{"id":"f73e4ede-18d6-4363-9192-6666923b5088","arxiv_id":"2501.13168","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"A 2D topological insulator-superconductor heterostructure is shown to exhibit Landau levels of Bogoliubov quasiparticles in the vortex state, with Chern number ±1 per level, leading to peaked magneto-optical conductivity and quantized thermal Hall effect.","lead":"This theoretical paper predicts that a two-dimensional topological insulator-superconductor heterostructure can host Landau levels of superconducting quasiparticles in a magnetic field. These levels should produce sharp optical absorption peaks and a quantized thermal Hall effect, giving concrete experimental signatures for a long-sought phenomenon.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Supercurrent absorption into an effective vector potential leaves a nonuniform flux-line field; the full sqrt(n) LL ladder and Chern numbers hinge on an unverified suppression of vortex-lattice scattering.","rationale":"The Reader's weakest assumption identifies the same nerve: the vortex-lattice supercurrent is treated as an effective vector potential that yields LLs rather than Bloch waves. I agree, and I sharpen the objection: A_eff = a + κm u_s has a nonuniform curl with delta-function flux lines at the vortex cores for any κ ≠ -1, so even the low-energy model Eq. (3) is not equivalent to a uniform magnetic field. The generalized chiral symmetry is sufficient for a zero-energy Landau level, but the paper needs the full set of LLs and their Chern numbers; that requires that scattering by the periodic flux lattice be negligible, which is not shown analytically. The numerical spectra in Fig. 1 are the only support, and the supplemental material containing the method is absent from arXiv v1. This is not an accusation of error—the physics may well be correct—but it is a load-bearing assumption that must be checked with a full model. Hence the conditional verdict should stand, pending the proposed numerical test.","tokens_in":12030,"tokens_out":14144,"duration_ms":159171,"concrete_test":"Run a real-space BdG calculation for the Hamiltonian in Eq. (8) on a vortex lattice using the full spatial profile of the pair potential (e.g., a self-consistent Abrikosov solution with finite core size), not the uniform-|Δ| / first-order-in-u_s reduction. Compute the QP spectrum and Chern numbers for l_B = 22, 45, and 80, and for a few core sizes ξ. Check whether the eigenstates remain dispersionless, whether the energies follow E_n = √n E_1, and whether each non-degenerate LL keeps Chern number ±1. If the √n ladder and integer Chern numbers survive with widths vanishing as l_B increases, the supercurrent-absorption assumption is validated; if the spectrum broadens into Bloch bands or the Chern numbers change, the central claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central LL claim rests on the reduction of the vortex lattice to the effective vector potential A_eff = a + κ m u_s in Eq. (3), with a = (1/2)∇φ and m u_s = a - eA. Even accepting the first-order-in-u_s reduction (which footnote [19] explicitly limits), A_eff is not a uniform-field vector potential: its curl is B_eff = (1+κ)π Σ_i δ(r-r_i) - κ eB, since each 2π phase winding contributes ∮a·dl = π. The total flux per magnetic unit cell is 2π for any κ, but the field is spatially nonuniform unless κ = -1, i.e. Δ0 = 0. A periodic array of flux lines with per-cell flux does not generically produce the Dirac-LL ladder E_n = √n E_1; Franz-Tesanovic (Ref. [5]) showed that spatially varying supercurrent converts would-be LLs into Bloch waves. The generalized chiral symmetry ζ†Hζ = -H protects a dispersionless zero mode, but it does not by itself enforce the entire √n spectrum, the ±1 Chern numbers of every LL, or the quantization of σ̃_xy in Eq. (13). The assertion that the supercurrent 'does not scatter the LLs' (text after Eq. (3)) is therefore the load-bearing step, and the only direct evidence is the numerical Fig. 1, whose construction is relegated to the missing Supplemental Material [17].","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Landau quantization of Bogoliubov quasiparticles in the mixed state of nodal superconductors. It first contrasts intrinsic and heterostructure Weyl superconductors, arguing that the presence or absence of a generalized chiral symmetry determines whether supercurrents act as an effective vector potential without scattering the Landau levels. It then proposes a two-dimensional topological-insulator/s-wave-superconductor heterostructure whose vortex lattice is claimed to host Dirac-like Landau levels En = sqrt(n)E1 with Chern number ±1, leading to peaked magneto-optical conductivity and a quantized thermal Hall conductivity κxy/T in the low-temperature limit. The analytic symmetry argument is coherent, but the quantitative claims—the LL spectrum, Chern numbers, conductivity peaks, and thermal Hall plateaus—are supported only by numerical calculations placed in a Supplemental Material that is not included with the arXiv v1.","tokens_in":12400,"tokens_out":5763,"duration_ms":64320,"significance":"If correct, the central proposition is significant: it would provide a concrete 2D superconducting platform in which the vortex lattice does not destroy Landau quantization, yielding a genuine Landau-level spectrum of Bogoliubov quasiparticles with measurable magneto-optical and thermal Hall signatures, and it would identify a useful symmetry criterion for other nodal superconductors. The paper also makes falsifiable predictions, namely the √n + √(n+1) optical peak positions and the quantized low-temperature values of κxy/T. The analytic symmetry-based distinction between the intrinsic and heterostructure Weyl cases is a valuable conceptual contribution. However, the verification of the central claims currently depends entirely on numerical results whose details are not available in the submitted manuscript, so the significance cannot be fully assessed as submitted.","major_comments":[{"comment":"The central numerical result—the Dirac-LL ladder En = sqrt(n)E1 in the vortex lattice of Hamiltonian (8)—is stated to follow from calculations described only by reference [17], 'See the Supplemental Material'. The arXiv v1 contains no Supplemental Material, no code, and no convergence checks. Because this figure is the sole direct evidence that the vortex lattice does not destroy Landau quantization in the proposed 2D system, the manuscript as submitted does not allow a referee or reader to verify the main claim. Please include the Supplemental Material or an equivalent appendix with the full real-space BdG Hamiltonian, the vortex phase profile, the magnetic unit cell construction, boundary conditions, k-point sampling, the values of L and l_B used, and finite-size scaling checks.","section":"LLs in the mixed state of 2D superconductors, Fig. 1(b)"},{"comment":"The statement that the supercurrent velocity 'does not scatter the LLs' is load-bearing but is not established. From a = (1/2)∇φ and m u_s = (1/2)∇φ − eA, the effective vector potential A_eff = a + κ m u_s has curl ∇×A_eff = (1+κ)π Σ_i δ(r−r_i) − κ eB. Unless κ = −1, which corresponds to Δ0 = 0, the effective magnetic field is spatially nonuniform with flux-line singularities, and a periodic nonuniform field generically converts would-be Landau levels into Bloch bands, as shown in Ref. [5]. The generalized chiral symmetry protects only the zero mode, not the full √n ladder or the Chern numbers of every level. Please provide a quantitative argument, such as an estimate of the vortex-lattice scattering potential relative to the level spacing or a controlled limit, showing that this scattering is suppressed in the parameters used in Fig. 1.","section":"Text after Eq. (3)"},{"comment":"The paper explicitly neglects virtual transitions of second order in the supercurrent velocity v_s. Near each vortex core v_s diverges as 1/r, so the neglected terms are not uniformly small over the magnetic unit cell. Please estimate the magnitude of these terms and their effect on the Landau-level spectrum and Chern numbers. If the numerical calculation already includes the full BdG Hamiltonian without this second-order truncation, that should be stated explicitly, and a comparison between the truncated effective theory and the full numerics should be shown.","section":"Footnote [19], second-order-in-v_s terms"},{"comment":"The magneto-optical and thermal Hall conductivities are the main observable predictions, but the operator HM appearing in the Landau-level basis is not defined in the main text, and the velocity operator of Eq. (11) is written in a form whose gauge invariance with respect to the vortex phase and the Peierls substitution is not demonstrated. Please provide HM in the magnetic unit cell, define the current operator used, and give a derivation or reference showing that the Kubo formula in the form of Eq. (10) yields a gauge-invariant conductivity when evaluated in the magnetic Brillouin zone.","section":"Equations (10)–(13)"}],"minor_comments":[{"comment":"In Fig. 1(a), the two-band model is plotted as E/E1 while the single-band model is plotted as E/(0.6E1), but both curves appear on the same vertical axis; this mixing of scales should be clarified to avoid a misleading visual comparison.","section":"Fig. 1 caption"},{"comment":"The reference to the Supplemental Material contains only the phrase 'See the Supplemental Material'; a detailed document should be provided with the submission, and the arXiv version should include it as an ancillary file.","section":"Reference [17]"},{"comment":"The captions quote l_B = 45 and l_B = 46 without stating the unit; since the lattice constant is set to 1, these values appear to be in units of a0, and the corresponding magnetic unit cell sizes and total flux should be stated explicitly.","section":"Fig. 1 parameters"},{"comment":"For the green dashed curve, the parameters β = 1.5 and Δ0 = 1 do not satisfy the gapless condition β^2 = 1 + Δ0^2, and the text states that the conductivity peaks shift, but no formula or numerical estimate is given for the shifted peak positions.","section":"Fig. 3"},{"comment":"The low-temperature limit in Eq. (14) assumes that σ̃xy(ξ) is evaluated at ξ = 0 and that no other energy scale enters; please state explicitly the condition that the chemical potential lies in a gap of the Bogoliubov spectrum, since this is required for the quantization of κxy/T.","section":"Eq. (12)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within scope for cond-mat.supr-con and the central idea is attractive. The main barrier to acceptance is verifiability: all quantitative claims—the Landau-level spectrum, Chern numbers, optical conductivity peaks, and thermal Hall plateaus—are documented only in a missing Supplemental Material, with no code, data, or convergence checks. If the Supplemental Material can be provided and the convergence and gauge-invariance concerns addressed, the paper would be suitable for publication. I see no circularity or citation-practice problem: the new 2D model is analyzed consistently, and the dependence on prior work is properly acknowledged."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this paper gives a clean symmetry criterion for when vortex-lattice supercurrents quantize Bogoliubov quasiparticles into Landau levels rather than Bloch waves, and it proposes a concrete 2D TI-superconductor heterostructure that should realize the effect. The intrinsic vs heterostructure Weyl superconductor distinction is genuinely new, and the experimental signatures (magneto-optical peaks, quantized thermal Hall) are specific enough to matter. The paper deserves serious peer review, but the posted version is not self-contained: the load-bearing numerics are all in a Supplemental Material that is not included on arXiv, and there is no code or convergence analysis.\n\nThe symmetry argument is the strongest part. In the heterostructure Weyl case the supercurrent enters as an effective vector potential and respects the generalized chiral symmetry, while in the intrinsic case it breaks it. That is a real step beyond the Franz-Tesanovic picture.\n\nNow the soft spots. The stress-test concern about A_eff not being a uniform field is legitimate and not addressed in the text. Even accepting the first-order-in-v_s reduction, which footnote 19 explicitly limits, the effective field is a periodic array of flux-line singularities plus a uniform piece. The generalized chiral symmetry protects the zeroth LL, but it does not by itself enforce the full sqrt(n) ladder, the ±1 Chern numbers, or the quantized sigma_xy. The paper asserts that the supercurrent does not scatter the LLs, and the only direct evidence is Fig. 1, whose construction sits in the missing SM. So the central claim is plausible but not verifiable from what is posted.\n\nAlso, the contrast between two-band and single-band d-wave superconductors is only shown numerically; no analytic argument explains why the second band changes the conclusion so drastically.\n\nWhat is here is honest: the limitations on v_s are acknowledged, references to the authors' own prior work are appropriate, and the model is physically motivated. I would want to see the full numerics before betting on it, but this is a serious candidate for publication after revision.\n\nRecommendation: send it to peer review, with the supplement and code made available so the claims can actually be checked. I'd bring it to a reading group and would cite it once the details are out.","headline":"Plausible symmetry-based route to Bogoliubov Landau levels in a 2D TI-SC heterostructure, but the central numbers are in a missing supplement.","tokens_in":12921,"tokens_out":2716,"would_cite":false,"duration_ms":25124,"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":"A two-dimensional topological-insulator–superconductor heterostructure is claimed to host Landau levels with Chern number ±1 in its vortex lattice, with quantized thermal Hall conductivity and peaked magneto-optical response as…","keywords":["Landau levels","mixed state","vortex lattice","Bogoliubov quasiparticles","generalized chiral symmetry","thermal Hall effect","topological insulator-superconductor heterostructure","magneto-optical conductivity"],"falsifier":"Direct numerical diagonalization of the full BdG Hamiltonian (8) on a finite vortex lattice, without the low-energy projection, would settle the main claim: if the spectrum shows avoided crossings or fails to follow $E_n=\\sqrt{n}E_1$ as the magnetic length $l_B$ is varied, the central claim collapses. Experimentally, measuring $\\kappa_{xy}/T$ at low temperature in a TI–s-wave superconductor heterostructure and finding values that do not match integer sums of the level Chern numbers (or plateau values that drift non-trivially with temperature) would falsify the quantized thermal Hall prediction.","tokens_in":11852,"feed_emoji":"🧲","tokens_out":11923,"duration_ms":101366,"temperature":0.7,"pith_summary":"The paper argues that, unlike the usual fate of quasiparticles in the vortex lattice of a gapless superconductor, a two-dimensional topological-insulator–superconductor heterostructure can show genuine Landau quantization of its Bogoliubov quasiparticles when a perpendicular magnetic field creates a vortex lattice. The decisive ingredient is a generalized chiral symmetry of the low-energy Hamiltonian: in the heterostructure the spatially varying supercurrent couples to quasiparticles as an effective vector potential and does not scatter them into Bloch waves, whereas in an intrinsic Weyl superconductor the same supercurrent breaks the symmetry and no Landau levels form. For the 2D system the paper obtains a Dirac-like ladder $E_n = \\sqrt{n}E_1$ with a dispersionless zeroth level, each non-degenerate level carrying Chern number $\\pm1$. It follows, the paper claims, that the mixed state of this superconductor shows a peaked magneto-optical conductivity and, in the low-temperature limit, a quantized transverse tensor $\\tilde{\\sigma}_{xy}$ and quantized thermal Hall ratio $\\kappa_{xy}/T$. A sympathetic reader would take this as a concrete route to Landau-quantized Bogoliubov quasiparticles and to quantized thermal Hall physics in a proximitized 2D superconductor.","feed_headline":"Landau levels appear in a 2D superconductor's vortex lattice","feed_subtitle":"Each level carries Chern number ±1, so the thermal Hall conductance quantizes and the magneto-optical response peaks.","key_machinery":"The machinery that carries the argument is the generalized chiral symmetry for tilted Dirac/Weyl cones, expressed by $\\zeta^\\dagger H_{\\mathrm{BW}}\\zeta = -H_{\\mathrm{BW}}$ with $\\zeta = (\\sigma_z - i\\eta_x\\sigma_y)/\\sqrt{1-\\eta_x^2}$. Together with this, the key structural move is to absorb the vortex-lattice phase into an effective vector potential $\\mathbf{A}_{\\mathrm{eff}} = \\mathbf{a} + \\kappa m\\mathbf{u}_s$ after the Anderson gauge transformation, where $\\mathbf{a} = \\frac{1}{2}\\nabla\\phi$ is the superconducting gauge field and $\\mathbf{u}_s$ the supercurrent velocity. In the heterostructure Weyl superconductor this effective potential couples to Bogoliubov–Weyl fermions and preserves the generalized chiral symmetry, which protects a dispersionless zeroth Landau level and yields the $\\sqrt{n}$ ladder; in the intrinsic Weyl superconductor the external vector potential drops out of the pairing-linearized Hamiltonian, the supercurrent term enters as $\\eta_x v_F m v_{s,x}\\sigma_z$ and breaks the symmetry, so only Bloch-wave-like quasiparticles survive. For the 2D TI–superconductor heterostructure, the zero-field Hamiltonian has a Bogoliubov–Dirac node at $\\beta^2 = 1+\\Delta_0^2$, and the numerical vortex-lattice solution shows that the same symmetry logic produces Dirac Landau levels with Chern numbers $\\pm1$.","core_discovery":"On its own terms, the paper's central claim is that a 2D superconducting system—a thin topological-insulator layer proximitized by an s-wave superconductor, described by a four-band Bogoliubov–de Gennes Hamiltonian with a Dirac node tuned by magnetization and pairing gap—hosts Landau levels of Bogoliubov quasiparticles in the mixed state. At the fine-tuned condition $\\beta^2 = 1+\\Delta_0^2$ the vortex-lattice spectrum is a doubly degenerate Dirac ladder $E_n=\\sqrt{n}E_1$ with a dispersionless zeroth level; when $\\beta$ and $\\mu$ are tuned the degeneracy splits into massive-Dirac ladders whose individual levels have Chern number $+1$ or $-1$, the same values as the normal-state Landau levels. Because Bogoliubov quasiparticles are charge-neutral, the electric Hall conductivity is not the quantized observable; instead the paper defines a transverse conductivity tensor $\\tilde{\\sigma}_{xy}(\\xi)$ built from Berry curvature of occupied levels and shows it develops integer plateaus, so that $\\kappa_{xy}/T \\to (\\pi^2/3)(k_B/\\hbar)^2\\tilde{\\sigma}_{xy}(0)$ is quantized at low temperature. The paper also predicts that optical transitions between the Bogoliubov levels produce magneto-optical conductivity peaks at $\\omega/E_1 = \\sqrt{n}+\\sqrt{n+1}$, analogous to graphene but with reduced amplitude, and that detuning away from the nodal condition shifts the peak positions. In short, the paper claims that the mixed state of this heterostructure is a genuine Landau-quantized nodal superconductor with measurable topological transport signatures.","pith_inferences":["Extension beyond the paper: the symmetry criterion could be used as a screening rule—for any candidate 2D nodal superconductor, checking whether the vortex-lattice effective Hamiltonian preserves the generalized chiral symmetry would predict whether a $\\sqrt{n}$ ladder appears before performing the full vortex-lattice diagonalization.","Extension beyond the paper: if the quantized $\\kappa_{xy}/T$ plateau is observed, the temperature scale at which quantization sets in would provide an independent measure of the first inter-level spacing $E_1$, cross-checking the magneto-optical peak positions.","Extension beyond the paper: because the effective vector potential contains the internal superconducting gauge field $\\mathbf{a}$, vortex lattices in these heterostructures might act as synthetic magnetic fields for neutral Bogoliubov quasiparticles, tunable independently of the external flux.","Extension beyond the paper: the claim that only the zeroth Bogoliubov level has nearly uniform Berry curvature singles out that level as the place to search for fractional quantum thermal Hall states; computing the full momentum-resolved Berry curvature of the first few levels would sharpen this prediction."],"forward_implications":["If the 2D heterostructure is realized as a topological-insulator layer on an s-wave superconductor in a perpendicular field, the vortex-lattice spectrum should show a $\\sqrt{n}$ Dirac ladder with a dispersionless zeroth level.","The magneto-optical conductivity should exhibit peaks at $\\omega/E_1 = \\sqrt{n}+\\sqrt{n+1}$ in the gapless case, with peak positions shifting when the nodal condition $\\beta^2 = 1+\\Delta_0^2$ is violated.","At low temperature, $\\kappa_{xy}/T$ should approach an integer multiple of $(\\pi^2/3)(k_B/\\hbar)^2$ because each filled Bogoliubov level contributes Chern number $\\pm1$ to the transverse tensor.","The contrast between intrinsic and heterostructure Weyl superconductors implies that Landau quantization in a vortex state depends on the generalized chiral symmetry of the low-energy Hamiltonian, not merely on the nodal structure.","Because the zeroth Bogoliubov level has nearly uniform Berry curvature, it is a candidate flat band for interaction-driven fractional quantum thermal Hall states."],"supporting_citations":[{"why":"Demonstrates that vortex-lattice quasiparticles in single-band d-wave superconductors form Bloch waves, the counter-case the paper's symmetry argument distinguishes.","marker":"[5]"},{"why":"Establishes a topologically protected zeroth Landau level in a heterostructure Weyl superconductor's vortex lattice, the case the 2D proposal extends.","marker":"[7]"},{"why":"Defines the generalized chiral symmetry for tilted Dirac cones that determines when a dispersionless zeroth level exists.","marker":"[9]"},{"why":"Supplies the multilayer heterostructure model that the proposed 2D Hamiltonian is a single layer of.","marker":"[14]"},{"why":"Contains the numerical vortex-lattice diagonalization and Chern-number details that the main text's figures rely on.","marker":"[17]"},{"why":"Gives the graphene magneto-optical transition rule used to identify the superconducting Landau-level peaks.","marker":"[58]"},{"why":"Provides the Berry-curvature expression for the transverse conductivity tensor that underlies the thermal Hall quantization.","marker":"[68]"},{"why":"Gives the low-temperature limit that turns Chern numbers into quantized values of the thermal Hall coefficient.","marker":"[69]"}],"fun_headline_variants":["Vortex lattice in 2D nodal superconductor yields Landau levels with Chern ±1","Quantized thermal Hall from Landau levels in a 2D nodal superconductor","Landau levels with Chern ±1 quantize thermal Hall in a superconductor","2D nodal superconductor shows Landau-quantized heat transport","Peaked magneto-optics and quantized Hall from superconductor Landau levels"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The claim rests on the assumption that the vortex-lattice supercurrent, after the Anderson gauge transformation, acts only as an effective vector potential that preserves the generalized chiral symmetry rather than scattering quasiparticles into Bloch waves, and the paper validates this assumption only numerically, with the details placed in the Supplemental Material.","fun_headline_variants_meta":{"raw":{"variants":["Vortex lattice in 2D nodal superconductor yields Landau levels with Chern ±1","Quantized thermal Hall from Landau levels in a 2D nodal superconductor","Landau levels with Chern ±1 quantize thermal Hall in a superconductor","2D nodal superconductor shows Landau-quantized heat transport","Peaked magneto-optics and quantized Hall from superconductor Landau levels"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000739,"raw_usage":{"total_tokens":3387,"prompt_tokens":1117,"completion_tokens":2270,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":733,"completion_tokens_details":{"reasoning_tokens":2160}},"tokens_in":733,"tokens_out":2270,"duration_ms":18028,"temperature":1.0,"reasoning_tokens":2160,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T16:24:16.392573+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Direct numerical diagonalization of the full BdG Hamiltonian (8) on a finite vortex lattice, without the low-energy projection, would settle the main claim: if the spectrum shows avoided crossings or fails to follow $E_n=\\sqrt{n}E_1$ as the magnetic length $l_B$ is varied, the central claim collapses. Experimentally, measuring $\\kappa_{xy}/T$ at low temperature in a TI–s-wave superconductor heterostructure and finding values that do not match integer sums of the level Chern numbers (or plateau values that drift non-trivially with temperature) would falsify the quantized thermal Hall prediction.","supporting_citations":[{"cited_title":"Franz and Z","cited_arxiv_id":null,"evidence_quote":"Demonstrates that vortex-lattice quasiparticles in single-band d-wave superconductors form Bloch waves, the counter-case the paper's symmetry argument distinguishes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes a topologically protected zeroth Landau level in a heterostructure Weyl superconductor's vortex lattice, the case the 2D proposal extends."},{"cited_title":"Kawarabayashi, Y","cited_arxiv_id":null,"evidence_quote":"Defines the generalized chiral symmetry for tilted Dirac cones that determines when a dispersionless zeroth level exists."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the graphene magneto-optical transition rule used to identify the superconducting Landau-level peaks."},{"cited_title":"Cvetkovic and O","cited_arxiv_id":null,"evidence_quote":"Provides the Berry-curvature expression for the transverse conductivity tensor that underlies the thermal Hall quantization."},{"cited_title":"Vafek, A","cited_arxiv_id":null,"evidence_quote":"Gives the low-temperature limit that turns Chern numbers into quantized values of the thermal Hall coefficient."}],"review_version":1}