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Thermal Hall tomography of chiral superconductivity in rhombohedral graphene

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Pith's one-line read The low-temperature thermal Hall conductance of a gapped chiral superconductor in rhombohedral graphene is quantized to $\kappa_{xy}/T = (\pi^2 k_B^2/6h)\,C_{\rm BdG}$, so measuring the plateau reads out the Bogoliubov–de Gennes Chern…

desk verdict A careful, honest proposal that connects thermal Hall quantization to BdG Chern tomography in rhombohedral graphene; the material-specific conclusion remains conditional on the occupied-vortex rule and the assumed pairing texture. read the letter →

arxiv 2608.12586 v1 pith:K2TOQE5P submitted 2026-08-12 cond-mat.mes-hall cond-mat.str-elcond-mat.supr-con

Kumar Ghosh This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.str-elcond-mat.supr-con
keywords thermalHalleffectchiralsuperconductivityrhombohedralgrapheneBogoliubov–deGennesChernnumberMajoranaedgemodesoccupied-vortexruletomographyBerrycurvature
verification ladder T0 review T1 audit T2 compute T3 formal

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 argues that the long-sought integer classifying a chiral superconductor—the Bogoliubov–de Gennes (BdG) Chern number—can be read directly from a low-temperature thermal Hall measurement in rhombohedral graphene, even though the normal state's Fermi surface is too intricate to reconstruct pocket by pocket. The key move is the occupied-vortex rule: for band-projected pairing, $C_{\rm BdG}$ equals the total winding of the gap zeros enclosed by the occupied regions of momentum space, so the measured plateau $\kappa_{xy}/T=(\pi^2 k_B^2/6h)\,C_{\rm BdG}$ acts as a topological checksum that needs no Fermi-surface tomography. The paper then shows that trigonal warping and finite Cooper pair momentum enter the BdG Hamiltonian only through an identity part that cannot affect the Chern number, and verifies numerically across 525 parameter points that the invariant survives while one inequality marks where a Bogoliubov Fermi surface removes the plateau. If the claim is right, the plateau's sign tracks imaged isospin domains, an even plateau would demand physics beyond single-valley pairing, and a written domain wall should carry $2|C_{\rm BdG}|$ Majorana channels—all testable with existing millikelvin thermometry.

What carries the argument

The load-bearing identity is the occupied-vortex rule, $C_{\rm BdG}=\sum_{i: k_i\in D_{\rm occ}}\ell_i$, which sums the winding numbers $\ell_i$ of the pairing gap zeros at momenta $k_i$ enclosed by the occupied region $D_{\rm occ}$ of the Fermi sea; it collapses an unreconstructed (possibly disconnected) Fermiology and a pairing texture into a single integer. Around it, the paper splits the intravalley BdG Hamiltonian as $H_{\rm BdG}=\xi_a(k)\tau_0 + \big(\xi_s(k)\tau_z+\mathrm{Re}\,\Delta_k\,\tau_x-\mathrm{Im}\,\Delta_k\,\tau_y\big)$, where the antisymmetric part $\xi_a$ (carrying trigonal warping and finite Cooper pair momentum $Q$) is proportional to the identity in Nambu space and therefore cannot enter the Berry curvature or the Chern number, while the symmetric part $\xi_s$ can change the invariant only if the direct gap $\eta(k)=\sqrt{\xi_s^2+|\Delta|^2}$ closes, i.e., when a pairing vortex crosses a Fermi sheet. Quantized transport is then controlled by the inequality $\delta_{\rm BdG}>0$; when it fails, a Bogoliubov Fermi surface removes the plateau while leaving the formal Chern number intact.

What would settle it

Measure $\kappa_{xy}/T$ at $T\ll T_c$ on a fully gapped rhombohedral graphene device with a prepared, imaged isospin domain: if the plateau is not an integer multiple of $4.732\times10^{-13}$ W/K² with sign following the domain, or if a direct self-consistent eight-band calculation gives a BdG Chern number disagreeing with the occupied-vortex charge, the central claim is falsified.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that in a gapped chiral superconductor realized in rhombohedral graphene, the zero-temperature thermal Hall conductance is exactly quantized as $\kappa_{xy}/T=(\pi^2k_B^2/6h)\,C_{\rm BdG}$, so the measured heat transport simply counts the chiral Majorana edge modes. The integer $C_{\rm BdG}$ can be computed without reconstructing the Fermi sea: for band-projected pairing it equals the occupied-vortex charge, the sum of winding numbers of the pairing gap zeros that lie inside the occupied regions of momentum space, which may be disconnected. The paper's analytic decomposition of the intravalley BdG Hamiltonian into symmetric and antisymmetric parts shows that trigonal warping and finite pair momentum are topologically inert—they enter only through the identity term $\xi_a\tau_0$—and direct Chern calculations across 525 parameter points confirm the invariant is unchanged; quantization is lost only when $\delta_{\rm BdG}\equiv\min_k[\sqrt{\xi_s^2+|\Delta|^2}-|\xi_a|]\le 0$, which is precisely the appearance of a Bogoliubov Fermi surface. The plateau therefore reads out the integer in units of the Majorana thermal quantum, its sign reverses with the imaged isospin domain, an even value excludes single-valley same-spin pairing, and a written domain wall between opposite domains should carry $2|C_{\rm BdG}|$ co-propagating Majorana channels.

Load-bearing premise

The central claim rests on importing the occupied-vortex rule from band-projected pairing on a Chern band to the realistic eight-band rhombohedral graphene model, and on assuming that the actual pairing texture contains the Berry-curvature-nucleated momentum-space vortices that the rule sums.

Editorial extensions

If this is right

  • At $T\ll T_c$, the plateau $\kappa_{xy}/T=C_{\rm BdG}\times 4.732\times10^{-13}$ W/K² returns the BdG Chern number directly, so odd versus even values classify the pairing (even excludes same-spin single-valley intravalley pairing).
  • Reversing an imaged isospin domain should reverse the sign of $\kappa_{xy}$ at fixed magnitude, providing a direct test of chirality inheritance from the parent state.
  • A written domain wall separating $C_{\rm BdG}=+C$ and $-C$ fully gapped domains should conduct $2|C|$ co-propagating Majorana channels, giving a switchable heat path with conductance $K_{DW}/T=|C|\pi^2k_B^2/3h$.
  • Because the occupied-vortex rule sums windings over whatever is occupied, the plateau height is independent of how the quantum-oscillation spectrum is eventually decomposed into pockets.
  • A gate-driven integer change of the plateau must coincide with a bulk gap closing, so simultaneous thermal Hall, longitudinal thermal, and spectroscopic measurements can separate a genuine topological transition from domain repopulation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extending the logic beyond rhombohedral graphene, the same thermal Hall plateau could serve as the topological readout in other chiral superconductor candidates with complex fermiology (e.g., UTe2 or kagome metals), provided a fully gapped chiral phase and a way to control or image domains; the plateau height, not the magnetic signal, would be the protected observable.
  • A testable extension is to report the pair ($\kappa_{xx}/T$, $\kappa_{xy}/T$) across the phase diagram: activated $\kappa_{xx}$ with an integer $\kappa_{xy}/T$ would certify both gapped and quantized, whereas a residual $\kappa_{xx}/T$ with non-integer $\kappa_{xy}/T$ would localize exactly the Bogoliubov Fermi surface window predicted by Eq. (7) on the same device.
  • If the multitone orbits indeed sit near the Berry-curvature ring, modest gate tuning should drive the vortex pair across a Fermi-sheet crossing, producing a gate-induced jump in the plateau accompanied by a gap closing and a crossover in the superfluid stiffness from activated to power-law; this is a falsifiable prediction of the vortex-nucleation picture.
  • The symmetric/antisymmetric decomposition is a general tool: for any superconductor with asymmetric dispersion $\varepsilon(k)\neq\varepsilon(-k)$, the antisymmetric part cannot affect the Chern number as long as the direct gap stays open, so thermal Hall quantization survives warping in a wider class of non-centrosymmetric superconductors.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript proposes 'thermal Hall tomography' for chiral superconductivity in rhombohedral graphene: at T→0 the thermal Hall conductance is quantized as κ_xy/T = (π²k_B²/6h) C_BdG, and C_BdG is argued to equal the occupied-vortex charge of Eqs. (9) and (10), so a measured plateau would read out the BdG Chern number without reconstructing the Fermi surface. The authors split the intravalley BdG Hamiltonian into symmetric and antisymmetric dispersion parts, show that the antisymmetric part is topologically inert, and validate the invariant with direct Fukui–Hatsugai–Suzuki, Wilson-loop, and DMRG computations on model systems, including an annular Fermi sea with a three-vortex gap texture. They also derive a curvature-resolved finite-temperature transport kernel, identify a Bogoliubov-Fermi-surface criterion for loss of quantization, and outline an experimental programme based on existing floating-contact Johnson-noise thermometry. The paper is explicit about several of its own limitations, including the hand-inserted vortex ansatz and the conditional nature of the above-Tc continuation.

Significance. If the material-specific premises were established, this would be a significant result: a direct, protected measurement of the BdG Chern number in a chiral superconductor, with a domain-sign correlation and a domain-wall heat-channel prediction that are both testable. The strengths include the exact T→0 quantization statement, the machine-checked FHS and Wilson-loop certificates with stated admissibility margins, the careful separation of the single-mass tanh envelope from the curvature-resolved Kubo kernel, and unusually honest statements about what each numerical check does and does not test. The force of the paper is currently limited by two material-specific assumptions: the transfer of the occupied-vortex rule to the realistic eight-band multiband model, and the assumed pairing-vortex texture in rhombohedral graphene. Neither is derived from the actual R4G Hamiltonian or directly computed here, so the central 'tomography without fermiology' claim remains conditional.

major comments (3)
  1. [§II.C and §II.D, Eqs. (9)–(12)] The occupied-vortex rule is imported from Ref. [13], where it was established for band-projected pairing on a Chern band, and is then applied to the eight-band R4G BdG problem without a derivation or a direct test in that model. The annular three-zero texture is an inserted analytic proxy, as the text itself says ('inserted by hand into an analytic gap ansatz, as a controlled proxy'), and the underlying single-band model tests topological bookkeeping rather than the multiband projection. The central claim that a measured plateau reads out the occupied-vortex charge therefore rests on an unverified material-specific premise: if the rule acquires corrections from band-off-diagonal pairing or from the disconnected multitone Fermi sea, or if the actual gap texture lacks these vortices, the plateau could be quantized yet differ from the occupied-vortex charge. The revision should either derive Eq. (9) for the realistic eight-band Hamiltonian or provide a direct numerical test with a self-consistent R4G gap texture.
  2. [§III.A and §II.D; §III.D, Eq. (28)] The vortex positions, windings, and their nucleation by the parent Berry curvature are assumed, not computed for rhombohedral graphene. The paper delegates nucleation to Refs. [10,13], and the Berry-ring comparison in §III.D relies on a semiclassical reading of the quantum-oscillation tones that the paper itself notes is disputed by the magnetic-breakdown interpretation [33]. A self-consistent eight-band calculation could place gap zeros elsewhere, produce no zeros, or change the winding; none of these outcomes is excluded by the 525-point scan of §III.A, which uses the fixed texture of Eq. (12) throughout. Because the experimental prediction depends on the actual pairing texture, the revision should include at least one material-specific computation of the gap texture or a sharp argument from the R4G parent state, rather than a further extension of the proxy model.
  3. [§II.F, Eq. (24)] The finite-pair-momentum thermal weight W_Q is asserted, with 'G continued oddly', rather than derived from Eq. (20). For the asymmetric branches E_± = ξ_a ± η, the Berry-curvature occupation and the band summation are not the same as in the particle–hole-symmetric case, and the replacement of G by the average of the two branch weights requires a derivation that goes beyond the ξ_a = 0 limit. Since the finite-temperature curves in Figs. 1 and 5(f) use this kernel, the temperature dependence above T = 0 is not yet fully supported. The T→0 plateau itself is unaffected, but the derivation should be supplied or the kernel should be stated as an approximation with a controlled error estimate.
minor comments (4)
  1. [§II.D, paragraph 2] The text refers to 'RNG' in the sentence about parent-band Berry curvature nucleating vortices; this should presumably be 'R4G' or 'rhombohedral graphene'.
  2. [§III.H, Eq. (35)] Equation (35) uses the tanh envelope even though §II.E and §II.F state that the tanh form is not the transport law for a dispersing BdG band; please add an explicit sentence clarifying whether Eq. (35) is the same single-mass comparison form applied under the quasiparticle-continuation hypothesis or a separate approximate transport law.
  3. [References [14] and [21]] References [14] and [21] appear to cite the same arXiv preprint (2411.02503) in different forms; they should be consolidated or clearly distinguished to avoid confusion about the sources of the eight-band results.
  4. [Appendix O, Fig. 5(e)] The Wilson-loop finite-width analysis is stated as consistent with an exponential envelope and inconsistent with a pure power law; the text already concedes that a polynomial prefactor is not excluded, so it would be helpful to state in the main text that this is a qualitative consistency check rather than a precision test of the c1 = 0 theorem.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation of the central plateau; only minor non-load-bearing self-citations for the finite-temperature envelope.

full rationale

The paper's central derivation chain is self-contained and not circular. The zero-temperature plateau, Eq. (1), is the standard class-D result cited to Refs. [1,25,30,34], and the finite-temperature transport law Eq. (21) is derived independently in Appendix B from the Qin-Niu-Shi Kubo formula, Eq. (20), without using the author's self-citations. The occupied-vortex rule, Eq. (9), is an external result from Ref. [13], and the paper checks it against direct Fukui-Hatsugai-Suzuki Chern number calculations rather than fitting it to itself: in Sec. III.B, 'The FHS and Q_analytic curves agree at every scan point, with max|C_FHS_BdG - Q_analytic| < 10^-14'. The symmetric/antisymmetric decomposition of Sec. II.B is analytical, and the 525-point scan in Sec. III.A is direct numerics with no parameter tuned to force the invariant. The only self-citations are Refs. [22-24], which support the finite-temperature Poisson kernel and the c1=0 finite-size statement; the paper explicitly demotes the single-mass envelope to comparison use: 'We use it only as a comparison envelope. It is not the finite-temperature transport law for a dispersing BdG band', and Appendix F limits the c1=0 theorem to the continuum single-mass determinant. The illustrative gap ratio and T* are explicitly not inputs to the zero-temperature claim, and the Delta_eff = 0.837 rescaling in Appendix D is openly a one-parameter comparison fit, not a prediction. The paper also states its own limitations, including Sec. II.D: the finite-momentum vortices are 'inserted by hand into an analytic gap ansatz, as a controlled proxy' and nucleation 'we rely on the self-consistent gap equations of Refs. [10,13]'; these are honest external premises, not circular reductions. Any concern that the occupied-vortex rule may fail for the realistic multiband R4G model is a correctness risk, not a circularity, because the rule's target conclusion is not used as its own input. The minor self-citations are non-load-bearing, so the score is 2 rather than 0.

Assumptions & free parameters 7 free parameters · 7 assumptions · 0 invented entities

The central claim rests on the imported occupied-vortex rule, the assumed chiral pairing texture, and the model dispersions. The many scanned parameters are not fitted to data; they test robustness. The only numbers chosen for the illustrative finite-temperature curves are explicitly declared non-inputs.

free parameters (7)
  • r_in, r_out (annulus radii) = 0.45, 1.35 (isotropic); 0.20, 0.55 (warped)
    Hand-chosen to model the annular Fermi sea produced by a Lifshitz transition; the topological invariant is independent of their values within the gapped regime.
  • Delta_0 (pairing amplitude) = 0.50 (isotropic), 2.0 (warped) in model units
    Chosen by hand; the Chern number is independent of the gap magnitude as long as the direct gap stays open.
  • q (gap-texture parameter) = 0.90 reference
    Sets the finite-momentum vortex pair position; chosen to place the pair inside the annulus at reference parameters.
  • w (trigonal warping strength) = scanned 0 to 6
    Scanned to test robustness; no single fitted value, but the scan range is a modeling choice.
  • |Q| (pair momentum) = scanned 0 to 0.30 in model units; reference 0.1 k_F from Ref. [14]
    Scanned; the reference value is taken from an eight-band calculation, not fitted here.
  • 2 Delta_0 / k_B T_c and T*/T_c = 20 and 5
    Illustrative parameters used only for Fig. 5f; the paper explicitly states they are not measured inputs.
  • u_D (interlayer potential for Berry-ring estimate) = 42.8 meV
    Taken from the modeling of Ref. [8]; used for the k_Omega comparison in Sec. III D, not fitted here.
assumptions (7)
  • domain assumption The occupied-vortex rule of Le Nir, Mitra and Kim: C_BdG equals the total winding of gap zeros enclosed by the occupied Fermi sea (Eqs. 9-10).
    Imported from Ref. [13], a preprint; the paper does not derive it and relies on it for the central reading of the plateau.
  • domain assumption The normal state of rhombohedral tetralayer graphene inside the superconducting dome is a multitone metal with two density-independent oscillation frequencies, and the parent state is an orbital ferromagnet with switchable isospin domains.
    Taken from Refs. [7,8]; the argument that the plateau bypasses the need for Fermi-surface reconstruction depends on this fermiology being unresolved and complicated.
  • domain assumption The superconducting gap has chiral p+ip or d+id texture with vortices whose positions and windings are inherited from the parent Berry curvature.
    Assumed from Refs. [9-13]; the paper explicitly states that vortex nucleation by the parent band is not demonstrated in this work.
  • domain assumption The minimal dispersion model Eq. (3) captures the low-energy band structure of N-layer rhombohedral graphene.
    Adopted from Refs. [11,18]; used for the Berry-ring radius and the quarter-metal benchmark.
  • standard math Fukui-Hatsugai-Suzuki discretization gives the exact Chern number when max|F_plaq| < pi.
    Standard numerical method; the paper verifies the admissibility margin at every reported integer.
  • standard math Coleman-Hill non-renormalization and the c1=0 theorem control finite-size and interaction corrections to the parity-odd kernel.
    Imported from Refs. [23,35]; used for the finite-temperature envelope, not for the zero-temperature plateau.
  • ad hoc to paper The quasiparticle continuation hypothesis above T_c, namely long-lived chirally oriented pairs, is a conditional assumption.
    The paper explicitly labels it as a hypothesis in Sec. III H and separates it from the primary claim.

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Pith. "Pith review of Thermal Hall tomography of chiral superconductivity in rhombohedral graphene." pith.science (2026). https://pith.science/paper/K2TOQE5P

@misc{pith2026260812586,
  author       = {Pith},
  title        = {Pith review of: Thermal Hall tomography of chiral superconductivity in rhombohedral graphene},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K2TOQE5P}},
  note         = {Machine review of arXiv:2608.12586}
}
abstract

A chiral superconductor carries chiral Majorana modes along its edges, and a single integer, the Bogoliubov--de Gennes Chern number, counts them. Thirty years of candidate materials have not yielded a measurement of that integer, because the magnetic signatures usually invoked are not topologically protected. Rhombohedral graphene makes the question both urgent and answerable: magnetic imaging resolves rewritable time-reversal-breaking domains inside the superconducting phase, while quantum oscillations reveal a normal state too intricate to reconstruct pocket by pocket. We show that the low-temperature thermal Hall conductance returns the integer directly, with no such reconstruction. For band-projected pairing it equals the pairing-vortex winding enclosed by the occupied regions of momentum space. Splitting the intravalley Hamiltonian into symmetric and antisymmetric parts isolates the trigonal warping and finite Cooper pair momentum of the real material: the antisymmetric part is topologically inert, direct Chern calculations across $525$ parameter points show the invariant preserved, and one inequality marks where a Bogoliubov Fermi surface removes quantization. The plateau $\kappa_{xy}/T=(\pi^2k_B^2/6h)\,C_{\rm BdG}$ then reads out the integer, its sign reverses with the imaged domain, a written domain wall should carry $2|C_{\rm BdG}|$ Majorana channels, and the thermometry required already resolves single thermal quanta in encapsulated graphene at millikelvin temperatures.

Figures

Figures reproduced from arXiv: 2608.12586 by the authors.

Figure 1
Figure 1. FIG. 1. Finite-temperature response framework. (a) The coherent quasiparticle contribution obtained from the curvature [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Trigonal warping and finite pair momentum preserve [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Momentum-space tomography made explicit. Gap [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Direct FHS validation of the even-Chern annular phase. Vortex-crossing scan over the finite-vortex pair radius 2 sin( [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Six-panel numerical validation summary. (a) FHS integer stability for the square-lattice [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. DMRG tensor-network benchmark on 28 converged ground states ( [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Tensor-network finite-width [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Ratio-method correlation length from DMRG energy shifts on consecutive cylinders, compared to the continuum [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.