note         = {Machine review of arXiv:1908.08459}
}

read the original abstract

We show that the conclusion reported in Ref. 1, that there are no spontaneous magnetic fields in multiband superconductors that break time reversal symmetry, is incorrect. We demonstrate that the state proposed in Ref. 1 is not a solution of the Ginzburg-Landau equations for the considered model. The reason is that in Ref. 1 one of the Ginzburg-Landau equations is neglected and substituted by the spurious zero current restriction. This restriction together with all of the Ginzburg-Landau equations leads to an overdetermined system which does not have a solution. This inconsistency invalidates all the results of the paper.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

20 extracted references · 20 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

author Y. N. Ovchinnikov and author D. Efremov , journal Physical Review B volume 99 , pages 224508 ( year 2019 )

work page 2019

[2] [2]

Grinenko , author P

author V. Grinenko , author P. Materne , author R. Sarkar , author H. Luetkens , author K. Kihou , author C. H. Lee , author S. Akhmadaliev , author D. V. Efremov , author S.-L. Drechsler , and author H.-H. Klauss , journal Phys. Rev. B volume 95 , pages 214511 ( year 2017 ), ://link.aps.org/doi/10.1103/PhysRevB.95.214511

work page doi:10.1103/physrevb.95.214511 2017

[3] [3]

Emerging superconductivity with broken time reversal symmetry inside a superconducting $s$-wave state

author V. Grinenko , author R. Sarkar , author K. Kihou , author C. Lee , author I. Morozov , author S. Aswartham , author B. B \"u chner , author P. Chekhonin , author W. Skrotzki , author K. Nenkov , et al. , journal arXiv preprint arXiv:1809.03610 ( year 2018 )

work page Pith review arXiv 2018

[4] [4]

Lee , author S.-C

author W.-C. Lee , author S.-C. Zhang , and author C. Wu , journal Phys. Rev. Lett. volume 102 , pages 217002 ( year 2009 ), ://prl.aps.org/abstract/PRL/v102/i21/e217002

work page 2009

[5] [5]

Maiti and author A

author S. Maiti and author A. V. Chubukov , journal Phys. Rev. B volume 87 , pages 144511 ( year 2013 ), ://link.aps.org/doi/10.1103/PhysRevB.87.144511

work page doi:10.1103/physrevb.87.144511 2013

[6] [6]

Carlstr\"om , author J

author J. Carlstr\"om , author J. Garaud , and author E. Babaev , journal Phys. Rev. B volume 84 , pages 134518 ( year 2011 ), ://prb.aps.org/abstract/PRB/v84/i13/e134518

work page 2011

[7] [7]

Stanev and author Z

author V. Stanev and author Z. Te s s anovi c \' c , journal Phys. Rev. B volume 81 , pages 134522 ( year 2010 ), ://prb.aps.org/abstract/PRB/v81/i13/e134522

work page 2010

[8] [8]

B\"oker , author P

author J. B\"oker , author P. A. Volkov , author K. B. Efetov , and author I. Eremin , journal Phys. Rev. B volume 96 , pages 014517 ( year 2017 ), ://link.aps.org/doi/10.1103/PhysRevB.96.014517

work page doi:10.1103/physrevb.96.014517 2017