{"id":"28427d0f-a29f-4bab-a4b3-09089f3d9e5d","arxiv_id":"1908.08459","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Silaev, Winyard and Babaev show that the zero-current state proposed by Efremov and Ovchinnikov is not a solution of the full Ginzburg-Landau equations, so its no-spontaneous-field conclusion is invalid.","lead":"This comment claims that a recent paper concluding multiband superconductors with broken time-reversal symmetry produce no spontaneous magnetic fields is wrong. The authors show that the earlier paper's proposed state fails to satisfy one of the Ginzburg-Landau equations.","discovery_kind":"replication","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sign error in Eq. (14) invalidates the linearized inconsistency proof.","rationale":"The reader flagged the linearization as the weakest assumption and accepted the paper with moderate confidence. However, a direct algebraic check reveals a concrete error in the key linearized relation, Eq. (14). Solving the zero-current condition (13) together with p1 - p2 = ∇φ̃12 yields a coefficient with a plus sign, not the minus sign printed. This error propagates into the coefficients of Eqs. (15) and (16), which contain the same denominator u2² m1 - u1² m2. The alleged overdetermination and the explicit axial-symmetry contradiction are therefore not trustworthy as written. The paper may still be right, and the error might be repairable, but the current proof does not establish the central claim. Because the conclusion could survive a sign correction, I recommend CONDITIONAL rather than outright rejection: the manuscript should be revised to correct Eq. (14) and the subsequent derivation, and the axial-symmetry calculation must be redone with the correct relation. The reader's weakest_assumption about nonlinear terms is a separate, less immediate issue; the primary concern is the internal algebraic validity of the linearized proof.","tokens_in":28,"tokens_out":25469,"duration_ms":361402,"concrete_test":"Re-derive Eq. (14) from Eqs. (12)-(13) and the definition p1 - p2 = ∇φ̃12. The result should be ∇φ̃12 = (1 + u1² m2/(u2² m1)) p1, not the printed minus-sign version. Then recompute the linearized equations (15), (16), and (17) using the corrected relation and re-examine the axially symmetric case: if the corrected system admits a nonzero solution for some parameter set, the claimed inconsistency is an artifact of the sign error; if the corrected system still forces all fields to vanish, the conclusion survives.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central contradiction relies on the relation between the phase gradients and p1, Eq. (14). Combining Eq. (13) with p1 - p2 = ∇φ̃12 gives (u1²/m1)p1 + (u2²/m2)(p1 - ∇φ̃12) = 0, so p1 = u2² m1/(u1² m2 + u2² m1) ∇φ̃12, i.e. ∇φ̃12 = (1 + u1² m2/(u2² m1)) p1. The manuscript instead prints ∇φ̃12 = (1 - u1² m2/(u2² m1)) p1. The minus sign is not a typo: the same wrong denominator (u2² m1 - u1² m2) appears in Eqs. (15) and (16), and the algebraic structure of the overdetermined system (15)-(17) depends on this sign. With the correct plus sign, the coefficients change, and the claimed inconsistency of the linearized system for axial symmetry is no longer established. Since the paper's entire proof that no zero-current EO state exists hangs on this linearized overdetermination, the sign error is load-bearing. The nonlinearity caveat identified by the reader is secondary; the immediate problem is that the linearized equations as written are not correctly derived.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":6500,"tokens_out":15734,"duration_ms":133418,"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":[{"comment":"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":"Section III, Eq. (14)"},{"comment":"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":"Section III, Eqs. (15)–(18)"},{"comment":"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.","section":"Section III and Conclusions"}],"minor_comments":[{"comment":"Equation (19) contains 'cos(4x)' where the angular variable should be θ; this is presumably a typo but should be corrected.","section":"Section III, Eq. (19)"},{"comment":"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.","section":"Section III, Eq. (17)"},{"comment":"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.","section":"References"},{"comment":"The PACS numbers field is empty and should either be completed or removed.","section":"Title page"}],"recommendation":"major_revision","confidential_remarks":"The Comment is appropriate for the journal only if the technical proof can be repaired. The sign error in Eq. (14) is load-bearing: it breaks the derivation of the overdetermined system and the axial contradiction. I recommend major revision rather than rejection because the overdetermination idea is plausible and the error is local, but the revised manuscript must redo the linearized reduction and verify whether the contradiction survives the corrected algebra."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nI 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.\n\nI 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.\n\nWhat 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.\n\nBottom 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.","headline":"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.","tokens_in":7019,"tokens_out":16494,"would_cite":false,"duration_ms":155301,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The zero-field ground state proposed for broken-time-reversal multiband superconductors is not a solution of the Ginzburg-Landau equations.","keywords":["multiband superconductivity","time-reversal symmetry breaking","s+is state","s+id state","Ginzburg-Landau theory","spontaneous magnetic fields","zero-current constraint","anisotropic derivative coupling"],"falsifier":"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.","tokens_in":1590,"feed_emoji":"🧲","tokens_out":2111,"duration_ms":98593,"temperature":0.7,"pith_summary":"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.","feed_headline":"No zero-field ground state survives the full Ginzburg-Landau equations","feed_subtitle":"The dropped phase-difference equation leaves an overdetermined system; fields must appear near defects.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"The commented paper: supplies the model free energy, the weak-inhomogeneity setup, and the zero-field state whose validity is being tested.","marker":"1"},{"why":"Shows that anisotropy in a multiband system mixes the magnetic mode with other modes, the mechanism invoked in the comment.","marker":"11"},{"why":"Demonstrates spontaneous magnetic field generation in broken-time-reversal systems, cited as a counterexample to the no-field claim.","marker":"13"},{"why":"Provides a counterexample to both earlier no-spontaneous-field claims, supporting the comment's conclusion.","marker":"14"},{"why":"Shows how anisotropic multiband systems couple the magnetic mode with other modes, providing the basis for the angular coupling argument.","marker":"17"},{"why":"Demonstrates that different anisotropies for different components mix the magnetic mode with other modes, another pillar of the comment's reasoning.","marker":"18"},{"why":"Demonstrates the existence of spontaneous magnetic fields in broken-time-reversal systems, cited as a direct counterexample.","marker":"19"}],"fun_headline_variants":["Zero-field state fails full Ginzburg-Landau equations","Neglected phase equation voids no-field claim","Overdetermined system: no solution for zero-field state","Spurious zero-current restriction invalidates prior result"],"cache_read_input_tokens":9216,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Zero-field state fails full Ginzburg-Landau equations","Neglected phase equation voids no-field claim","Overdetermined system: no solution for zero-field state","Spurious zero-current restriction invalidates prior result"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000176,"raw_usage":{"total_tokens":1245,"prompt_tokens":856,"completion_tokens":389,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":472,"completion_tokens_details":{"reasoning_tokens":326}},"tokens_in":472,"tokens_out":389,"duration_ms":4332,"temperature":1.0,"reasoning_tokens":326,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:38:09.498440+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The commented paper: supplies the model free energy, the weak-inhomogeneity setup, and the zero-field state whose validity is being tested."},{"cited_title":"Silaev , author J","cited_arxiv_id":null,"evidence_quote":"Shows that anisotropy in a multiband system mixes the magnetic mode with other modes, the mechanism invoked in the comment."},{"cited_title":"Vadimov and author M","cited_arxiv_id":null,"evidence_quote":"Demonstrates spontaneous magnetic field generation in broken-time-reversal systems, cited as a counterexample to the no-field claim."},{"cited_title":"Magnetic signatures of domain walls in $s+is$ and $s+id$ superconductors: observability and what that can tell us about the superconducting order parameter","cited_arxiv_id":"1908.07969","evidence_quote":"Provides a counterexample to both earlier no-spontaneous-field claims, supporting the comment's conclusion."},{"cited_title":"Silaev , author T","cited_arxiv_id":null,"evidence_quote":"Shows how anisotropic multiband systems couple the magnetic mode with other modes, providing the basis for the angular coupling argument."},{"cited_title":"Winyard , author M","cited_arxiv_id":null,"evidence_quote":"Demonstrates that different anisotropies for different components mix the magnetic mode with other modes, another pillar of the comment's reasoning."},{"cited_title":"Chiral p-wave superconductors have complex coherence and magnetic field penetration lengths","cited_arxiv_id":"1905.07296","evidence_quote":"Demonstrates the existence of spontaneous magnetic fields in broken-time-reversal systems, cited as a direct counterexample."}],"review_version":1}