{"id":"bc7f99e9-73d5-4a5b-a408-76389f39480d","arxiv_id":"2505.16930","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"A CFJ Lorentz-violating term in Ginzburg-Landau theory produces a modified London equation whose solutions transition from Meissner screening to anomalous vortex phases when kλL exceeds about 0.39.","lead":"The paper adds a Lorentz-violating Carroll-Field-Jackiw term to the Ginzburg-Landau model of superconductivity and derives a modified London equation. Numerical solutions show that, as the new term grows, the ordinary Meissner state gives way to phases with in-plane fields, vortices and an anti-vortex.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sign error in Eq. (7)/(10): the supercurrent term has the wrong sign, so the derived London equation (14) is not a consequence of the free energy; the numerical phase diagram is unsupported.","rationale":"Reader already rejected the paper, and the same central defect is present in the derivation: the current equation has the wrong sign. The reader's formal weakest_assumption (dropping spatial components of k_AF and unspecified boundary conditions) is real but secondary; the sign error alone prevents Eq. (14) from being a consequence of the stated free energy (3). The numerical results cannot rescue the central claim because they solve an equation that has not been derived. A corrected derivation might still yield exotic solutions, but the paper's central claim as written is unsupported. Therefore the REJECT verdict is unchanged, and I partially agree with the reader's framing (the rationale identifies the sign error, though the formal weakest_assumption does not).","tokens_in":11419,"tokens_out":18808,"duration_ms":139455,"concrete_test":"Analytical check: substitute the standard 1D Meissner profile B(x)=B0 e^{-x/λ_L} ẑ, A(x)=-λ_L B0 e^{-x/λ_L} ŷ into Eq. (10) with k=0. Since ∇×∇×A = -(1/λ_L²)A, the equation is satisfied only with a negative sign on the supercurrent term; the written positive sign makes Eq. (10) false, so Eq. (14) does not follow.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing concern is a sign error in the current equation, which invalidates the derivation of Eq. (14). Minimizing the free energy (3) with respect to A for constant Ψ gives δF/δA = (1/4π)∇×∇×A + 2K(2e/ℏc)²|Ψ|²A = 0, so the correct relation is (1/4π)∇×∇×A = -2K(2e/ℏc)²|Ψ|²A. Equation (7) and its constant-k version (10) have the opposite sign on the supercurrent term. Taking the curl of Eq. (10) using ∇×∇×∇×A = -∇²B and ∇×A = B therefore yields ∇²B = -(1/λ_L²)B - 2k∇×B, not the positive-sign Eq. (13) or (14). The inconsistency is visible without numerics: for k=0 the standard Meissner profile B=B0 e^{-x/λ_L}ẑ, A=-λ_L B0 e^{-x/λ_L}ŷ gives ∇×∇×A = -(1/λ_L²)A, opposite to the positive coefficient in Eq. (10). Since Eq. (14) is the basis for the reported threshold kλ_L ≈ 0.39 and the anti-vortex phases, the central claim is not supported by the derivation as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11756,"tokens_out":4804,"duration_ms":37610,"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":[{"comment":"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.","section":"Eqs. (7)-(10) and (13)-(14)"},{"comment":"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.","section":"Paragraph before Eq. (2)"},{"comment":"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.","section":"Numerical results, Figs. 2-3"}],"minor_comments":[{"comment":"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?","section":"Text after Eq. (6)"},{"comment":"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.","section":"Fig. 2 caption and Sec. III"},{"comment":"The first sentence of the Conclusion contains a typo: 'In summa,' should be 'In summary,'.","section":"Conclusion"},{"comment":"References [9] and [10] are assigned the same URL; the bibliographic details should be corrected.","section":"References [9] and [10]"}],"recommendation":"reject","confidential_remarks":"The sign error in the current equation is load-bearing and invalidates the main result. The numerical claims are also not verifiable from the main text without boundary conditions and convergence information. This is not a case where a local fix would leave the central claim intact; the derived London equation, the threshold, and the phase diagram all depend on the sign of the CFJ contribution. A corrected derivation and more complete numerical reporting would be needed before reconsideration."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know that the paper's central claim does not follow from its own free energy. There's a sign error in the current equation: for k=0, Eq. (10) gives ∇²B = -(1/λ_L²)B, which contradicts their quoted London equation (11). Varying the free energy (3) with respect to A gives a minus sign on the 2K(2e/ℏc)²|Ψ|²A term. Taking the curl correctly leads to ∇²B = (1/λ²)B - 2k∇×B, not the plus sign in Eq. (13)/(14). The stress-test note is right. So the threshold kλ_L≈0.39 and the anti-vortex phase are unsupported as written.\n\nWhat is new: adding the CFJ vector to GL and studying the London limit is a direct extension of the SME-GL program, and I don't think the scalar-sector papers did this. The dimensionless coupling kλ_L is a clean single parameter, and the numerical plots do suggest a transition away from Meissner behavior if the equation is corrected. The paper is also honest about the toy-model status; the UTe2 connection is explicitly speculative and not used to fit anything.\n\nThe soft spots are proportional. The sign error is load-bearing, not cosmetic. The numerics lack convergence checks, error bars, and stated boundary conditions; the supplemental material is referenced but not provided. The static-limit reduction to only (k_AF)^0 is asserted without justifying the gauge choice or dropping spatial components. These are fixable, but they are not minor.\n\nWho gets value: someone working on Lorentz-violating versions of GL might want to see the corrected derivation and re-run the numerics. This version is not citable for the phase diagram. It does deserve a serious referee, mainly to force the correction and the numerical details.\n\nMy recommendation: engage with it on revision. Ask for a corrected current equation, a check of the k sign in the London equation, and reproducible numerics. As is, I'd reject.","headline":"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.","tokens_in":12221,"tokens_out":19477,"would_cite":false,"duration_ms":141627,"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":"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.","keywords":["superconductivity","Ginzburg-Landau theory","London equation","Lorentz violation","Carroll-Field-Jackiw term","Meissner effect","vortices","UTe2"],"falsifier":"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.","tokens_in":1708,"feed_emoji":"🧲","tokens_out":1965,"duration_ms":60992,"temperature":0.7,"pith_summary":"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.","feed_headline":"Above a threshold, a Lorentz-violating term kills the Meissner effect","feed_subtitle":"A modified London equation predicts in-plane fields and a central anti-vortex for strong coupling.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines the Carroll-Field-Jackiw coefficient (k_{AF})^\\mu and its Lorentz/CPT-violating couplings, the term this paper adds to the free energy.","marker":"[3]"},{"why":"Provides the method of building a modified Ginzburg-Landau theory from Lorentz-violating scalar sector terms, which this paper extends to the gauge sector.","marker":"[34]"},{"why":"Earlier application of Lorentz-inspired Ginzburg-Landau models to anisotropic superconductors, supplying the comparative baseline for the new CFJ-type term.","marker":"[35]"},{"why":"Standard reference for the Ginzburg-Landau and London equations from which the modified equation is derived.","marker":"[47]"},{"why":"Standard reference for superconductivity phenomenology and characteristic lengths used in the derivation.","marker":"[48]"},{"why":"Review of UTe2 as a candidate spin-triplet, stratified superconductor, the material context for the proposed signatures.","marker":"[58]"}],"fun_headline_variants":["Lorentz-violating term induces anti-vortex superconductivity","Carroll-Field-Jackiw term flips superconductivity's response","Lorentz violation beyond threshold spawns anti-vortices","Strong Lorentz violation leads to anti-vortex and currents"],"cache_read_input_tokens":14336,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Lorentz-violating term induces anti-vortex superconductivity","Carroll-Field-Jackiw term flips superconductivity's response","Lorentz violation beyond threshold spawns anti-vortices","Strong Lorentz violation leads to anti-vortex and currents"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000205,"raw_usage":{"total_tokens":1374,"prompt_tokens":910,"completion_tokens":464,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":393}},"tokens_in":526,"tokens_out":464,"duration_ms":4458,"temperature":1.0,"reasoning_tokens":393,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:54:17.656020+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Colladay and V","cited_arxiv_id":null,"evidence_quote":"Defines the Carroll-Field-Jackiw coefficient (k_{AF})^\\mu and its Lorentz/CPT-violating couplings, the term this paper adds to the free energy."},{"cited_title":"Furtado, R","cited_arxiv_id":null,"evidence_quote":"Provides the method of building a modified Ginzburg-Landau theory from Lorentz-violating scalar sector terms, which this paper extends to the gauge sector."},{"cited_title":"Araújo, I","cited_arxiv_id":null,"evidence_quote":"Earlier application of Lorentz-inspired Ginzburg-Landau models to anisotropic superconductors, supplying the comparative baseline for the new CFJ-type term."},{"cited_title":"Tinkham, Introduction to Superconductivity (McGraw-Hill, 1996), 2nd ed","cited_arxiv_id":null,"evidence_quote":"Standard reference for the Ginzburg-Landau and London equations from which the modified equation is derived."},{"cited_title":"De Gennes, Superconductivity Of Metals And Alloys , Advanced Books Classics Series (Westview Press, 1999), ISBN 9780813345840, URL https://books.google.com","cited_arxiv_id":null,"evidence_quote":"Standard reference for superconductivity phenomenology and characteristic lengths used in the derivation."},{"cited_title":"Aoki, J.-P","cited_arxiv_id":null,"evidence_quote":"Review of UTe2 as a candidate spin-triplet, stratified superconductor, the material context for the proposed signatures."}],"review_version":1}