{"id":"06714635-ee93-4ef8-9710-4d37a3c3b16d","arxiv_id":"2411.08969","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In a phase-disordered superconductor above the BKT transition, the dc Hall conductivity is predicted to equal the ac Hall conductance C5, which is close to the normal-state anomalous Hall value.","lead":"The paper studies what happens to the anomalous Hall effect in a two-dimensional superconductor above its superconducting transition temperature, when vortices and antivortices are free to move. It argues that the measured Hall signal in this phase disordered state is set by the same ac Hall conductance as the normal and superconducting states, not by the screened electric charge of the vortex cores.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The cancellation in Eq. (17) that removes the C4 Hall contribution holds only for q→0 at fixed time; the steady-state dc limit (t→∞) leaves a finite screening-current amplitude, so the central claim dc Hall = C5 is not established by the calculation.","rationale":"The reader's weakest_assumption—the flux–antiflux analogy—is real, but the sharper problem is internal: Eq. (17) is used to claim δj_L→0 as q→0, which is true only pointwise in time. Because the amplitude of the cos(vqt) component is finite after the q^2 cancellation, the long-time dc response contains a C4−C5 correction, unless a separate dissipative mechanism is added. No such mechanism appears in the effective action Eq. (3); the action is real and non-dissipative, so the cos(cqt) term does not decay and there is no steady-state limit. This makes the central claim—dc Hall conductivity equals C5—unsupported by the calculation as written. I credit the numerical static vortex-charge calculation (Fig. 2) and the transparent derivation of σLT in Eq. (4); those parts are coherent. The paper's own wording ('beyond the validity of the formalism', 'conjecture') supports the need for a conditional verdict. The recommended verdict remains conditional, but the condition should be a proper dc/steady-state treatment or a direct time-dependent simulation, not only a microscopic derivation of the flux-vortex analogy.","tokens_in":10598,"tokens_out":28788,"duration_ms":263679,"concrete_test":"Compute the retarded Laplace response from Eq. (16) for the flux-antiflux driving field B(q,t)=2i e^{-q^2R^2} sin(v q_x t)Θ(t), take the t→∞ limit at fixed q, and then send q→0. Extract the coefficient of cos(v q_x t) in δj_L and form the ratio j_L/E_T. If this ratio is C5 + (C4−C5)c^2/(v^2−c^2) rather than C5, then the cancellation in Eq. (17) is an artifact of the short-time limit and the central claim is not established.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Even granting the flux–antiflux analogy, the central cancellation comes from taking q→0 at fixed time. In Eq. (17), for q_x=q the q^2 factors cancel, leaving δj_L = 2i(C4−C5)v c^2 e^{-q^2R^2}[cos(cqt)−cos(vqt)]/(c^2−v^2). For fixed t the bracket is O(q^2t^2), so δj_L→0; but for t→∞ at any nonzero q the amplitude is finite, and the two limits do not commute. A dc Hall conductivity is a steady-state (t→∞) quantity, so the claimed elimination of the C4 contribution rests on an order-of-limits choice, not on the physics. Moreover, the effective action Eq. (3) is non-dissipative: the cos(cqt) screening oscillation never decays, and no steady state is defined. The authors explicitly concede that 'studying vortex formation systematically is beyond the validity of the formalism in this work' and label the flux–vortex mapping a conjecture. The numerical Fig. 2 establishes only the static core-charge relation, not the dynamical cancellation. Thus the calculation shows at most that the instantaneous long-wavelength current after pair creation is C5E_T; it does not prove the dc Hall conductivity equals C5.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the anomalous Hall effect in a superconductor above the BKT transition, motivated by recent experiments in tetralayer graphene. The authors introduce a gauge-invariant effective action for a two-dimensional superconductor with response coefficients C1, C2, C4, C5, where C4 is the Streda coefficient controlling the charge response to magnetic fields and C5 is the high-frequency ac Hall coefficient. They numerically compute the charge difference between a vortex and an antivortex in a BdG model with a BHZ normal part and px+ipy pairing, showing that the charge difference is proportional to the Berry-curvature Hall conductivity. The central claim is that in the vortex plasma phase above BKT, the dc Hall conductivity is determined by C5 rather than C4, because the screening cloud of a moving vortex-antivortex pair produces a cancellation in Eq. (17) in the long-wavelength limit.","tokens_in":10842,"tokens_out":5243,"duration_ms":47538,"significance":"If the central claim were established, the paper would resolve an apparent contradiction between vortex-charge-based Hall response and the unscreened ac Hall conductivity, and would provide a concrete prediction for the phase-disordered state of a chiral superconductor. The numerical BdG calculation is a clear strength: it provides a parameter-free check of the Streda-type relation between vortex charge and Berry curvature. The effective-action expansion is also a useful framework for organizing the response coefficients. However, the main dynamical result rests on a conjectured flux-antiflux analogy and on a delicate order-of-limits; the numerical verification does not extend to the dynamical cancellation. Thus the significance is conditional on a more rigorous treatment of the dc limit.","major_comments":[{"comment":"The cancellation of the C4-screening contribution δj_L is obtained by taking q, q_x → 0 at fixed time t, since for fixed t the numerator of Eq. (17) is O(q^2 t^2). A dc Hall conductivity, however, is a long-time steady-state transport coefficient. For any nonzero q, the expression [cos(cqt) − cos(vq_x t)] in Eq. (17) does not vanish as t → ∞; it oscillates with amplitude that does not decay in that limit. Therefore the limits q → 0 and t → ∞ do not commute, and Eq. (17) establishes at most that the instantaneous long-wavelength screening current vanishes immediately after pair creation. To support the claim that the dc Hall conductivity equals C5, the authors need a proper dc limit, e.g., by including dissipation or deriving a Kubo formula with the order of limits specified.","section":"Vortex charge screening, Eq. (17)"},{"comment":"The effective action Eq. (3) is non-dissipative, as noted right after Eq. (1). Consequently, the screening response in Eq. (17) is an undamped oscillation, and the vortex plasma described by the duality relations (13)–(14) has no intrinsic mechanism to relax to a steady state. The term 'dc Hall conductivity' therefore requires an external definition of the zero-frequency limit; the manuscript does not supply one within the effective-action calculation. The authors should either introduce a dissipative term into the action or explicitly state that the result applies only to a transient response, and adjust the abstract and conclusion accordingly.","section":"Effective action of a Hall superconductor; Vortex charge screening"},{"comment":"The dynamical cancellation relies on the conjecture that a vortex-antivortex pair is equivalent to a flux-antiflux pair; the authors state that 'studying vortex formation systematically is beyond the validity of the formalism' and that the mapping is a conjecture. The numerical results in Fig. 2 validate only the static core-charge relation, not the time-dependent screening of a moving vortex. The paper should provide additional evidence for the dynamical analogy, such as a time-dependent BdG or TDGL simulation of vortex-antivortex pair creation, or alternatively temper the claim to state that the equality with C5 is a conjecture supported by the static vortex-charge relation.","section":"Conclusion; Vortex charge screening"}],"minor_comments":[{"comment":"The footnote uses first-person singular 'I' while the paper has two authors; it should be 'we'.","section":"Footnote 1"},{"comment":"The phrase 'Combining with Eq. ??' contains an unresolved equation reference; the intended equation is likely Eq. (8).","section":"Section 'Hall response of the BKT phase'"},{"comment":"The word 'Heavisider' should be 'Heaviside'.","section":"Section 'Vortex charge screening'"},{"comment":"Eq. (18) appears garbled: the vector notation is unclear (e.g., the left-hand side is missing the current symbol 'j' and the q-dependence is written in a confused way). A cleaner vector expression would improve readability.","section":"Eq. (18)"},{"comment":"The text refers to 'Appendix. A' and 'Appendix. B' but the appendices are not explicitly labeled in the manuscript; adding labels would help the reader.","section":"Appendices"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the journal's scope and the numerical vortex-charge calculation is a useful contribution. However, the central dc-Hall claim is currently not established by the calculation due to the order-of-limits issue and the conjectural nature of the flux-antiflux analogy. The authors may be able to fix this by supplying a rigorous dc limit or by reframing the claim as a transient or ac response; this is reflected in my major comments."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe thing to know: this paper makes a specific, testable claim that the dc Hall conductivity in the phase-disordered state above BKT equals the high-frequency ac Hall coefficient C5, not the screened vortex charge C4. If true, that gives a clean prediction for tetralayer graphene experiments. What is genuinely new is the screening argument: vortex core charge is set by C4 (Streda), but the screening cloud conspires to cancel that contribution in the long-wavelength transport, leaving C5. The BdG numerics in Fig. 2 are a real plus—they verify that the static vortex-antivortex charge difference tracks the Berry-phase Hall coefficient in a noninteracting model, with error bars and convergence checks.\n\nWhere it gets soft: the central cancellation in Eq. (17) depends on the order of limits. The paper concludes that the screening current vanishes \"as q, qx → 0,\" but that is q → 0 at fixed time. For a dc conductivity you need t → ∞ at finite q, and in that limit the oscillatory bracket is finite—the two limits do not commute. The effective action Eq. (3) is non-dissipative, so the cos(cqt) screening oscillation never decays and a steady-state dc limit is not actually defined within the calculation. The authors are upfront about this: they call the flux-antiflux mapping a conjecture and state that \"studying vortex formation systematically is beyond the validity of the formalism.\" So the claim \"dc Hall = C5\" is physically plausible and consistent with their scenario, but it is not proven by the calculation presented. I would also flag the broken cross-reference to Eq. (??) and a mild sign-convention ambiguity in the Hall current definition; both are minor.\n\nIs it worth referee time? Yes. The question is timely, the algebra is internally consistent, the numerical static check is reproducible, and the authors are unusually clear about what is conjecture. A serious referee should ask for either a microscopic derivation of the screening cancellation or a dynamical simulation of vortex-antivortex pairs in an interacting model—the reader's suggestion is right. The paper should not be desk-rejected, but the central claim should not be taken as established until that dynamical question is answered.\n\nRecommendation: send to peer review, with the expectation that the authors either strengthen the dynamical part or explicitly reframe the dc-equals-ac result as a conjecture supported by analogy. This is the kind of paper that benefits from a firm referee.","headline":"A testable and honest scenario for dc Hall in a phase-disordered superconductor, but the central dc-equals-ac claim rests on a conjectural flux-vortex mapping and an unresolved order-of-limits issue; worth a serious referee rather than desk rejection.","tokens_in":11370,"tokens_out":2017,"would_cite":false,"duration_ms":19586,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"In the vortex-plasma phase above the superconducting transition, the dc Hall conductivity is controlled by the ac Hall response of the superconducting and normal phases, not by the screened vortex-core charge.","keywords":["anomalous Hall effect","vortex charge","BKT transition","phase-disordered superconductor","chiral superconductivity","Berry curvature","Streda formula","flux-flow Hall effect"],"falsifier":"A numerical experiment that creates separated vortex-antivortex pairs in a microscopic model (say by solving time-dependent Bogoliubov-de Gennes or time-dependent Ginzburg-Landau equations with explicit phase winding) and extracts the dc Hall conductivity of the resulting vortex plasma would settle the claim if its result is compared with the ac coefficient $C_5$ of the same model: agreement confirms the cancellation, while a result near $C_4$ falsifies it.","tokens_in":10351,"feed_emoji":"🌀","tokens_out":12982,"duration_ms":175528,"temperature":0.7,"pith_summary":"Motivated by experiments suggesting chiral superconductivity in tetralayer graphene, this paper asks what happens to the anomalous Hall effect when a two-dimensional superconductor is heated above its Berezinskii-Kosterlitz-Thouless transition into the resistive phase-disordered state, where current is carried by a plasma of vortices and antivortices. The paper argues that the dc Hall conductivity in this state is not set by the screened charge at the vortex cores (the Streda coefficient $C_4$), but by the high-frequency ac Hall conductivity $C_5$ of the superconducting and normal phases, which are essentially the same. If correct, this means the anomalous Hall effect measured above $T_c$ directly reflects the Berry-phase Hall response of the band structure, giving a clear experimental signature of a putative chiral superconductor.","feed_headline":"Melted superconductor dc Hall equals ac Hall, not vortex charge","feed_subtitle":"The result makes normal-state Hall value a direct fingerprint of the melted chiral superconducting phase.","key_machinery":"The load-bearing object is the gauge-invariant superconducting effective action $S_{\\mathrm{eff}} = \\int [-C_1 b_0^2 + C_2 b^2 - (C_3 b - C_4(\\hat{z}\\times b))\\cdot\\nabla b_0 + C_5(\\hat{z}\\times b)\\cdot\\dot{b}]$, written in terms of the gauge-invariant fields $b_\\alpha = A_\\alpha - \\partial_\\alpha \\phi$. The coefficients $C_1,C_2$ are compressibility and superfluid stiffness; $C_4$ is the Streda response coefficient that sets the charge induced by a flux line (and hence the vortex core charge), while $C_5$ is the high-frequency ac Hall conductivity. The argument turns on the difference $C_4 - C_5$: the naive dc Hall response of the vortex plasma would follow from $C_4$, but the screening contribution to the longitudinal current, Eq. (17), is proportional to $C_4 - C_5$ and cancels in the long-wavelength limit, so the surviving Hall current is $\\vec{j}_{L,0} = C_5(\\hat{z}\\times\\vec{E}_T)$. In non-interacting systems $C_4 = C_5$ and both equal the Berry-curvature Hall value; interactions renormalize $C_4$ (by screening) but not $C_5$.","core_discovery":"The central claim is that in the phase-disordered (vortex plasma) state above the BKT transition, the dc Hall conductivity matches the ac Hall conductance $C_5$ rather than the vortex-core charge coefficient $C_4$. Using a gauge-invariant effective action for a superconductor with Hall terms $C_4(\\hat{z}\\times b)\\cdot\\nabla b_0$ and $C_5(\\hat{z}\\times b)\\cdot\\dot{b}$, the vortex-antivortex charge difference is $2C_4\\Phi_0$ (the Streda-type response), so flux-flow reasoning would predict a dc Hall effect proportional to $C_4$. However, when a flux-antiflux pair moves apart, the interaction-screening contribution to the longitudinal current is proportional to $(C_4 - C_5)$ and vanishes in the long-wavelength limit (Eq. 17), leaving the current $\\vec{j}_{L,0}=C_5(\\hat{z}\\times\\vec{E}_T)$. Thus the Hall response of the vortex plasma is controlled by the unscreened ac coefficient $C_5$, which is close to the normal-state anomalous Hall value. The paper verifies numerically, in a gapped Dirac lattice model with $p_x+ip_y$ pairing, that the vortex-antivortex charge difference is indeed tied to the Berry phase via $C_4$.","pith_inferences":["If the flux-antiflux analogy survives microscopic vortex cores, the dc-vs-ac Hall comparison offers a clean experimental protocol: measure the Hall effect in the same device in the normal state, in the gapped superconductor at microwave frequencies, and in the BKT-fluid regime, and check that the dc value tracks the ac value.","The cancellation mechanism suggests that any vortex-transport regime with efficient screening (e.g., thermally activated vortex motion near $T_c$ or in disordered films) will exhibit a dc Hall coefficient closer to $C_5$ than to $C_4$; this could be tested with time-dependent Ginzburg-Landau simulations that resolve finite-size vortex cores.","The angular dependence of the far-field vortex current (the factor $q_y/q_x$ in Eq. 18) implies logarithmic-in-system-size corrections to the dc Hall current in finite samples, which may show up as a weak sample-size dependence in small Hall bars.","The same effective action applies to any chiral superconductor, so the predicted equality dc=$C_5$ provides a way to extract the coefficient $C_5$ from transport alone, without needing the ac measurement."],"forward_implications":["In the resistive vortex-plasma phase above $T_{BKT}$, the dc anomalous Hall conductivity should equal the ac Hall conductivity of the gapped superconductor, recovering the normal-state anomalous Hall value.","Hall measurements just above the superconducting transition of a candidate chiral superconductor can therefore be compared directly with the normal-state Hall value; agreement supports the flux-vortex analogy, while a value set by $C_4$ would contradict it.","The vortex-antivortex charge asymmetry, which separately controls local charge accumulation, is set by $C_4$ and is numerically confirmed here to follow the Fermi-surface Berry phase in a $p_x+ip_y$ superconductor.","The screening charge that hides $C_4$ is not lost: it reappears as longitudinal, radially propagating charge-density (plasmon-like) crescents around the moving vortices, so the cancellation is dynamical rather than a removal of Hall physics.","Because $C_5$ is an unscreened interband response, the dc Hall of the vortex plasma is robust to weak interactions, unlike the screened vortex-core charge."],"supporting_citations":[{"why":"Supplies the gauge-invariant effective-action framework for the superconducting response from which Eq. (3) and the coefficients $C_j$ are derived.","marker":"[39]"},{"why":"Supplies the BKT vortex-plasma duality relations that turn vortex motion into a voltage and define the phase-disordered state.","marker":"[21]"},{"why":"Supplies the flux-flow model and vortex diffusion conductivity that convert vortex-and-antivortex motion into the dc Hall signal.","marker":"[22]"},{"why":"Sets the non-interacting relation between Fermi-surface Berry curvature and the anomalous Hall conductivity, which fixes $C_4=C_5$ in that limit.","marker":"[33]"},{"why":"Conjectured that vortices in chiral superconductors carry fractional charges whose difference would produce a Hall effect, the idea this paper tests and refines.","marker":"[36]"},{"why":"Establishes the chiral ac Hall response of a superconductor and identifies the coefficient that later becomes $C_5$.","marker":"[37]"},{"why":"Shows that gauge-invariant treatment suppresses the low-wave-vector chiral conductivity, the starting point for separating $C_4$ from $C_5$.","marker":"[38]"},{"why":"Gives the Streda formula connecting flux-induced charge density to the Hall coefficient, used here to relate vortex charge to $C_4$.","marker":"[42]"},{"why":"Supplies the gapped Dirac lattice model used for the numerical calculation of $p_x+ip_y$ superconductivity and vortex charge.","marker":"[43]"},{"why":"Analyzed Streda-type charge response from vortex charges in chiral $p$-wave superconductors, the analogue extended to the BKT plasma here.","marker":"[44]"}],"fun_headline_variants":["Melted superconductor dc Hall equals ac Hall, not vortex charge","Hall effect in melted superconductors: ac wins over vortices","Phase-disordered superconductor: dc Hall mirrors ac Hall","Vortex charge fails to explain Hall effect in melted superconductor","Melted superconductor's Hall response matches ac coefficient"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central premise is that a real vortex-antivortex pair in the BKT plasma behaves like a time-dependent flux-antiflux pair whose screening dynamics are captured by the linear-response effective action, a conjecture the authors explicitly flag; if finite-size vortex cores with phase winding screen charge differently, or if the $q\\to 0$ limit does not capture the true dc limit of the vortex plasma, the predicted dc-equals-ac Hall equality can fail.","fun_headline_variants_meta":{"raw":{"variants":["Melted superconductor dc Hall equals ac Hall, not vortex charge","Hall effect in melted superconductors: ac wins over vortices","Phase-disordered superconductor: dc Hall mirrors ac Hall","Vortex charge fails to explain Hall effect in melted superconductor","Melted superconductor's Hall response matches ac coefficient"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000521,"raw_usage":{"total_tokens":2546,"prompt_tokens":997,"completion_tokens":1549,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":613,"completion_tokens_details":{"reasoning_tokens":1459}},"tokens_in":613,"tokens_out":1549,"duration_ms":12411,"temperature":1.0,"reasoning_tokens":1459,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T21:12:57.582820+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A numerical experiment that creates separated vortex-antivortex pairs in a microscopic model (say by solving time-dependent Bogoliubov-de Gennes or time-dependent Ginzburg-Landau equations with explicit phase winding) and extracts the dc Hall conductivity of the resulting vortex plasma would settle the claim if its result is compared with the ac coefficient $C_5$ of the same model: agreement confirms the cancellation, while a result near $C_4$ falsifies it.","supporting_citations":[{"cited_title":"Minnhagen, Reviews of modern physics 59, 1001 (1987)","cited_arxiv_id":null,"evidence_quote":"Supplies the BKT vortex-plasma duality relations that turn vortex motion into a voltage and define the phase-disordered state."},{"cited_title":"Bardeen and M","cited_arxiv_id":null,"evidence_quote":"Supplies the flux-flow model and vortex diffusion conductivity that convert vortex-and-antivortex motion into the dc Hall signal."},{"cited_title":"Haldane, Physical review letters 93, 206602 (2004)","cited_arxiv_id":null,"evidence_quote":"Sets the non-interacting relation between Fermi-surface Berry curvature and the anomalous Hall conductivity, which fixes $C_4=C_5$ in that limit."},{"cited_title":"Goryo, Physical Review B 61, 4222 (2000)","cited_arxiv_id":null,"evidence_quote":"Conjectured that vortices in chiral superconductors carry fractional charges whose difference would produce a Hall effect, the idea this paper tests and refines."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the chiral ac Hall response of a superconductor and identifies the coefficient that later becomes $C_5$."},{"cited_title":"Roy and C","cited_arxiv_id":null,"evidence_quote":"Shows that gauge-invariant treatment suppresses the low-wave-vector chiral conductivity, the starting point for separating $C_4$ from $C_5$."},{"cited_title":"Streda, Journal of Physics C: Solid State Physics 15, L717 (1982)","cited_arxiv_id":null,"evidence_quote":"Gives the Streda formula connecting flux-induced charge density to the Hall coefficient, used here to relate vortex charge to $C_4$."},{"cited_title":"Ariad, Y","cited_arxiv_id":null,"evidence_quote":"Analyzed Streda-type charge response from vortex charges in chiral $p$-wave superconductors, the analogue extended to the BKT plasma here."}],"review_version":1}