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Berry curvature of Bogoliubov quasiparticles can generate an intrinsic Nernst response in a clean, vortex-free superconductor—spontaneously in the chiral p-wave case.

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2026-08-04 13:33 UTC pith:5YFLIHSO

load-bearing objection 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. the 3 major comments →

arxiv 2509.26638 v2 pith:5YFLIHSO submitted 2025-09-30 cond-mat.supr-con cond-mat.mes-hall

Intrinsic Nernst Effect from Berry Curvature in Superconductors

Tzu-Chi Hsieh , Cong Xiao , Yi-Ting Hsu This is my paper
classification cond-mat.supr-con cond-mat.mes-hall
keywords Nernst effectBerry curvatureBogoliubov quasiparticleschiral p-wave superconductivityIsing spin-orbit couplingRashba spin-orbit couplingtransition metal dichalcogenidestopological superconductivity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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.

Core claim

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

What carries the argument

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

Load-bearing premise

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.

What would settle it

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.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [Semiclassical wavepacket approach / Eq. (3), footnote [32]] 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.
  2. [Experimental setup and estimation / Table I] 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
  3. [Intravalley chiral p-wave paired state / spin Nernst claim] 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.
minor comments (5)
  1. [Footnotes [32] and [39]] Both footnotes contain unresolved placeholder citations '[ ? ]'. These should be completed before publication.
  2. [Model section] Typographical issues: 'Numbu basis' should be 'Nambu basis'; 'ferromagnetic subtract' should be 'ferromagnetic substrate'; 'intravally' should be 'intravalley'.
  3. [Experimental setup] "where the radii is smaller than the penetration depth" is ungrammatical; should be "where the radius is smaller than the penetration depth".
  4. [Intervalley s-wave paired state] 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).
  5. [General] 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.

Circularity Check

0 steps flagged

No significant circularity: the transport formula and Berry-curvature decomposition are imported from the authors' prior work, but they are parameter-free derivations whose assumptions do not include the predicted pairing-dependent Nernst signals.

full rationale

The central quantitative predictions are obtained by applying a published semiclassical transport formula, Eq. (3), to two BdG models. Although Eq. (3) is taken from Ref. [25], coauthored by C. Xiao, and Eq. (8) is attributed to Ref. [9], coauthored by Y.-T. Hsu, these are not circular inputs: both are parameter-free formal results about Berry-curvature contributions to thermoelectric transport and BdG Berry curvature, respectively. Their stated assumptions concern the semiclassical wavepacket treatment and the real-form structure of BdG Hamiltonians, not the specific s-wave/p-wave pairing states whose Nernst response is predicted. The present paper's contribution is to evaluate these formulas in models with Ising/Rashba spin-orbit coupling, producing the field-activated s-wave effect, the spontaneous p-wave effect, sign-reversal tests, and flux estimates. These are calculations from the model, not fits to data, and they do not reduce to the inputs by construction. The acknowledged footnote '[ ? ]' correction to the current operator and the lack of a full screening theory are genuine quantitative limitations, but they are not circularity. No other circularity pattern (definitional, fitted-input, uniqueness, renaming) is present.

Axiom & Free-Parameter Ledger

7 free parameters · 5 axioms · 0 invented entities

The central claim is a linear-response calculation in minimal BdG models. The framework (Eq. (3), two-fluid counterflow) and the Berry-curvature decomposition (Eq. (8)) are taken from co-authored prior work (Refs [25], [9]); the model parameters are chosen by hand to illustrate regimes rather than fitted to data; the p-wave order is assumed, not derived; and no new physical entities are introduced.

free parameters (7)
  • Superconducting gap Δ = Δ = 1 (meV) in numerics; k_B T set equal to Δ
    Phenomenological order parameter introduced without a microscopic pairing mechanism or self-consistency; all quantitative αH values scale with the choice of Δ and T = Δ.
  • Zeeman field h = h = 0.5 (meV) in Fig. 2
    Chosen to break time-reversal symmetry weakly for the s-wave case; the paper notes the Pauli limit caps h.
  • Ising SOC βso = βso = 7 (meV) in Fig. 2(a), scanned 0-20 elsewhere
    Material parameter of the minimal model, chosen by hand for the demonstration; the optimization conditions (βso = λso k0) depend on it.
  • Hopping t = t = 500 meV·a²
    Sets the band curvature scale of the minimal model; chosen arbitrarily.
  • Chemical potentials μ1, μ2, μ3 = μ1 = 4 − βso, μ2 = 0, μ3 = 2 + βso (meV)
    Chosen to access three doping regimes (below/inside/above the spin-orbit gap).
  • Temperature = T such that k_B T = Δ (T = 1 in meV units)
    Operating point chosen 'to ensure sufficient quasiparticle excitations'; at T ≈ 0.57 Tc the quasiparticle damping is not modeled.
  • Ring radius R and ΔT = R = 60 nm, ΔT = 1 K
    Experimental parameters for the flux estimate B = μ0I/2R; the estimate scales as 1/R.
axioms (5)
  • domain assumption αH of the superconducting state is given by Eq. (3), the quasiparticle grand-potential-weighted Berry-curvature integral, with Cooper-pair counterflow accounted for
    Imported from Ref. [25] (co-author C. Xiao); validity stated as strong type-II superconductors with penetration depth much larger than sample scale. Not re-derived here.
  • domain assumption Berry curvature decomposes into particle-hole and spin-space twists, Eq. (8), and reduces to Eq. (9) for the two models
    From Ref. [9] (co-author Y.-T. Hsu) for BdG Hamiltonians that are real in a rotated frame; relies on rotational symmetry around K/K′.
  • domain assumption Well-defined quasiparticle wavepackets at k_B T = Δ
    Semiclassical transport presumes long-lived quasiparticles; no estimate of damping near Tc is given.
  • domain assumption Chiral intravalley p-wave pairing (same/opposite chirality) is the realized order
    Explicitly phenomenological; the paper states 'we treat the superconducting gap Δ as a phenomenological order parameter' and 'we expect fully gapped chiral p-wave pairing is favored over nodal p-wave' — the existence of the state is taken from Ref. [20] (co-authored).
  • domain assumption Ring-geometry transverse current is unscreened and equals αH ∇T, yielding B_th = μ0I/2R
    The paper argues the ring circumvents bar-geometry screening; it concedes 'a microscopic theory that integrates the screening effect into the semiclassical framework has yet to be developed' for thicker samples.

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Cite this review

Pith. "Pith review of Intrinsic Nernst Effect from Berry Curvature in Superconductors." pith.science (2026). https://pith.science/paper/5YFLIHSO

@misc{pith2026250926638,
  author       = {Pith},
  title        = {Pith review of: Intrinsic Nernst Effect from Berry Curvature in Superconductors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5YFLIHSO}},
  note         = {Machine review of arXiv:2509.26638}
}
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read the original abstract

The Nernst effect in superconductors is typically linked to fluctuating Cooper pairs above $T_c$ or vortex motion below $T_c$. We show instead that Berry curvature of Bogoliubov quasiparticles can generate an intrinsic Nernst response in a clean, vortex-free superconducting state. Focusing on two-dimensional systems with Ising spin-orbit coupling, relevant to transition-metal dichalcogenides, we identify two regimes: an intervalley $s$-wave paired state where a weak magnetic field activates the effect, and an intravalley chiral $p$-wave paired state that exhibits a spontaneous charge or spin Nernst response without a field. We propose an experimental setup that circumvents screening and provide estimates of the signal magnitude. Our results establish the Nernst effect as a direct probe of Berry curvature and pairing symmetry in two-dimensional spin-orbit-coupled superconductors.

Figures

Figures reproduced from arXiv: 2509.26638 by Cong Xiao, Tzu-Chi Hsieh, Yi-Ting Hsu.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic of the proposed geometry for measur [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Properties of intervalley [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Properties of intravalley chiral [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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