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REVIEW 2 major objections 5 minor 75 references

Optical signatures of Euler superconductors

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2501.00960 v2 pith:SD4OGORK submitted 2025-01-01 cond-mat.supr-con cond-mat.mes-hallphysics.optics

Chun Wang Chau , Wojciech J. Jankowski , Robert-Jan Slager This is my paper · ORCID
classification cond-mat.supr-concond-mat.mes-hallphysics.optics
keywords Eulerclassopticalconductivitymultibandsuperconductivityquantumgeometryd-wavepairingBogoliubovquasiparticlesnonlinearphotoconductivitymultigaptopology
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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

2 major / 5 minor

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.

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 (2)
  1. [Sec. II (Eq. 6), Sec. VI (Eq. 22), Appendix J] 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.
  2. [Appendix D, Eq. (D12) and Eq. (28)] 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.
minor comments (5)
  1. [Abstract and Introduction] 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.
  2. [Sec. III, Eq. (10)] 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.
  3. [Eq. (A10)] 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.
  4. [Appendix D, around Eq. (D5)] 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.
  5. [Appendix J] 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).

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the e²/(16ℏ) jump follows from the external Euler-node quantum-metric scaling plus the BdG factorization, and is verified numerically; only a minor self-citation to Ref. [28] remains.

full rationale

The central derivation is self-contained rather than circular. The paper's load-bearing analytic input is the 1/r quantum-metric scaling of an Euler node, quoted in Appendix A: "it was shown in Ref. [28] that the multiband quantum metric for a rotationally symmetric Euler nodes with patch invariant χ obtains ξ^x_ab ξ^x_ba + ξ^y_ab ξ^y_ba = χ²/q⁴ q_μ q_ν(2δ_μν − 1)." This is a prior result by overlapping authors, so there is a minor self-citation. It is not, however, a circularity: the scaling is a parameter-free geometric property of the Euler node, it does not contain the superconducting optical conductivity as an input, and the BdG Green's-function factorization (Eq. 22) together with the one-loop diagrams (Eqs. 24-27) converts that input into a genuinely new prediction — the (1−cos ϕ) jump at ω = 2Δ0 with coefficient e²/(32ℏ)|χ| (Eq. 28). The jump is then independently confirmed by numerical lattice-model conductivity (Fig. 4), with no fitting of the target quantity. The stated restrictions (intraband pairing, no Δ–A coupling, flat-band assumption in the clean limit) limit the regime of validity, but they are explicit assumptions rather than definitions that presuppose the claimed result. No uniqueness theorem is imported from the authors' prior work, and no fitted parameter is renamed as a prediction. Accordingly, no circular step is present; the score reflects only the minor, non-load-bearing self-citation to Ref. [28] for the Euler-node quantum-metric scaling.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The central prediction rests on a mean-field BdG model with intraband, uniform, vector-potential-independent pairings, plus the known quantum-metric scaling of Euler nodes. No new physical entities are introduced. The main free control is the pairing phase ϕ, which selects the d-wave channel.

free parameters (6)
  • Order-parameter phase difference ϕ = π (d-wave pairing)
    The topological jump in linear conductivity is ∝ (1−cos ϕ), vanishing for ϕ=0 and maximal for ϕ=π; ϕ is not determined by the model and is chosen to realize d-wave pairing.
  • Uniform pairing magnitude Δ0 = 2 (model units)
    Sets the gap scale 2Δ0 at which the jump occurs; the analytical derivation assumes |Δ_a|=Δ0 for all bands.
  • On-site energy M = 2
    Separates the Euler band pair from higher bands; the jump approaches the quantized value e²/(16ℏ) as M is increased (Appendix K).
  • Splitting parameter m = 0 (χ=1 node)
    Controls whether the Euler node is unsplit (m=0) or split into half-integer nodes (0<m<mc); the jump persists but is no longer quantized when the node splits.
  • Parity-breaking term t2 = 2 (for second-order results)
    Introduced ad hoc in Sec. V to break inversion symmetry and activate second-order photoconductivities; not needed for the first-order jump.
  • Broadening η = 10^-2
    Numerical regularization for delta functions in response formulas; the third-order response scales as 1/η^2, while the first-order jump is η-independent in the clean limit.
assumptions (7)
  • domain assumption Mean-field BdG description with intraband pairing only: Δ(k)=Σ_a Δ_a |u_a⟩⟨u_a|.
    Invoked in Sec. II (Eq. 6) and Sec. VI; the entire derivation of the jump relies on this decomposition and the resulting factorization of the Green's function (Eq. 22).
  • ad hoc to paper Uniform pairing magnitude |Δ_a|=Δ0 for all bands.
    Assumption (i) in Sec. VI; enables the closed-form expression Eq. (27) and the jump Eq. (28).
  • ad hoc to paper One of the Euler bands is flat (ϵ_a=0) and touches a dispersive band.
    Assumption (iv) in Sec. VI; gives the clean e²/(16ℏ) value. Generalization to dispersive bands changes the prefactor to (α+−α−)²/(α+²+α−²) (Appendix D.1).
  • domain assumption Order parameter does not couple to the vector potential A.
    Stated in Sec. II after Eq. (5); standard for mean-field optical response of superconductors but not exact when the superconducting phase fluctuates.
  • domain assumption Zero temperature limit.
    Assumption (ii) in Sec. VI; the intersector response is isolated at T=0.
  • standard math Quantum metric scaling |ξ|² ~ χ²/q⁴ for an Euler node, taken from Ref. [28].
    Eq. (A10) in Appendix A; this input from prior literature (same group) determines the r-dependence that produces the jump.
  • domain assumption Hamiltonian is real due to C2T/PT symmetry and P symmetry is preserved for the linear response.
    Assumption (iii) in Sec. VI; required for the Euler class to be well-defined and for σxy to vanish.

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Cite this review

Pith. "Pith review of Optical signatures of Euler superconductors." pith.science (2026). https://pith.science/paper/SD4OGORK

@misc{pith2026250100960,
  author       = {Pith},
  title        = {Pith review of: Optical signatures of Euler superconductors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SD4OGORK}},
  note         = {Machine review of arXiv:2501.00960}
}
abstract

We study optical manifestations of multigap band topology in multiband superconductors with a nontrivial topological Euler class. We introduce a set of lattice models for non-Abelian superconductors with the Euler invariant signified by a nontrivial quantum geometry. We then demonstrate that the topological Bogoliubov excitations realized in these models provide for a characteristic first-order optical response distinct from those of the other known topological superconductors. We find that the spectral distribution of the optical conductivity universally admits a topological jump originating from the Euler class in the presence of $d$-wave superconducting pairings, and naturally differs from the features induced by the quantum geometry in the noninteracting bands without pairing terms. Further to uncovering observable signatures in first-order optical conductivities, we showcase that the higher-order optical responses of the non-Abelian Euler superconductor can result in enhanced nonlinear currents that fingerprint the exotic topological invariant. Finally, by employing a diagrammatic approach, we generalize our findings beyond the specific models of Euler superconductors.

Discussion (0). Continue with ORCID to comment.

Reference graph

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