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Bogoliubov Fermi surfaces in chiral superconducting rhombohedral graphene break thermal-Hall quantization and boost low-temperature Nernst responses.

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Bogoliubov Fermi surfaces from band warping drive nonquantized thermal Hall conductivity and strongly enhanced low-T spin/orbital Nernst responses in chiral p-wave rhombohedral graphene.

T0 review reviewed 2026-07-30 challenge →

load-bearing objection Solid BdG numerics showing that warping-induced Bogoliubov Fermi surfaces spoil low-T thermal-Hall quantization and boost Nernst signals in chiral p-wave rhombohedral graphene; useful diagnostics, moderate novelty, main caveat is hand-set Δ. the 3 major comments →

arxiv 2607.23764 v1 pith:K5HZDSLE submitted 2026-07-26 cond-mat.supr-con

Berry curvature effects of chiral superconducting rhombohedral graphene

Jian-Hua Zeng , Zhi Wang , Qian Niu This is my paper
classification cond-mat.supr-con
keywords chiral superconductivityrhombohedral grapheneBerry curvatureBogoliubov Fermi surfacethermal Hall effectspin Nernst effectorbital Nernst effecttrigonal warping
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 reading

This paper asks how the geometry of superconducting quasiparticles—Berry curvature and orbital magnetic moment—shows up in heat and spin transport when rhombohedral graphene hosts a chiral p-wave superconductor. In a clean, fully gapped chiral state the low-temperature thermal Hall conductivity sits at a quantized value set by the Chern number, and Nernst signals are thermally suppressed. Realistic trigonal warping of the normal-state bands breaks the particle-hole symmetry of the pairing problem and can leave gapless Bogoliubov Fermi surfaces. Those surfaces rearrange which quasiparticle states are occupied, so the thermal Hall response loses its integer quantization and the spin- and orbital-Nernst coefficients become large already at low temperature, with magnitudes and signs that track the size and location of the surfaces. The authors argue these transport fingerprints can therefore serve as experimental markers that Bogoliubov Fermi surfaces are present.

Core claim

In chiral p-wave superconducting rhombohedral graphene, normal-state trigonal warping generates Bogoliubov Fermi surfaces that qualitatively remodel anomalous Hall transport of the Bogoliubov quasiparticles: the low-temperature thermal Hall conductivity κ_xy/κ_0 deviates from the Chern-number quantized value, while the spin- and orbital-Nernst responses are strongly enhanced and become sensitive in both magnitude and sign to the presence, size, and momentum-space location of those surfaces.

What carries the argument

A two-band Bogoliubov–de Gennes Hamiltonian for a single spin-valley sector, from which the quasiparticle Berry curvature Ω_n(k) and orbital magnetic moment m_n(k) are computed and then inserted into the thermal-Hall, spin-Nernst, and orbital-Nernst integrals.

Load-bearing premise

The superconducting gap is taken as a fixed-amplitude chiral p-wave pairing on one projected band, rather than being determined self-consistently together with warping and competing orders; if the real pairing is not that simple, the reported fingerprints need not appear.

What would settle it

Measure the low-temperature thermal Hall conductivity and spin/orbital Nernst coefficients while tuning electron density or displacement field across regimes where Bogoliubov Fermi surfaces are predicted to appear, disappear, or change size; non-quantized κ_xy/κ_0 together with density-dependent Nernst peaks and sign changes would support the claim, while strict quantization and vanishing low-T Nernst would refute it.

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

If this is right

  • Low-temperature thermal Hall conductivity that sits off the integer value of κ_0 is a direct transport signature of Bogoliubov Fermi surfaces in this platform.
  • Strong low-T spin and orbital Nernst signals, including density-driven peaks and sign reversals, track the presence and momentum location of those surfaces.
  • Orbital Nernst response appears only when the normal-state dispersion is asymmetric, so it diagnoses warping-induced quasiparticle orbital magnetism.
  • Electron-density and displacement-field sweeps that open or close Bogoliubov Fermi surfaces should produce correlated, non-monotonic changes in all three transport coefficients.

Where Pith is reading between the lines

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

  • If confirmed, the same transport protocol could be used to hunt Bogoliubov Fermi surfaces in other warped multilayer graphene and moiré superconductors without requiring spectroscopic gap maps.
  • A self-consistent treatment that lets the gap open or close with density might shrink the predicted Nernst windows, so quantitative comparison with experiment will need interaction-renormalized pairing.
  • The sign-sensitive orbital Nernst channel offers a possible bulk probe of the orbital magnetization texture that recent magnetic experiments on rhombohedral tetralayer graphene have begun to explore.
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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. The paper studies Berry-curvature-related transport of Bogoliubov quasiparticles in chiral p-wave superconducting rhombohedral tetralayer graphene. Using a two-band BdG Hamiltonian projected onto a single spin-valley sector, the authors compute the quasiparticle Berry curvature (Eq. 5), orbital magnetic moment (Eq. 6), thermal Hall conductivity (Eq. 9), and spin/orbital Nernst coefficients (Eqs. 10–11) in two regimes: a fully gapped state from a self-consistent screened-Coulomb model without trigonal warping (Sec. III, Appendix A), and a gapless state with Bogoliubov Fermi surfaces (BFS) induced by trigonal warping (Sec. IV, Appendix B). The central claims are that BFS (i) drive the low-temperature κ_xy/κ_0 away from the quantized Chern-number value via partial cancellation of opposite-sign Berry curvature of the two BdG bands, and (ii) strongly enhance the low-temperature spin and orbital Nernst responses, with magnitudes and signs sensitive to BFS size and momentum-space location. The formalism is standard and applied consistently; the gapped-state limits reproduce expected Chern-number quantization, and Appendix C provides a gapped reference within the warped model.

Significance. If the fingerprints hold, the paper provides concrete, falsifiable transport diagnostics — nonquantized low-T κ_xy, enhanced and density-tunable spin/orbital Nernst signals with sign changes — for detecting Bogoliubov Fermi surfaces in superconducting rhombohedral tetralayer graphene, a system with strong current experimental interest (Han et al., Nature 643, 654 (2025)). The derivation chain is parameter-light: all transport coefficients follow from geometric integrals over the stated BdG eigenstates with no fitting to a target, and the paper includes useful internal controls (gapped vs gapless comparisons, Chern-number limits, Appendix C reference calculations at larger Δ). The main limitation is that the gapless regime is explored with a non-self-consistent pairing amplitude, so the fingerprints are conditional on BFS surviving realistic pairing — a caveat the authors should state more sharply.

major comments (3)
  1. [Appendix B / Sec. IV] Appendix B / Sec. IV: the gapless-state calculations use Eq. (B1) with the pairing amplitude Δ fixed by hand (4–8 meV for BFS cases, 10–15 meV for the gapped references in Appendix C), while the only self-consistent gap calculation (Appendix A) neglects trigonal warping. This is load-bearing for the central claim: from Eq. (4), the existence, size, and position of the Bogoliubov Fermi surfaces are controlled by the competition between |Δ_k| and |ξ_a(k)|, so the reported fingerprints (nonquantized κ_xy, enhanced α_SN/α_ON) are conditional on Δ landing in a window the manuscript does not determine. Appendix C demonstrates sensitivity to Δ but does not establish which regime a physical interaction selects. At minimum, the manuscript should (i) state explicitly that the warped-state Δ is not self-consistently determined and frame the fingerprints as conditional on BFS survival, and (ii) show
  2. [Appendix B, Eq. (B1)] Eq. (B1) takes Δ_k = Δa(k_x + i k_y), a purely linear-in-k gap whose magnitude grows without bound across the momentum integration window. This differs qualitatively from the self-consistent η_1(k) of Appendix A (Eqs. A7–A9), which has nontrivial k-dependence set by the Fermi surface. Since ξ_a(k) also grows with k, whether zero-energy contours exist at the window edge — and whether the Berry-curvature/orbital-moment integrals converge — depends on this unbounded linear form. The text states only that 'the cutoff [is] chosen such that the results are converged' (end of Sec. II). Please quantify this: give the window size, show that the BFS contours and transport coefficients are insensitive to it, and discuss how a self-consistent η_1(k) that decays away from the Fermi surface would modify the gapless structure.
  3. [Sec. IV.B, Fig. 3(b)] In Sec. IV.B and Fig. 3(b), κ_xy/κ_0 is described as returning 'close to the quantized value' when the Bogoliubov Fermi surfaces disappear at intermediate densities. With warping present, ξ_a(k) ≠ 0 everywhere, so even in the gapped windows the spectrum is not the symmetric case of Sec. III, and the residual deviation from the integer could be either (a) a genuine small non-quantization from near-zero-energy states or (b) a finite-temperature effect at k_B T = 0.015 Δ_eff. Since the quantized-vs-nonquantized contrast is the paper's headline signature, the authors should separate these: e.g., show the T→0 extrapolation in the gapped windows and confirm the deviation there is numerically zero within integration accuracy.
minor comments (5)
  1. [Fig. 4] Fig. 4 caption: the units of α_SN and α_ON are not stated in the captions or axes (only the 10^{-2}–10^{-4} scale factors appear). Please give explicit units, and likewise for κ_0 in Fig. 3.
  2. [Figs. 1, 3, 4] Δ_eff ≡ min_k[E_+(k) − E_-(k)] is used to normalize temperature in both gapped and gapless cases. In the gapless cases this minimum band separation is not the excitation gap (which is zero), so the meaning of k_B T/Δ_eff differs between Figs. 1 and 3–4. A sentence clarifying this would prevent misreading of the temperature scales.
  3. [Fig. 2] Fig. 2 panels (c), (d), (g), (h): the color-bar quantities render as '++(k)' in the extracted text; please check that the Ω^z_+(k) and m^z_+(k) labels compile correctly in the final version.
  4. [Sec. II, Eq. (6)] Sec. II, Eq. (6): the orbital-moment formula is taken from Ref. [64]. Since the interband matrix element involves τ_z ∂_k H^d_k rather than the usual velocity operator, a brief comment on its relation to the conventional normal-state orbital moment (and the reason the identity component drops out) would help readers who know only the Xiao–Chang–Niu formula.
  5. [Sec. V] The discussion of detectability is qualitative throughout. Since thermal Hall measurements in graphene devices are demanding, a short estimate of the absolute magnitude of κ_xy and the Nernst coefficients in physical units at representative T would strengthen the connection to experiment.

Circularity Check

0 steps flagged

No significant circularity: transport fingerprints are numerical outputs of stated BdG integrals, not forced by definition or self-citation.

full rationale

The paper’s load-bearing chain is: (i) write a two-band BdG Hamiltonian with chiral p-wave pairing on a projected conduction band (Eqs. 1–4; Apps. A–B); (ii) evaluate Berry curvature, orbital moment, and thermal/spin/orbital Nernst coefficients from the standard geometric integrals (Eqs. 5–11); (iii) compare fully gapped spectra (ξ_a=0 or large fixed Δ) to gapless spectra with trigonal warping that produce Bogoliubov Fermi surfaces. The claimed nonquantized low-T κ_xy/κ_0 and enhanced low-T Nernst signals are direct numerical consequences of which BdG states sit near zero energy and how they sample Ω_n and m_n—they are not fitted to data, not normalized to equal a target, and not declared unique by an authors-only theorem. Self-citations ([46], [57], [64], and graphene-SC setups [39], [41]) supply methods and model ingredients; the BFS-induced modulation is an output of those integrals, not an input. Hand-chosen Δ in the warped model is a modeling limitation, not circularity. No step reduces Eq. X to Eq. Y by construction in the sense of the circularity taxonomy.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

The claim rests on standard BdG Berry-transport machinery plus domain modeling choices for rhombohedral tetralayer graphene (single-sector chiral p-wave, two-band projection, warping from an eight-band continuum model). Free parameters are representative pairing amplitudes, densities, and displacement/potential values used to place the system in gapped or BFS regimes; hoppings and screening constants are taken from cited literature. No new particles or forces are invented.

free parameters (4)
  • Pairing amplitude Δ (warping model) = 4–15 meV (case-dependent)
    In Sec. IV/Appendix B, Δ is chosen by hand (4, 8 meV gapless; 10, 15 meV gapped) to open or close BFS rather than solved self-consistently with warping.
  • Electron density n_e and layer potential U or D = n_e ~ (2–7)×10^11 cm^−2; U=70–100 meV; D=30–60 meV
    Representative densities and displacement/potential biases select single vs annular Fermi surfaces and topological vs trivial regimes; results are scanned in n_e but values are model choices.
  • Screened-Coulomb parameters ε, r_K (gapped model) = ε=5ε0, r_K=3 nm
    Rytova–Keldysh screening inputs for self-consistent p-wave gap in Appendix A.
  • Circular-band fit parameters k0, m(D) = D-dependent fits (not tabulated here)
    Appendix A treats k0 and m as D-dependent fitting parameters following Ref. [39] for the warping-free dispersion.
axioms (6)
  • domain assumption Two-band BdG Hamiltonian on the lowest conduction band in one spin-valley sector captures the relevant quasiparticle geometry and transport.
    Sec. II and Appendices A–B project to a two-component Nambu spinor; multi-band and inter-valley processes are omitted.
  • domain assumption The superconducting order is chiral p-wave intravalley pairing (Δ_k ∝ e^{iθ} or Δ a(k_x+ik_y)).
    Stated as the natural candidate from recent theory/experiment context; used throughout Sec. III–IV.
  • domain assumption Standard Kubo/Berry formulas for quasiparticle Berry curvature, orbital moment, thermal Hall, and spin/orbital Nernst apply to BdG bands as written in Eqs. (5)–(11).
    Imported from cited geometric-SC literature; no re-derivation of response theory.
  • domain assumption Trigonal warping makes ξ_k ≠ ξ_−k and can generate Bogoliubov Fermi surfaces for finite pairing.
    Sec. I–II and Appendix B; follows prior rhombohedral-graphene SC analyses [37,41].
  • domain assumption Continuum eight-band rhombohedral tetralayer hoppings and the circular N-layer dispersion are adequate normal-state inputs.
    Appendices A–B adopt parameters from Refs. [39,41].
  • standard math Linear response integrals with equilibrium Fermi/grand-potential factors describe the anomalous thermal and Nernst conductivities.
    Eqs. (9)–(11); standard semiclassical/Berry transport structure.

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Cite this review

Pith. "Pith review of Berry curvature effects of chiral superconducting rhombohedral graphene." pith.science (2026). https://pith.science/paper/K5HZDSLE

@misc{pith2026260723764,
  author       = {Pith},
  title        = {Pith review of: Berry curvature effects of chiral superconducting rhombohedral graphene},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K5HZDSLE}},
  note         = {Machine review of arXiv:2607.23764}
}
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read the original abstract

We study the Berry curvature effects of the Bogoliubov quasiparticles in chiral superconducting rhombohedral graphene. Using a two-band Bogoliubov-de Gennes Hamiltonian to describe the superconducting quasiparticles, we calculate the momentum-space Berry curvature, the orbital magnetic moment, the anomalous thermal-Hall and the anomalous spin- and orbital-Nernst transport of the chiral $p$-wave superconducting states. We investigate the impact of the normal-state energy band warping in rhombohedral graphene, which can cause the Bogoliubov Fermi surface for quasiparticle excitations. We find that the Bogoliubov Fermi surface qualitatively modulates the anomalous Hall transports, inducing a deviation of the thermal Hall conductivity from the quantized value and strongly enhancing the spin- and orbital-Nernst responses.

Figures

Figures reproduced from arXiv: 2607.23764 by Jian-Hua Zeng, Qian Niu, Zhi Wang.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Quasiparticle spectrum along [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Normal-state (gray line) and Bogoliubov Fermi surfaces, [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Temperature dependence of the thermal Hall conductivity [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Temperature dependence of the spin Nernst coefficient [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) shows the quasiparticle spectra for these two [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

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This paper was first reviewed by grok-4.5 on July 30, 2026.