{"id":"6bbb8a86-778c-41d2-9fc7-6db5602f0d9d","arxiv_id":"2505.23278","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"In gapless superconductors, a longitudinal normal current can convert into a transverse supercurrent via quasiparticle Berry curvature.","lead":"This paper predicts a new Hall effect in gapless superconductors, where a longitudinal current of ordinary electrons is converted sideways into a lossless supercurrent. The effect is traced to the quantum geometry, or Berry curvature, of the material's electron bands.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (7)'s Berry-curvature term is the standard quasiparticle anomalous Hall current; the paper neither derives its identification as a Cooper-pair supercurrent nor separates supercurrent from quasiparticle current in the four-probe transport calculation.","rationale":"The paper proposes a novel effect: conversion of a longitudinal normal current into a transverse supercurrent in a gapless superconductor via quasiparticle Berry curvature. The central equation, Eq. (7), has two terms: a Drude term and a Berry-curvature term -f_n E × Ω_n. The authors identify the latter as the dissipationless transverse supercurrent. This is the most load-bearing assertion, and it is also the reader's selected weakest assumption. My independent reading confirms that the identification is not supported by the text. The expression -f_n E × Ω_n is exactly the standard intrinsic anomalous Hall conductivity for Bloch/BdG quasiparticles; in the normal-state limit it describes a quasiparticle anomalous Hall current. A true supercurrent response to a dc electric field is the London inductive term, which is singular at zero frequency and arises from the diamagnetic kernel, not from the Berry-curvature contribution. The paper's thermodynamic derivation of Eq. (7) is relegated to an absent supplement, so the crucial step—whether the Berry term comes from the Cooper-pair momentum dependence of the spectrum—cannot be checked. Furthermore, the four-probe transport calculation computes a total charge current via Eq. (9) without separating the normal quasiparticle channel from the Andreev pair channel, so it cannot establish that the transverse current is a supercurrent. The proposed concrete test—decomposing the transverse current into diagonal and off-diagonal Green's-function contributions—would settle this. If the off-diagonal (Andreev) component dominates, the supercurrent interpretation is credible; if the diagonal component dominates, the effect reduces to a quasiparticle Hall effect and the paper's title claim is refuted. Because the manuscript currently lacks both the derivation and the decomposition, the reader's CONDITIONAL verdict is appropriate; no change is needed.","tokens_in":9198,"tokens_out":14107,"duration_ms":171043,"concrete_test":"Reproduce the four-probe calculation of Fig. 3(f) and decompose the transverse current I3 (or I4) in Eq. (9) into the diagonal contribution (G^r_ee and G^r_hh terms, quasiparticle current) and the off-diagonal contribution (G^r_eh, G^r_he terms, Andreev/pair current). If the transverse current is dominated by the off-diagonal (Andreev) channel, the supercurrent interpretation is supported; if it is dominated by the diagonal normal channel, the effect is a quasiparticle anomalous Hall current and the central claim is refuted. Also run the same decomposition for a normal-state control (Δ=0) and for a gapped superconductor (Δ=0.01t) to establish the crossover.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the second term of Eq. (7), -f_n E × Ω_n, is a dissipationless transverse supercurrent produced by quasiparticle Berry curvature. This identification is load-bearing and is asserted rather than derived. Equation (7) is the standard Boltzmann/semiclassical expression for the intrinsic anomalous Hall effect of Bogoliubov quasiparticles; the same term with Δ=0 gives a normal-state anomalous Hall current in a time-reversal-broken metal. A dc electric field in a superconductor generates a supercurrent through the inductive London kernel (a δ-function at ω=0, or 1/ω in ac), not through a finite dc Berry-curvature conductivity. The thermodynamic derivation from J=∂F/∂A is delegated to a supplement that is not present in the reviewed artifact, so we cannot verify whether the Berry term originates from the dependence of the quasiparticle spectrum on the Cooper-pair momentum (which would justify a supercurrent reading) or from the ordinary quasiparticle band geometry (which would make it a quasiparticle Hall current). The four-probe NEGF calculation of Eq. (9) computes the total charge current in the transverse leads and provides no decomposition into the normal (electron–hole diagonal) and pair (Andreev, off-diagonal) channels. Without such a decomposition, or an independent derivation, the paper's central claim 'dissipationless transverse supercurrent' is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a new 'superconducting Hall effect' (ScHE) in a two-dimensional gapless superconductor described by the BdG Hamiltonian in Eq. (2), with Rashba spin-orbit coupling, an in-plane Zeeman field, and hexagonal warping. It argues that in the segmented Fermi-surface regime, a longitudinal normal current driven by a small electric field is converted into a transverse dissipationless supercurrent via the quasiparticle Berry curvature. The central formula is Eq. (7), whose second term, -fn E × Ωn, is interpreted as the intrinsic supercurrent contribution. The claim is supported by (i) a thermodynamic derivation from J = ∂F/∂A, (ii) two-probe N-S transport calculations reproducing the segmented Fermi-surface transmission features, and (iii) a four-probe NEGF calculation showing a transverse current that depends on the magnetic-field orientation and the warping strength. An experimental realization in Bi2Te3/NbSe2 heterostructures is proposed.","tokens_in":9599,"tokens_out":7236,"duration_ms":81521,"significance":"If the identification of the Berry-curvature term as a Cooper-pair supercurrent is correct, the ScHE would be a conceptually new transport phenomenon, distinct from previously studied superconducting Hall and diode effects. It would also be a concrete case where the band geometry of Bogoliubov quasiparticles controls the condensate current, and it yields a falsifiable angular dependence (JH = 0 at θ = 0 and for λ = 0). The paper makes a useful connection to the experimentally realized gapless superconductor Bi2Te3/NbSe2 and proposes a specific measurement geometry. However, the central claim is presently an interpretation layered on Eq. (7), and the key derivation is not contained in the main text; the significance is therefore conditional on the authors closing that gap.","major_comments":[{"comment":"Equation (7) is the load-bearing result, but its derivation is not in the main text; the text says 'The derivation detail is provided in the Supplementary Material.' The main text also states that Eq. (3) is suitable for equilibrium systems, then replaces fn by gn to capture the driven system. This out-of-equilibrium step is asserted rather than derived. Because the second term of Eq. (7) is the entire basis for the ScHE, the authors must either present the derivation in the main text or provide a rigorous justification for applying the equilibrium thermodynamic expression to a steady state with a finite electric field. Please clarify whether the Berry term originates from the dependence of the quasiparticle spectrum on the Cooper-pair momentum or from the ordinary band geometry; these two origins lead to different physical interpretations.","section":"Theoretical framework, Eq. (7)"},{"comment":"The identification of -fn E × Ωn as a 'dissipationless transverse supercurrent carried by finite-momentum Cooper pairs' is not established. In the Δ→0 limit, the same expression is the standard intrinsic anomalous Hall current of quasiparticles in a time-reversal-broken metal, and a dc electric field in a superconductor normally produces the inductive London response rather than a finite dc Berry-curvature conductivity. The paper needs to show that the transverse current is carried by the pair (off-diagonal) component of the BdG current operator. A concrete way would be to derive Eq. (7) from the microscopic current operator and separate the normal (electron-hole diagonal) and anomalous (Andreev off-diagonal) contributions. Without such a separation, the central claim is unsupported.","section":"Eq. (7), second term"},{"comment":"The four-probe calculation computes the total transverse charge current I3 (or I4) through the Landauer-Büttiker formula; it does not decompose this current into quasiparticle and Cooper-pair contributions. The sentence 'Eq. (9) can also calculate both normal current and supercurrent' is therefore not demonstrated by the presented results. Figure 3(f) shows that a transverse current appears with the expected angular dependence, but it does not establish that this current is a supercurrent. Please provide a channel-resolved decomposition, or a complementary calculation such as the dependence of IH on Δ at fixed θ, to distinguish a supercurrent from a quasiparticle Hall current.","section":"Quantum transport formalism, Eq. (9), Fig. 3(f)"}],"minor_comments":[{"comment":"Typos include 'suppercurrent' in the Fig. 3(f) caption, 'B filed' below Eq. (1), 'gaped' in the introduction, 'the the' in the Mechanism section, 'tranverse' in the four-probe paragraph, and 'identity' for 'identify' in the Conclusion.","section":"Throughout"},{"comment":"The main text does not specify the integration domain, the temperature, or the definition of t for the JH calculation, so the quantitative JH values cannot be reproduced from the main text alone. These parameters should be stated explicitly.","section":"Fig. 1(f)"},{"comment":"The text repeatedly refers to the Supplementary Material for essential details, including the derivation of Eq. (7), the symmetry argument for Fig. 2, and the circular-disc measurement setup. For a self-contained manuscript, please ensure that all load-bearing material appears in the main text or is available to the reviewers.","section":"Supplementary Material references"},{"comment":"The electron charge e is not shown in Eq. (7); please state the convention (e = 1) or restore the explicit charge factors in the Drude and Berry terms.","section":"Eq. (7)"}],"recommendation":"major_revision","confidential_remarks":"The central issue is the gap between the standard anomalous-Hall form of Eq. (7) and the claimed supercurrent interpretation. I am not recommending rejection because the thermodynamic formalism might be valid if properly derived and if the current decomposition is provided; however, as the manuscript stands, the key step cannot be verified by the reader. The missing Supplementary Material is essential, not cosmetic, and should be included in any resubmission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper predicts a superconducting Hall effect (ScHE) in a gapless superconductor with segmented Fermi surfaces: a longitudinal normal current gets deflected by quasiparticle Berry curvature into a transverse supercurrent. The novelty is applying the known Berry-curvature transport formula to this specific superconducting state, and the authors back it with a concrete model, clear Fermi-surface plots, and two calculations (thermodynamic and four-probe NEGF) that agree qualitatively. The experimental proposal using Bi2Te3/NbSe2 is concrete and testable. That part is genuinely useful.\n\nThe soft spot is the identification of the transverse term in Eq. (7) as a supercurrent. Eq. (7) is the standard semiclassical anomalous Hall expression for quasiparticles; with Δ=0 it gives the normal-state AHE. The paper says the second term “induces a dissipationless transverse supercurrent,” but that label is never derived. The thermodynamic derivation from J=∂F/∂A is delegated to a supplement we don't have, and the jump from equilibrium free energy to a nonequilibrium distribution in the relaxation-time approximation is hand-waved. The NEGF four-probe calculation computes total charge current through the transverse leads; there is no decomposition into normal (electron-hole) versus pair (Andreev) channels. So the core claim — that the transverse current is carried by Cooper pairs rather than by quasiparticles — is unsupported as written. The paper even says the Cooper pair velocity is not well defined in the BdG Hamiltonian, which makes the supercurrent assignment more puzzling, not less.\n\nThat said, the flaw is not obviously fatal. If the supplement shows the Berry term comes from the dependence of the quasiparticle spectrum on Cooper-pair momentum, the interpretation would be justified. And the qualitative agreement between the two methods is a point in favor. The paper is honest about its thermodynamic starting point and does not overclaim beyond the pending interpretation.\n\nThis deserves peer review because the idea is new, tied to an experimentally realized phase, and the authors have done enough groundwork to make a referee's job meaningful. But the referee should demand the full derivation and a clear way to distinguish supercurrent from quasiparticle Hall current — either analytically or via a current decomposition in the NEGF calculation. Without that, the central effect remains a label on a known formula.\n\nI would bring this to a reading group if someone else is presenting, and I would not cite it yet until the interpretation is tightened.","headline":"The paper makes a testable prediction for a Hall effect in a recently realized gapless superconductor, but the central claim that the transverse current is a supercurrent is asserted rather than demonstrated, so the paper needs a serious referee and a heavy revision.","tokens_in":10026,"tokens_out":1935,"would_cite":false,"duration_ms":21932,"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":"A gapless superconductor converts a longitudinal normal current into a dissipationless transverse supercurrent through quasiparticle Berry curvature.","keywords":["superconducting Hall effect","gapless superconductor","Berry curvature","segmented Fermi surface","transverse supercurrent","finite-momentum Cooper pairs","proximity superconductivity","quantum transport"],"falsifier":"If an angular-dependent Hall measurement on a gapless superconductor with hexagonal warping shows no dissipationless transverse current when the in-plane magnetic field is rotated away from $\\theta = 0^\\circ$, or if a system with $\\lambda = 0$ shows a transverse current, the central claim is falsified. Equivalently, a calculation showing that the $-f_n \\mathbf{E} \\times \\boldsymbol{\\Omega}_n$ term is a normal dissipative Hall current rather than a supercurrent would invalidate the ScHE.","tokens_in":9023,"feed_emoji":"⚡","tokens_out":4921,"duration_ms":44874,"temperature":0.7,"pith_summary":"The paper tries to establish that in a gapless superconductor with segmented Fermi surfaces, a longitudinal normal current can be converted into a transverse dissipationless supercurrent, an effect it calls the superconducting Hall effect (ScHE). It argues that the origin is intrinsic: the Berry curvature of Bogoliubov quasiparticles produces an anomalous transverse velocity, captured by the term $-f_n \\mathbf{E} \\times \\boldsymbol{\\Omega}_n$ in the current density. The claim is supported by two independent approaches, a thermodynamic free-energy derivation for bulk systems and a four-probe quantum transport calculation. If true, this would be a new kind of Hall effect in which the transverse response is carried by Cooper pairs rather than quasiparticles, observable by rotating the in-plane magnetic field in standard Hall measurements.","feed_headline":"Superconducting Hall effect turns normal current to supercurrent","feed_subtitle":"Berry curvature of quasiparticles turns longitudinal current into a dissipationless transverse supercurrent.","key_machinery":"The load-bearing object is the quasiparticle Berry curvature $\\boldsymbol{\\Omega}_n$ of the Bogoliubov-de Gennes bands, combined with the thermodynamic current formula $\\mathbf{J} = \\partial F/\\partial \\mathbf{A}$. In linear response under the relaxation-time approximation, this yields Eq. (7), whose second term, $-f_n \\mathbf{E} \\times \\boldsymbol{\\Omega}_n$, is the intrinsic Berry-curvature contribution producing the transverse supercurrent. The segmented Fermi surface, enabled by spin-orbit coupling, an in-plane Zeeman field, and hexagonal warping with coefficient $\\lambda$, supplies the anisotropic geometry in which quasiparticle and Cooper-pair transport are allowed in orthogonal directions.","core_discovery":"The central claim is that gapless superconductors with segmented Fermi surfaces host a superconducting Hall effect: a longitudinal electric field drives a quasiparticle current along the gapless direction, while the Berry curvature of those quasiparticles deflects electron and hole motion, and the deflected carriers combine into finite-momentum Cooper pairs that flow as a dissipationless transverse supercurrent. The paper derives a current-density expression, $\\mathbf{J} = \\sum_{n\\mathbf{k}} [\\tau \\mathbf{E} \\cdot \\nabla_{\\mathbf{k}} f_n \\mathbf{v}_n - f_n \\mathbf{E} \\times \\boldsymbol{\\Omega}_n]$, where the second term is identified as the intrinsic Berry-curvature contribution to the supercurrent, and confirms the same behavior in a four-probe transport calculation. The effect vanishes when the hexagonal warping vanishes, vanishes for one special magnetic-field orientation, and grows as the field is rotated, matching the symmetry of the quasiparticle Berry curvature.","pith_inferences":["The ScHE may turn angle-resolved Hall measurements on gapless superconductors into a direct probe of the quasiparticle Berry curvature distribution, in the same way that anomalous Hall measurements map band geometry in metals.","A dissipationless transverse supercurrent that can be switched by rotating the magnetic field suggests a possible mechanism for lossless current routing controlled by field orientation, although the paper does not explore device applications.","The thermodynamic-plus-Berry-curvature framework could extend naturally to other partially gapped superconducting states, including finite-momentum or altermagnet-based superconductors, to predict similar transverse supercurrent responses.","If the transverse response is truly dissipationless, it should be distinguishable from a normal Hall current by measuring the absence of transverse voltage drop, a testable experimental signature."],"forward_implications":["If the superconducting Hall effect is real, rotating the in-plane magnetic field in a gapless superconductor should produce a transverse supercurrent that is zero at $\\theta = 0^\\circ$ and grows as $\\theta$ deviates.","The transverse supercurrent should vanish when hexagonal warping is absent, because the quasiparticle Berry curvature also vanishes for $\\lambda = 0$.","Two-probe normal-metal/superconductor transport should show Andreev reflection dominating when the gap opens along the transverse direction, reproducing the segmented Fermi surface geometry.","Four-probe quantum transport should reproduce the thermodynamic prediction qualitatively, with the transverse current carried as a supercurrent rather than a normal dissipative current.","Angular-dependent Hall measurements on a circular disc of a proximitized gapless superconductor, such as a Bi2Te3/NbSe2 heterostructure, should reveal the signatures of both the tilted gap and the ScHE."],"supporting_citations":[{"why":"Provides the theoretical gapless superconducting phase with a partial Fermi surface that the paper builds on.","marker":"[19]"},{"why":"Supplies the concept that electrons and holes combine into finite-momentum Cooper pairs, which carries the transverse supercurrent.","marker":"[21]"},{"why":"Reports the experimental discovery of the segmented Fermi surface in Bi2Te3/NbSe2, the material platform proposed for detecting the ScHE.","marker":"[22]"},{"why":"Defines the Berry curvature and anomalous velocity formalism used to derive the intrinsic term in Eq. (7).","marker":"[5]"},{"why":"Provides the semiclassical Berry-curvature treatment of quasiparticle dynamics in superconductors that underpins the transport framework.","marker":"[36]"},{"why":"Uses the thermodynamic supercurrent formula $J = \\partial F/\\partial A$ that the paper adapts for the gapless state.","marker":"[20]"},{"why":"Gives the multi-probe scattering formula used in Eq. (9) to compute the transverse current in the four-probe setup.","marker":"[44]"}],"fun_headline_variants":["Berry curvature drives a superconducting Hall effect","Gapless superconductors convert normal current to supercurrent","New Hall effect in gapless superconductors: current converted","Supercurrent from normal current via Berry curvature in gapless superconductors","Transverse supercurrent from quasiparticle Berry curvature"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that the equilibrium thermodynamic relation $\\mathbf{J} = \\partial F/\\partial \\mathbf{A}$, with the equilibrium distribution replaced by a nonequilibrium one under the relaxation-time approximation, still gives the correct current, and that the Berry-curvature term represents a Cooper-pair supercurrent rather than an ordinary quasiparticle Hall current.","fun_headline_variants_meta":{"raw":{"variants":["Berry curvature drives a superconducting Hall effect","Gapless superconductors convert normal current to supercurrent","New Hall effect in gapless superconductors: current converted","Supercurrent from normal current via Berry curvature in gapless superconductors","Transverse supercurrent from quasiparticle Berry curvature"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0007,"raw_usage":{"total_tokens":3125,"prompt_tokens":874,"completion_tokens":2251,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":490,"completion_tokens_details":{"reasoning_tokens":2174}},"tokens_in":490,"tokens_out":2251,"duration_ms":15737,"temperature":1.0,"reasoning_tokens":2174,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:48:15.781606+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"If an angular-dependent Hall measurement on a gapless superconductor with hexagonal warping shows no dissipationless transverse current when the in-plane magnetic field is rotated away from $\\theta = 0^\\circ$, or if a system with $\\lambda = 0$ shows a transverse current, the central claim is falsified. Equivalently, a calculation showing that the $-f_n \\mathbf{E} \\times \\boldsymbol{\\Omega}_n$ term is a normal dissipative Hall current rather than a supercurrent would invalidate the ScHE.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the semiclassical Berry-curvature treatment of quasiparticle dynamics in superconductors that underpins the transport framework."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Uses the thermodynamic supercurrent formula $J = \\partial F/\\partial A$ that the paper adapts for the gapless state."}],"review_version":1}