{"id":"e4c32c06-8e0b-47ef-a96a-8c4bc294e52f","arxiv_id":"2501.00960","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Euler-class superconductors with d-wave pairing exhibit a topological jump in the linear optical conductivity at 2Δ0, bounded by the total patch Euler class, plus a √(ω−2Δ0) third-order enhancement.","lead":"This paper predicts that superconductors whose band structure carries a nontrivial Euler class, a multigap topological invariant, produce a distinctive jump in the optical conductivity at twice the superconducting gap. The jump size is controlled by the Euler invariant and the d-wave pairing phase, giving a new optical fingerprint for exotic topological superconductors.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central quantization in Eq. (28) assumes a band-diagonal intraband pairing without Δ–A coupling (Eqs. 6, 22); generic physical d-wave pairing produces interband Δ_ab that breaks the BdG factorization and may remove the e²/(16ℏ) jump.","rationale":"The central prediction is a universal, Euler-class-quantized jump in Re σ. The derivation in Eqs. (24)–(28) and Appendices C–D is internally coherent: under the stated ansatz, the BdG Green's function factorizes, the divergent quantum geometry gives a finite step through the integral, and the numerical Lieb and kagome results match. I checked the factor flow from Eq. (D1) to Eq. (D8) and found no obvious algebraic slip leading to the e²/16ℏ value. The weakest point is not the flat-band assumption, since Appendix D1 shows the jump survives with a dispersion-dependent prefactor and supplies the bound in Eq. (D13). The truly load-bearing condition is band-diagonal, A-independent pairing. The paper acknowledges this in Sec. II (Eq. 6), Sec. VI assumption (i), and Appendix J, where it notes that orbital interactions generally produce interband pairing in the band basis. Because the same assumption is what makes the Green's function decompose and lets the 1/r connection appear in the conductivity, generic d-wave pairing or gauge-consistent Δ–A coupling could regularize or shift the jump. This is a known limitation rather than an internal contradiction, so the reader's CONDITIONAL verdict remains appropriate; the proposed self-consistent orbital-to-band computation is the minimal check that would settle the universality claim.","tokens_in":32991,"tokens_out":13457,"duration_ms":136045,"concrete_test":"Set up the full three-band Lieb model with a microscopic orbital interaction (e.g., a nearest-neighbor or onsite attractive V) that yields a d-wave self-consistent solution in the orbital basis. Compute the self-consistent Δ_orb(k), project onto normal bands to obtain the off-diagonal band pairing Δ_f,-(k), and evaluate max_k |Δ_f,-|/Δ0. Then compute Re σ_xx(ω) from the full BdG Green's function, including gauge-consistent ∂Δ/∂A vertices, for χ = 1 and compare with the band-diagonal ansatz: if a step of e²/16ℏ appears at ω = 2Δ0 only when Δ_f,- is artificially set to zero, the quantization is an artifact of the assumption; if it persists for generic d-wave interactions, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the factorization G = Σ_a G_a ⊗ P_a (Eq. 22), which follows only if the order parameter is band-diagonal, Δ(k) = Σ_a Δ_a |u_a⟩⟨u_a|, and if ∂Δ/∂A = 0 (Eq. 5 and Sec. VI, assumption i). Under that ansatz, each BdG quasiparticle is tied to one normal band, and the 1/r singularity of ξ^μ_ab enters Eq. (27) directly, producing the jump in Eq. (28). The paper motivates ϕ = π as an orbital-space d-wave pairing (Eq. 10), but projecting a generic orbital interaction onto bands generates off-diagonal components Δ_ab(k). The authors themselves acknowledge in Appendix J that even an on-site interaction becomes interband in the band basis. With Δ_ab ≠ 0, the BdG problem no longer decouples by band, the pole structure of G changes, and there is no obvious reason the e²/16ℏ jump survives. Likewise, a gauge-invariant treatment of the pairing phase under A adds Δ-dependent current vertices beyond J = τ_z ⊗ h^μ; omitting them can modify the vertex that produces the jump. These are stated assumptions, but they are exactly the conditions under which the universal claim has been demonstrated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies optical responses of superconductors whose normal-state band structure carries a nontrivial Euler class. It introduces a Lieb-lattice model with mean-field intraband pairing, computes the first-, second-, and third-order optical conductivities, and identifies a discrete jump in Re σ(ω) at ω = 2Δ0 whose value is e²/(16ℏ) for an integer Euler node χ = 1 with flat band and d-wave-like phase difference φ = π, bounded by the total patch Euler class. The analytical derivation is carried out in a diagrammatic one-loop formalism under explicit assumptions: band-diagonal intraband pairing, uniform |Δ_a| = Δ0, zero temperature, particle-hole symmetric flat band, and no dependence of the order parameter on the vector potential. The same formalism yields a continuous √(ω−2Δ0) enhancement in the third-order photoconductivity.","tokens_in":33330,"tokens_out":6756,"duration_ms":70531,"significance":"If the central claim holds, the paper provides a concrete and falsifiable optical signature that distinguishes Euler superconductors from ordinary topological superconductors and links multigap topology to quantum-geometric optical response in the superconducting state. The authors give explicit lattice models, numerical benchmarks, a diagrammatic derivation, an explicit bound in Eq. (D13), and a kagome-lattice generalization, which are substantial strengths. The analytical derivation in Appendix D is internally consistent in the flat-band/isolated-node limit, and the use of the known quantum-metric scaling from Ref. [28] as an external input is a legitimate, non-circular step. The main caveat is that the headline 'universal' jump is demonstrated only under a restrictive band-diagonal pairing ansatz; for generic orbital interactions the factorization underlying the derivation fails, as the authors themselves acknowledge in Appendix J.","major_comments":[{"comment":"The central derivation of the e²/(16ℏ) jump relies on the factorization G = Σ_a G_a ⊗ P_a, which is valid only when the order parameter is band-diagonal, Δ(k) = Σ_a Δ_a |u_a(k)⟩⟨u_a(k)|, and when ∂Δ/∂A = 0. For a generic orbital d-wave or on-site interaction, projecting onto Bloch bands produces interband components Δ_ab(k) = ⟨u_a|Δ_orb|u_b⟩. The authors explicitly state in Appendix J that an on-site interaction becomes both interband and intraband in the band basis. Since the abstract and introduction describe the jump as universal for Euler superconductors, the manuscript should either show that interband pairing does not destroy the jump (at least perturbatively) or substantially qualify the universality claim to the intraband ansatz. This is a load-bearing restriction, not a minor caveat.","section":"Sec. II (Eq. 6), Sec. VI (Eq. 22), Appendix J"},{"comment":"The value e²/(16ℏ) is not protected by topology alone. Equation (D12) shows that for dispersive Euler bands with curvatures α_+ and α_−, the jump is (e²/32ℏ)|χ|(α_+−α_−)²(1−cos φ)/(α_+²+α_−²), which equals e²/(16ℏ) only in the flat-band limit and can vary continuously, reaching the larger value e²/(8ℏ) for α_+ = −α_−. The manuscript sometimes refers to this as 'quantization' (e.g., the title of Appendix K). I recommend that the text consistently distinguish the robust existence of a discontinuity at 2Δ0 from the fine-tuned numerical value obtained under the flat-band assumption.","section":"Appendix D, Eq. (D12) and Eq. (28)"}],"minor_comments":[{"comment":"The word 'universally' in the abstract overstates the demonstrated scope; it should be qualified to the intraband, uniform-|Δ| pairing setting defined in Sec. II.","section":"Abstract and Introduction"},{"comment":"The order parameter in Eq. (10) contains an s-wave component on the A orbital; calling the whole pairing 'd-wave' is accurate only for the B–C sector. A sentence clarifying this decomposition would prevent overinterpretation of the cuprate analogy.","section":"Sec. III, Eq. (10)"},{"comment":"The index structure of the quantum-metric scaling formula in Eq. (A10) is terse; stating the summation convention and referencing the explicit derivation in Ref. [28] more precisely would help readers verify the subsequent integrals.","section":"Eq. (A10)"},{"comment":"The restoration of e and ℏ in the final expressions is not shown explicitly; adding the conversion from the units e=ℏ=1 to e²/ℏ would improve reproducibility of the reported constants.","section":"Appendix D, around Eq. (D5)"},{"comment":"The admission that an orbital on-site interaction becomes interband in the band basis is important enough to be mentioned in the main text near Sec. VI, not only in an appendix, because it directly delimits the validity of Eq. (22).","section":"Appendix J"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically solid within its stated assumptions, and the diagrammatic machinery is a useful contribution. The main issue is the gap between the 'universal' language in the abstract and the restrictive intraband-pairing ansatz. I would ask the authors to either extend the analysis to generic interband pairing or clearly reframe the claim as a property of the intraband-pairing Euler superconductor. The citation of Ref. [28] by overlapping authors is appropriate and not a concern. The paper fits the scope of the journal, provided the scope of the claim is made commensurate with the proof."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead the Euler superconductor optical paper. The one thing to know: the central claim is a discrete jump in the linear optical conductivity at 2Δ0, Re[σ] = (e²/32ℏ)|χ|(1−cos φ), which for the model d-wave pairing (φ=π, χ=1) is e²/16ℏ. The paper derives this from the 1/r quantum-metric singularity of the Euler node, verifies it numerically in Lieb and kagome models, and also gets a continuous √(ω−2Δ0) enhancement in the third-order response with a set of selection rules. That is genuinely new: earlier work had optical signatures of the Euler class in the normal state, but not in the superconducting state with these phase-controlled features. The diagrammatic machinery in the appendices is careful and the numerical match with Eq. (28) in the flat-band limit is credible.\n\nThe soft spot is the pairing ansatz. The derivation depends on the BdG Green's function factorizing as G = Σ_a G_a ⊗ P_a (Eq. 22), which requires band-diagonal intraband pairing of uniform magnitude and no coupling of the order parameter to the vector potential. The authors state this in Sec. VI, but Appendix J contains the more damning remark: even an on-site interaction in the orbital basis produces interband pairing in the band basis. So the clean e²/16ℏ quantization is not what a generic d-wave superconductor would show; it is a property of a specially engineered band-diagonal pairing. The paper partially softens this by showing the jump persists with a dispersion-dependent prefactor and by proving the bound Re[σ] ≤ (e²/8ℏ) Σ|χ_i| (Eq. D13). So the existence of a jump tied to the Euler class is robust within the ansatz, but the universal constant is not.\n\nThe abstract's word 'universally' overstates what is actually demonstrated. The second-order response, to their credit, does not give a Euler signature; they say so explicitly. No code or data is shipped, which is a minor reproducibility issue given the density of numerical claims.\n\nOverall, this is a serious theory paper with an honest limitations section and a novel, checkable prediction under stated assumptions. The caveat is real but not fatal: it narrows the applicability rather than invalidating the calculation. Worth sending to a good referee, and worth having in the literature as a reference for the bound and the phase-controlled effect. I would not desk-reject it.","headline":"A serious, well-executed theory paper predicting a phase-controlled optical jump from the Euler class; the headline quantization rests on a band-diagonal pairing ansatz that limits its universality, but the paper earns serious refereeing.","tokens_in":33813,"tokens_out":6168,"would_cite":true,"duration_ms":55335,"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 superconductor whose normal-state bands carry a patch Euler class $\\chi$ shows a discrete jump in the real optical conductivity at twice the pairing gap, equal to $(e^2/32\\hbar)|\\chi|(1-\\cos\\phi)$ for $d$-wave pairing.","keywords":["Euler class","optical conductivity","multiband superconductivity","quantum geometry","d-wave pairing","Bogoliubov quasiparticles","nonlinear photoconductivity","multigap topology"],"falsifier":"Compute the converged $\\eta\\to0$ limit of $\\mathrm{Re}[\\sigma_{xx}(\\omega\\to 2\\Delta_0^+)]$ for the Lieb lattice model at $t=1$, $M=2$, $m=0$, $\\Delta_0=2$, with chemical potential at the Euler node and $d$-wave pairing ($\\phi=\\pi$). The paper's claim fixes this value at $e^2/(16\\hbar)$; a different converged value, or an optical measurement on a clean realization of the same model showing no step at $2\\Delta_0$, would falsify the central claim.","tokens_in":32795,"feed_emoji":"⚛️","tokens_out":10743,"duration_ms":95757,"temperature":0.7,"pith_summary":"This paper establishes that the multigap Euler class, a topological invariant of pairs of bands in real-valued Hamiltonians, leaves a quantized optical fingerprint once those bands are made superconducting. For a $d$-wave pairing configuration, the real part of the linear optical conductivity acquires a discrete jump at $\\omega = 2\\Delta_0$ with magnitude proportional to the Euler class, and the jump is bounded by the total nodal Euler class. The result is derived for a tunable Lieb lattice model, verified on a kagome lattice model, and generalized with a one-loop diagrammatic calculation that does not rely on the specific lattice. The paper also finds a continuous $\\sqrt{\\omega - 2\\Delta_0}$ enhancement of the third-order photoconductivity, while the second-order response carries no direct topological signature because the relevant intersector quantum metric vanishes exactly at the Euler node. If correct, this gives a phase-sensitive optical route to identifying an exotic multiband invariant in superconducting materials.","feed_headline":"Euler topology leaves a quantized jump in superconductor optics","feed_subtitle":"Optical conductivity jumps by e²/16ℏ at twice the pairing gap when Euler bands pair in a d-wave channel.","key_machinery":"The load-bearing object is the factorization $G(k_0,k)=\\sum_a G_a(k_0,k)\\otimes P_a(k)$ of the Bogoliubov-de Gennes Green's function, valid for purely intraband pairing with uniform gap $|\\Delta_a|=\\Delta_0$ and no coupling of the order parameter to the vector potential. It lets every optical diagram be written in terms of normal-state band projectors and the non-Abelian Berry connection $\\xi^\\mu_{ab}=\\langle a|\\partial_\\mu b\\rangle$. The Euler class $\\chi$ is the quantized obstruction (Stokes boundary term) carried by the pair of bands, and the quantum-metric bound $g_{aa}^{xx}+g_{bb}^{xx}+g_{aa}^{yy}+g_{bb}^{yy}\\ge 2|\\mathrm{Eu}_{ab}|$ controls how much optical weight the geometry can deliver. Around an Euler node $\\xi^\\mu_{ab}$ is purely imaginary and diverges as $1/r$, and the $d$-wave pairing with $\\phi=\\pi$ activates the interband dipole vertices that turn this divergence into the conductivity jump.","core_discovery":"At the core is the claim that the superconducting state inherits the Euler class through singular quantum geometry: the non-Abelian Berry connection between the two Euler bands scales as $1/r$ near the node. With intraband pairing of uniform magnitude, the BdG Green's function factorizes as $G(k_0,k)=\\sum_a G_a(k_0,k)\\otimes P_a(k)$, and the intersector part of the conductivity reduces to an integral over $\\epsilon_{ab}^2 |\\xi^\\mu_{ab}|^2$. The $1/r$ divergence of $\\xi^\\mu_{ab}$ converts into a finite step when the photon energy crosses the quasiparticle gap: $\\mathrm{Re}[\\sigma(\\omega\\to 2\\Delta_0^+)] = (e^2/32\\hbar)|\\chi|(1-\\cos\\phi)$, which equals $e^2/(16\\hbar)$ for a single $\\chi=1$ Euler node, and is bounded by $\\mathrm{Re}[\\sigma]\\le (e^2/8\\hbar)\\sum_i |\\chi_i|$. The step appears only when the order-parameter phase difference between the Euler bands is $\\phi=\\pi$, i.e. for $d$-wave pairing; at $\\phi=0$ the vertex coupling vanishes. At third order a four-vertex loop diagram dominates and produces a continuous onset $\\propto\\sqrt{\\omega-2\\Delta_0}$, and the paper shows the linear jump persists beyond the flat-band limit with a dispersion-dependent prefactor.","pith_inferences":["Beyond the paper, the factor $(1-\\cos\\phi)$ suggests the conductivity step could be used as a relative phase detector for multiband order parameters: tuning $\\phi$ continuously should interpolate the jump between zero and its maximum, a testable prediction the paper does not emphasize.","Beyond the paper, a natural next step is to average the conductivity over disorder while preserving $PT$ on average; if the $e^2/(16\\hbar)$ step survives only as a broadened plateau, it still offers a robust observable, but if it is destroyed, the quantization is a clean-limit property rather than a topological quantization.","Beyond the paper, engineered platforms such as ultracold atoms or acoustic metamaterials, where the pairing phase and chemical potential are externally tunable, may test the jump more directly than bulk crystals, because the model parameters and node pinning can be controlled.","Beyond the paper, the bound $\\mathrm{Re}[\\sigma]\\le (e^2/8\\hbar)\\sum_i|\\chi_i|$ could be inverted as an optical sum-rule estimate of the total patch Euler class from a single conductivity measurement near $2\\Delta_0$; the paper does not propose this inverse use."],"forward_implications":["A clean $\\chi=1$ Euler superconductor with $d$-wave pairing should show a step of $e^2/(16\\hbar)$ in $\\mathrm{Re}[\\sigma_{xx}]$ at $\\omega=2\\Delta_0$, with the step absent for phase-locked $s$-wave pairing ($\\phi=0$).","Splitting the Euler node into half-integer patch nodes removes the quantized value but keeps the jump bounded by $e^2/(8\\hbar)$ times the total patch Euler class, and the jump vanishes once the nodes annihilate.","Third-order photoconductivity provides a second, non-quantized fingerprint: a continuous $\\sqrt{\\omega-2\\Delta_0}$ onset dominated by a four-vertex one-loop diagram.","In noncentrosymmetric Euler superconductors, second-order injection currents are governed by band dispersion rather than by the Euler class, so they cannot serve as direct topological probes.","The same signatures appear in a kagome-lattice realization and, by the diagrammatic argument, in any multiband superconductor with an Euler node under the stated assumptions."],"supporting_citations":[{"why":"Defines the Euler class as the multigap invariant of real two-band subspaces and relates it to band-node charges.","marker":"[22]"},{"why":"Supplies the patch Euler class with Stokes boundary term and the non-Abelian braiding framework used in Eq. (4).","marker":"[26]"},{"why":"Provides the quantum-geometric tensor and the metric bound controlling the optical weight carried by the Euler bands.","marker":"[27]"},{"why":"Establishes the normal-state optical manifestations and bounds of the Euler class that this paper extends to the superconducting state.","marker":"[28]"},{"why":"Gives the effective $k\\cdot p$ Hamiltonian for an Euler node with dispersion $\\propto r^{2|\\chi|}$, used in the analytic conductivity integrals.","marker":"[55]"},{"why":"Provides the generalized velocity operators and optical response formulae for superconductors used in the numerical calculations.","marker":"[63]"},{"why":"Supplies the gauge-invariant diagrammatic one-loop formulation and vertex-correction analysis adapted to the band basis here.","marker":"[66]"},{"why":"Establishes the diagrammatic framework for nonlinear optical response used to evaluate the four-vertex third-order loop.","marker":"[69]"}],"fun_headline_variants":["Euler superconductors show quantized optical step","Topological Euler class leaves jump in optical response","Superconductor optics reveals Euler invariant step","Quantized conductivity jump from Euler topology"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central claim rests on assuming purely intraband pairing with uniform gap magnitude $|\\Delta_a|=\\Delta_0$, no coupling of the order parameter to the vector potential, and (for the clean $e^2/(16\\hbar)$ value) a flat band touching a dispersive band, so if any of these is relaxed the factorization underlying the jump fails and the quantization can be modified.","fun_headline_variants_meta":{"raw":{"variants":["Euler superconductors show quantized optical step","Topological Euler class leaves jump in optical response","Superconductor optics reveals Euler invariant step","Quantized conductivity jump from Euler topology"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000231,"raw_usage":{"total_tokens":1530,"prompt_tokens":1035,"completion_tokens":495,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":651,"completion_tokens_details":{"reasoning_tokens":439}},"tokens_in":651,"tokens_out":495,"duration_ms":5075,"temperature":1.0,"reasoning_tokens":439,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:38:52.641378+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the converged $\\eta\\to0$ limit of $\\mathrm{Re}[\\sigma_{xx}(\\omega\\to 2\\Delta_0^+)]$ for the Lieb lattice model at $t=1$, $M=2$, $m=0$, $\\Delta_0=2$, with chemical potential at the Euler node and $d$-wave pairing ($\\phi=\\pi$). The paper's claim fixes this value at $e^2/(16\\hbar)$; a different converged value, or an optical measurement on a clean realization of the same model showing no step at $2\\Delta_0$, would falsify the central claim.","supporting_citations":[{"cited_title":"Bouhon, T","cited_arxiv_id":null,"evidence_quote":"Defines the Euler class as the multigap invariant of real two-band subspaces and relates it to band-node charges."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the patch Euler class with Stokes boundary term and the non-Abelian braiding framework used in Eq. (4)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the normal-state optical manifestations and bounds of the Euler class that this paper extends to the superconducting state."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the effective $k\\cdot p$ Hamiltonian for an Euler node with dispersion $\\propto r^{2|\\chi|}$, used in the analytic conductivity integrals."}],"review_version":1}