{"id":"d0acbaf7-a21c-4925-9f78-eb7a2b81e335","arxiv_id":"2608.12586","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"Thermal Hall conductance at low temperature returns the BdG Chern number in rhombohedral graphene through the occupied-vortex rule, with trigonal warping and finite pair momentum shown to be topologically inert.","lead":"This paper argues that the low-temperature thermal Hall conductance of a chiral superconductor directly reveals its topological invariant, the Bogoliubov-de Gennes Chern number, in rhombohedral graphene. It combines an existing vortex-counting rule with a new decomposition of the Hamiltonian to make the measurement practical, and proposes a concrete experimental protocol.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central claim depends on the occupied-vortex rule and Berry-curvature-nucleated gap zeros in R4G, neither of which is derived for the realistic multiband model.","rationale":"The reader's weakest-assumption analysis identifies exactly the load-bearing step: the occupied-vortex rule is imported from band-projected Chern-band models and the pairing-texture vortices are inserted by hand rather than obtained from R4G. The present stress-test agrees. The standard class-D quantization Eq. (1), the finite-temperature Kubo kernel Eq. (21), and the numerical certificates (FHS, Wilson loop, DMRG) are not the weak point; they support the toy-model bookkeeping but do not certify the material-specific reduction. The paper itself honestly flags the delegation in Sec. II.D, which counts in its favor, but the central claim is nevertheless conditional on a self-consistent multiband calculation or a direct experimental readout in a regime where the longitudinal conductance certifies the gap. Because the reader already assigned CONDITIONAL and this concern does not move that assessment, the recommended verdict is UNCHANGED.","tokens_in":33924,"tokens_out":4846,"duration_ms":58220,"concrete_test":"Run a self-consistent mean-field calculation on the eight-band R4G model of Ref. [14] inside the superconducting dome (e.g., n_e ≈ 0.5×10^12 cm^-2, D = 42.8 meV, with Q up to 0.1 k_F and other parameters as in that work), starting from the bare band Hamiltonian without inserting any gap ansatz. From the converged normal self-energy and pairing function Δ(k), locate all zeros and compute their windings; define the occupied region as the set where the paired-band normal-state energy is negative, including all disconnected sheets; compute the direct BdG Chern number by FHS on at least a 301×301 k-grid with max plaquette flux below π. Repeat at five densities spanning the dome.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central assertion—that a measured plateau in R4G reads out C_BdG by occupied-vortex counting—rests on two unproven, material-specific premises. (P1) Eq. (9)/(10), imported from Ref. [13] where it was derived for band-projected pairing on a Chern band, remains valid for the realistic eight-band, spin-valley-resolved BdG problem with a disconnected multitone Fermi sea. The paper does not derive or test this; its annular 'multitone' proxy is a single-band ξ_s<0 region with an inserted three-zero texture, which checks topological bookkeeping but not the multiband projection. (P2) The actual R4G pairing texture has vortex-like zeros at Berry-curvature-nucleated positions, with the windings and locations assumed in Secs. II.C–III. The paper is explicit in Sec. II.D that the vortices are 'inserted by hand into an analytic gap ansatz, as a controlled proxy' and that nucleation is delegated to Refs. [10,13], not computed for R4G. If either premise fails, C_BdG may be quantized and measured correctly while differing from the occupied-vortex charge, so the central 'tomography without fermiology' conclusion would not follow. A self-consistent eight-band calculation could yield gap zeros elsewhere, no zeros, or a rule correction from band off-diagonal pairing; none of these are excluded by the presented numerics.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":34291,"tokens_out":5168,"duration_ms":53761,"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":[{"comment":"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.","section":"§II.C and §II.D, Eqs. (9)–(12)"},{"comment":"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.","section":"§III.A and §II.D; §III.D, Eq. (28)"},{"comment":"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.","section":"§II.F, Eq. (24)"}],"minor_comments":[{"comment":"The text refers to 'RNG' in the sentence about parent-band Berry curvature nucleating vortices; this should presumably be 'R4G' or 'rhombohedral graphene'.","section":"§II.D, paragraph 2"},{"comment":"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.","section":"§III.H, Eq. (35)"},{"comment":"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.","section":"References [14] and [21]"},{"comment":"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.","section":"Appendix O, Fig. 5(e)"}],"recommendation":"major_revision","confidential_remarks":"The paper is well within the journal's scope and is unusually candid about its own limitations, which I view as a strength. The main risk is that the central claim depends on two unproven material-specific inputs, the transfer of the occupied-vortex rule to the eight-band model and the assumed pairing texture in rhombohedral graphene; these are exactly the points where the revision must add either a derivation or a direct numerical computation. I do not see these as unfixable in principle, so I recommend major_revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Kumar Ghosh has written a careful and unusually honest proposal. The core idea: the low-temperature thermal Hall plateau in rhombohedral graphene should read out the BdG Chern number in units of the Majorana quantum, without reconstructing the multitone Fermi surface. The new content is the application of the occupied-vortex rule to this material, the symmetric/antisymmetric decomposition that puts trigonal warping and finite pair momentum into a topologically inert identity term, and a 525-point numerical sweep verifying the invariant under both. The experimental protocol, including domain-resolved sign reversal and a domain-wall heat channel, is concrete.\n\nThe strengths are real. The analytic argument is clean, the numerics are careful, and the paper is unusually explicit about what it does not prove. The DMRG run is framed as a benchmark of an explicitly paired quadratic Hamiltonian, not as evidence for spontaneous chirality. The distinction between the exact transport kernel and the single-mass tanh envelope is a useful correction to a common sloppiness. The experimental section is grounded in existing thermometry and imaging.\n\nThe soft spot is the material-specific bridge. The occupied-vortex rule is imported from Le Nir, Mitra and Kim, where it was derived for band-projected pairing on a Chern band; the paper does not derive it for the eight-band rhombohedral problem. The numerical test uses a single-band annular proxy with a three-zero gap texture inserted by hand, and the nucleation of those vortices by Berry curvature is delegated to other work. The paper says this plainly, but it leaves the central prediction conditional: if the actual R4G pairing texture lacks those vortices, or if the rule fails on a multitone Fermi sea, the plateau would still be quantized but would not equal the occupied-vortex charge. A self-consistent eight-band calculation, or the experiment itself, is the way to close that gap.\n\nMinor points: the finite-Q kernel in Eq. (24) is asserted rather than derived, though the T→0 plateau is insensitive to it. The self-citations to Refs. [23,24] back the finite-size envelope, not the plateau itself, so I do not see a circularity problem.\n\nThis paper deserves a serious referee. It is important, well-executed, and honest about its limits. I would send it out, with the instruction to push on the occupied-vortex rule's validity in the multiband model and on the pairing texture assumption.","headline":"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.","tokens_in":34783,"tokens_out":5248,"would_cite":true,"duration_ms":49364,"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":"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…","keywords":["thermal Hall effect","chiral superconductivity","rhombohedral graphene","Bogoliubov–de Gennes Chern number","Majorana edge modes","occupied-vortex rule","thermal Hall tomography","Berry curvature"],"falsifier":"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.","tokens_in":33717,"feed_emoji":"🌡️","tokens_out":16258,"duration_ms":113683,"temperature":0.7,"pith_summary":"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.","feed_headline":"Read out a chiral superconductor's topological integer","feed_subtitle":"A single thermal Hall plateau replaces Fermi surface reconstruction and reveals the hidden integer.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the occupied-vortex rule, the central identity that equates the BdG Chern number with the winding of gap zeros enclosed by the occupied Fermi sea.","marker":"[13]"},{"why":"Provides the self-consistent eight-band R4G model with finite pair momentum up to 0.1 kF and the location of Bogoliubov Fermi surfaces.","marker":"[14]"},{"why":"Establishes that ring-concentrated Berry curvature nucleates momentum-space vortices and can produce higher odd Chern numbers.","marker":"[10]"},{"why":"Gives the exact Berry-trashcan p+ip solution tying the gap orientation to parent Berry curvature.","marker":"[9]"},{"why":"Supplies the imaging of rewritable isospin domains inside the superconducting phase, giving the sign axis and domain-wall tests.","marker":"[7]"},{"why":"Demonstrates floating-contact Johnson-noise thermometry resolving individual thermal quanta, establishing the measurement's feasibility.","marker":"[5]"},{"why":"Reports the multitone quantum-oscillation data that make the normal state too intricate to reconstruct, motivating the topological checksum.","marker":"[8]"},{"why":"Provides the Lifshitz-reconstruction scenario with annular Fermi surfaces and the minimal dispersion used as the benchmark.","marker":"[11]"}],"fun_headline_variants":["Thermal Hall plateau counts Majorana modes in graphene","Chiral superconductor's Chern number from heat flow","Rhombohedral graphene thermal Hall reads topological integer","No Fermi surface recon: thermal Hall gives Chern number","Quantized thermal Hall in rhombohedral graphene reveals integer"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Thermal Hall plateau counts Majorana modes in graphene","Chiral superconductor's Chern number from heat flow","Rhombohedral graphene thermal Hall reads topological integer","No Fermi surface recon: thermal Hall gives Chern number","Quantized thermal Hall in rhombohedral graphene reveals integer"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000228,"raw_usage":{"total_tokens":1559,"prompt_tokens":1113,"completion_tokens":446,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":729,"completion_tokens_details":{"reasoning_tokens":368}},"tokens_in":729,"tokens_out":446,"duration_ms":4286,"temperature":1.0,"reasoning_tokens":368,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:04:51.606657+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Chiral superconductors from parent states with non-uniform Berry curvature: Momentum-space vortices, BdG topology, and thermal Hall conductivity","cited_arxiv_id":"2605.21618","evidence_quote":"Supplies the occupied-vortex rule, the central identity that equates the BdG Chern number with the winding of gap zeros enclosed by the occupied Fermi sea."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that ring-concentrated Berry curvature nucleates momentum-space vortices and can produce higher odd Chern numbers."},{"cited_title":"Banerjee, M","cited_arxiv_id":null,"evidence_quote":"Demonstrates floating-contact Johnson-noise thermometry resolving individual thermal quanta, establishing the measurement's feasibility."}],"review_version":1}