{"id":"a62b76ab-5ed4-4496-9ec4-8f95fae8f94d","arxiv_id":"2509.26638","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"Berry curvature of superconducting quasiparticles produces an intrinsic Nernst signal: spontaneous in chiral p-wave pairs, magnetic-field-activated in s-wave pairs with spin-orbit coupling, measurable as a small ring flux.","lead":"This paper predicts that the Nernst effect — a sideways electric current produced by heat flow — can appear inside a clean superconductor purely from the geometry of its quantum energy bands, with no vortices or fluctuating Cooper pairs involved. If real, the effect reveals the pairing symmetry of the superconductor and provides a new bulk probe for identifying topological superconductors.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The predicted Nernst signal rests on an imported transport formula (Eq. 3); the gap-induced current-operator correction admitted in footnote [32] could invalidate the quantitative predictions unless the Kubo formula reproduces Eq. (3).","rationale":"The reader's verdict is CONDITIONAL, and the weakest assumption identified by the reader is exactly the validity of Eq. (3), which is imported from Ref. [25] and is subject to the admitted current-operator correction in footnote [32]. My stress-test confirms this as the most load-bearing concern: all numerical estimates and the experimental detection strategy hinge on Eq. (3). The paper presents symmetry arguments for why a Nernst effect should exist, which are plausible and internally consistent, but the magnitude and even the sign of the effect depend on the correct microscopic current operator. The unresolved citation in footnote [32] is a red flag that the authors themselves are aware of the potential issue. I see no reason to change the verdict from CONDITIONAL: the paper is promising and likely correct in its qualitative predictions, but it must either provide a microscopic derivation of Eq. (3) from the Kubo formula, or estimate the size of the gap-induced current-operator corrections, before the quantitative flux predictions can be trusted. The concrete test I propose is a direct independent check of the central formula, which would settle the concern.","tokens_in":11562,"tokens_out":8709,"duration_ms":79994,"concrete_test":"Compute the intrinsic transverse thermoelectric conductivity αH for the 2×2 chiral p-wave BdG Hamiltonian in Eq. (6) using the Kubo formula for the current–current correlation function, retaining the full momentum-dependent velocity operator v = ∂H/∂k (including the ∂Δ(k)/∂k terms), and compare the result with Eq. (3) over the same parameter regime (T = Δ, βso < 0, λso = h = 0). If the Kubo result deviates by more than 10%, the paper's central predictions are not the physical Nernst response; if it matches, the imported formula is validated and the concern is closed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claim—αH ≈ (0.03–0.3)α0 and the corresponding 1–10 nT flux—follows directly from Eq. (3), which is taken verbatim from Ref. [25] (co-authored by one of the present authors) and is not re-derived in this paper. The paper's footnote [32] explicitly concedes that for general electron–electron interactions the quasiparticle current operator may acquire contributions from the gap function, and the supporting citation is an unresolved placeholder '[ ? ]'. This is not a cosmetic issue: the Nernst coefficient is proportional to the integrand of Eq. (3), involving the quasiparticle charge ρnk, the Berry curvature Ωnk, and the grand-potential derivative. In a p-wave superconductor, Δ(k) is momentum-dependent, so the physical velocity operator v = ∂H/∂k contains a term ∝ ∂Δ(k)/∂k that is not captured by the simple two-fluid decomposition of Ref. [25]. If such a correction is sizable at the operating point kBT = Δ ≈ 0.57 Tc, the predicted αH and flux values are not the physical response. The paper does not estimate the magnitude of this correction, and the acknowledged lack of a screening theory further separates the predicted flux from an experimental measurement. This is a foundational, unresolved assumption rather than a computational error.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript proposes an intrinsic Nernst effect in clean, vortex-free superconducting states, driven by the Berry curvature of Bogoliubov quasiparticles. Using a semiclassical wavepacket two-fluid framework and a two-valley model with Ising and Rashba spin-orbit coupling, the authors study intervalley s-wave pairing, where a Zeeman field activates the effect, and intravalley chiral p-wave pairing, where a spontaneous Nernst response appears at zero field. They propose a ring-geometry measurement of the resulting non-quantized magnetic flux, estimate signals of 1–10 nT for a 60 nm ring and ΔT = 1 K, and argue that the sign and field-dependence of the Nernst signal can diagnose pairing symmetry and quasiparticle Berry curvature.","tokens_in":11787,"tokens_out":6634,"duration_ms":62079,"significance":"If the central transport formula is valid, this is a valuable proposal: it offers a bulk, boundary-independent probe of chiral topological superconductivity, distinguishes s-wave from chiral p-wave pairing through the field dependence and sign of the Nernst signal, and gives concrete, falsifiable experimental predictions in currently accessible platforms using NV-center or SQUID magnetometry. The manuscript is clear and internally coherent: the symmetry decomposition of Berry curvature in Eq. (8), the reduction to Eq. (9), and the explicit BdG Hamiltonians in Appendix A are consistent with the stated model. The main weakness is that the quantitative predictions inherit Eq. (3) verbatim from a co-authored earlier paper, and the manuscript itself flags a possible gap-induced correction to the current operator without resolving it. This must be addressed before the numerical estimates can be considered self-contained.","major_comments":[{"comment":"Equation (3) is the key input for all quantitative results, yet it is quoted from Ref. [25] and not derived in this manuscript. The paper's footnote [32] concedes that for general electron-electron interactions the gap function may contribute to the current operator, but the supporting citation is an unresolved placeholder '[ ? ]'. For the p-wave gap Δ(k) ∝ kx + iky, a velocity correction ∝ ∂Δ/∂k would enter directly into the integrand of Eq. (3) and hence into αH. Since the paper operates at kBT ≈ Δ ≈ 0.57 Tc, where thermally excited quasiparticles near kF dominate, such a correction could be sizable. The authors should either prove that this correction vanishes for the BdG mean-field Hamiltonians in Eqs. (5)–(6), or provide a quantitative estimate of its magnitude and adjust the predictions accordingly. Without this, the claimed αH range 0.03–0.3 α0 in Table I is not self-contained.","section":"Semiclassical wavepacket approach / Eq. (3), footnote [32]"},{"comment":"The proposed ring geometry is intended to circumvent screening, and the predicted flux values follow from the vacuum Biot–Savart relation Bth = μ0 I/(2R). However, the same section states that \"a microscopic theory that integrates the screening effect into the semiclassical framework has yet to be developed.\" In the closely analogous superconducting Seebeck ring experiments cited by the authors [17], penetration-depth and screening effects substantially affect the measured flux. The manuscript should specify precisely the regime in which the unscreened formula applies (for example, ring radius much smaller than penetration depth and no backflow correction to αH in the ring), or provide a minimal screening model. As written, the 1–10 nT numbers are not yet tied to a well-defined physical observable unless the authors commit to the ideal strongly type-II limit and justify why screening is","section":"Experimental setup and estimation / Table I"},{"comment":"The manuscript claims that for opposite-chirality intravalley p-wave pairing, the charge Nernst signal vanishes but a spontaneous spin Nernst effect appears. No definition of a spin Nernst conductance is given, no spin-current analogue of Eq. (3) is written, and no detection protocol or estimate is provided. Since this is presented as a concrete result of the model, the authors should either derive the spin Nernst observable explicitly or clearly label this as a qualitative prediction requiring further development.","section":"Intravalley chiral p-wave paired state / spin Nernst claim"}],"minor_comments":[{"comment":"Both footnotes contain unresolved placeholder citations '[ ? ]'. These should be completed before publication.","section":"Footnotes [32] and [39]"},{"comment":"Typographical issues: 'Numbu basis' should be 'Nambu basis'; 'ferromagnetic subtract' should be 'ferromagnetic substrate'; 'intravally' should be 'intravalley'.","section":"Model section"},{"comment":"\"where the radii is smaller than the penetration depth\" is ungrammatical; should be \"where the radius is smaller than the penetration depth\".","section":"Experimental setup"},{"comment":"The statement that αH changes sign between upper and lower BdG bands is clear, but the text would benefit from an explicit formula or plot showing the sign of Ω per band; currently the reader must infer this from Fig. 2(a).","section":"Intervalley s-wave paired state"},{"comment":"The spin Nernst effect in the opposite-chirality p-wave case is mentioned only qualitatively; please clarify whether it corresponds to a measurable spin current and how it would be detected.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper is a plausible and well-structured theoretical proposal with a clearly identified experimental signature. The main obstacle is that the central quantitative formula is imported from a co-authored previous work, and the manuscript itself acknowledges an unresolved correction. I do not see this as grounds for rejection, but the revision must close this gap. I would also encourage the authors to be explicit about the ideal limit in which the ring-flux estimate is quantitative, rather than deferring screening to future work."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper is worth reading. It proposes that Berry curvature of Bogoliubov quasiparticles can produce an intrinsic Nernst signal in the vortex-free superconducting state, and backs that with concrete model calculations and a measurement setup that looks feasible. The key new result is the dichotomy: a chiral p-wave state with the same chirality in both valleys gives a spontaneous Nernst flux at zero field, while an intervalley s-wave state gives a field-activated signal. The ring geometry is a sensible way around the screening that kills the usual bar-geometry response, and the predicted flux, 1–10 nT for a 60 nm ring and ΔT = 1 K, is within reach of NV-center magnetometry.\n\nThe paper does several things well. It works through a realistic two-valley model with Ising and Rashba spin–orbit coupling rather than an abstract lattice model. The sign reversal of αH between the upper and lower bands in the s-wave case is a nice fingerprint that would distinguish a Berry-curvature contribution from fluctuation or vortex backgrounds. The numerical estimates are transparent and the optimization conditions are clearly explained.\n\nThe main soft spot is that the transport formula, Eq. (3), is taken verbatim from Ref. [25] and not re-derived or checked against Kubo. The text itself acknowledges in footnote [32] that for general electron–electron interactions the gap function may contribute to the current operator, and the supporting citation is an unresolved '[ ? ]'. Since the predicted flux is proportional to αH, a sizable correction would shift the numbers. I don't see the symmetry-based dichotomy failing, but the quantitative claim does rest on this imported formula. The absence of a comparison to the vortex/fluctuation Nernst background that dominates real TMD experiments near Tc is also a gap; an experimentalist needs to know the intrinsic signal can be isolated. Finally, the authors note that a microscopic theory of screening in the ring has yet to be developed; the estimates assume the ideal type-II limit.\n\nPartly for completeness, the manuscript contains two other unresolved '[ ? ]' placeholders, one near the Berry-curvature expression in the appendix and one in the standard Ω formula in footnote [39]. Those are easy to fix but should not be in a submission.\n\nI think the paper deserves a serious referee. The central physics is coherent and the proposal is concrete. My recommendation is to send it to review, with the request that the authors either verify Eq. (3) against a Kubo calculation or clearly justify its regime, address the gap-function issue, and clean up the citations. This would be useful for both theorists and experimentalists working on topological superconductivity and 2D superconductors.","headline":"A clean proposal for a Berry-curvature Nernst probe of pairing symmetry in 2D superconductors, with quantitative predictions that depend on an imported transport formula and a few unresolved citations.","tokens_in":12431,"tokens_out":5137,"would_cite":true,"duration_ms":45904,"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":"Berry curvature of Bogoliubov quasiparticles can generate an intrinsic Nernst response in a clean, vortex-free superconductor—spontaneously in the chiral p-wave case.","keywords":["Nernst effect","Berry curvature","Bogoliubov quasiparticles","chiral p-wave superconductivity","Ising spin-orbit coupling","Rashba spin-orbit coupling","transition metal dichalcogenides","topological superconductivity"],"falsifier":"Measure the flux through a small ring (radius ≈ 60 nm, radial temperature difference ≈ 1 K) made of a candidate chiral p-wave superconductor in zero applied magnetic field; if no spontaneous flux of order 10 nT appears above the noise floor, the predicted spontaneous Nernst effect would be ruled out.","tokens_in":11301,"feed_emoji":"🧲","tokens_out":2935,"duration_ms":26102,"temperature":0.7,"pith_summary":"The paper argues that the Nernst effect in superconductors need not come from vortex motion or fluctuating Cooper pairs; Berry curvature of thermally excited Bogoliubov quasiparticles can produce an intrinsic transverse thermoelectric response in a clean, vortex-free state. In a two-valley model with Ising and Rashba spin-orbit coupling, intervalley s-wave pairing gives a field-activated Nernst response, while intravalley chiral p-wave pairing with the same chirality in both valleys gives a spontaneous response at zero field. The measurable signature is a small, non-quantized magnetic flux through a ring subjected to a radial temperature gradient, estimated at 1–10 nT. If correct, this turns the Nernst effect into a bulk probe of pairing symmetry and quasiparticle Berry curvature.","feed_headline":"Berry curvature gives superconductors a spontaneous Nernst signal","feed_subtitle":"Chiral p-wave pairing could be detected in a ring as a 1–10 nT flux from a radial temperature drop.","key_machinery":"The central object is the semiclassical transport formula for the Hall component of the thermoelectric conductance, αH = (e/ℏ) Σ_n ∫ d²k/(2π)² (dg_nk/dT) Ω_nk ρ_nk, in which Ω_nk is the Berry curvature of a Bogoliubov quasiparticle band, ρ_nk is its charge expectation value, and g_nk is the grand potential. The Berry curvature is decomposed into separable contributions from the twist of the wavefunction in particle-hole space and in spin space; this decomposition is what allows the contrasting behaviors of s-wave and chiral p-wave pairing. The Cooper-pair counterflow is essential: because quasiparticle charge is not conserved, the conserved total current includes a compensating Cooper-pair c","core_discovery":"The central claim is that the anomalous Nernst conductance of a superconductor is given by a Berry-curvature integral over Bogoliubov quasiparticle bands, αH = (e/ℏ) Σ_n ∫ d²k/(2π)² (dg_nk/dT) Ω_nk ρ_nk, where Cooper-pair counterflow has been accounted for in a semiclassical wavepacket transport theory. Applying this to a two-valley model with Ising and Rashba spin-orbit coupling, the paper finds two regimes: an intervalley s-wave paired state in which an external Zeeman field is required to activate a Nernst signal, with a predicted sign reversal as the chemical potential is tuned across the spin-orbit gap; and an intravalley chiral p-wave paired state in which same-chirality pairing at the","pith_inferences":["The same Berry-curvature mechanism suggests that nodal superconductors, which have a greater density of low-energy quasiparticles, would show a larger Nernst response than the fully gapped cases examined here—an extension the paper mentions as a future direction.","If the transport formula survives a microscopic treatment of magnetic-field screening, the ring-flux measurement would provide a boundary-independent complement to quantized thermal Hall for identifying chiral topological order in two-dimensional superconductors.","The spin Nernst effect predicted for opposite-chirality p-wave pairing could be used to detect a differently broken time-reversal configuration without producing a charge flux, offering a route to distinguishing chirality arrangements in the same material class."],"forward_implications":["A spontaneous magnetic flux appearing through a ring in zero applied field would serve as a bulk signature of chiral p-wave superconductivity, independent of boundary modes.","For a conventional s-wave superconductor with both Ising and Rashba spin-orbit coupling, an applied out-of-plane field would activate a Nernst signal whose sign reversal with chemical potential fingerprints the Berry-curvature origin.","The induced flux is non-quantized even for a Chern-number-carrying chiral superconductor, because the Nernst response comes from thermally activated quasiparticles near the gap, not from edge modes.","Estimated fluxes of roughly 1 nT (s-wave, field-activated) and 10 nT (chiral p-wave, spontaneous) for a 60 nm ring with ΔT = 1 K are within reach of NV-center and SQUID magnetometry."],"fun_headline_variants":["Chiral p-wave pairing gives a spontaneous Nernst signal","Berry curvature drives Nernst effect in superconductors","Superconductors without vortices can still produce Nernst","Spontaneous Nernst from chiral p-wave pairing"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The predictions stand on the inherited transport formula that identifies the measured Nernst conductance with the Berry-curvature integral of Eq. (3), assuming Cooper-pair counterflow is fully accounted for and that corrections to the quasiparticle current operator and magnetic-field screening are negligible at the operating point kBT ≈ Δ ≈ 0.57 Tc.","fun_headline_variants_meta":{"raw":{"variants":["Chiral p-wave pairing gives a spontaneous Nernst signal","Berry curvature drives Nernst effect in superconductors","Superconductors without vortices can still produce Nernst","Spontaneous Nernst from chiral p-wave pairing"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000871,"raw_usage":{"total_tokens":3594,"prompt_tokens":715,"completion_tokens":2879,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":459,"completion_tokens_details":{"reasoning_tokens":2812}},"tokens_in":459,"tokens_out":2879,"duration_ms":15399,"temperature":1.0,"reasoning_tokens":2812,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T13:33:17.966421+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the flux through a small ring (radius ≈ 60 nm, radial temperature difference ≈ 1 K) made of a candidate chiral p-wave superconductor in zero applied magnetic field; if no spontaneous flux of order 10 nT appears above the noise floor, the predicted spontaneous Nernst effect would be ruled out.","supporting_citations":[],"review_version":1}