{"id":"b9310b9b-da80-43d4-8f44-5a60f3042356","arxiv_id":"2511.03790","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The longitudinal amplitude mode in a clean d-wave superconductor oscillates at twice the anti-nodal pairing gap, and its amplitude decays as 1/t^2.","lead":"This paper calculates how the amplitude ('Higgs') mode of a d-wave superconductor oscillates and decays after a weak perturbation, finding a frequency fixed by the anti-nodal pairing gap and a 1/t^2 decay. It gives concrete predictions for time-resolved pump-probe experiments on cuprates and other d-wave superconductors.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Neglect of off-diagonal-in-momentum pairing terms, acknowledged by the authors as unjustified for nodal d-wave, is the tightest point: if those terms act at q=0, the 1/t^2 decay becomes exponential.","rationale":"The reader identified the same weakest assumption. I agree with the CONDITIONAL verdict. The calculation is internally coherent: the susceptibility and pseudospin methods agree, and the paper is transparent about the model. However, the physical claim that the d-wave Schmid-Higgs mode has frequency 2√2Δ0 and decays as 1/t^2 depends on the neglected off-diagonal momentum terms. The authors' own caveat in Sec. II is explicit: nodal quasiparticles should, a priori, cause exponential decay. Their dismissal of this concern for q=0 is not supported by an explicit estimate or calculation. A concrete numerical test with a non-separable interaction would settle whether the 1/t^2 law survives inclusion of these processes. Because the authors flag the limitation themselves and the central result is conditional on this assumption, the appropriate verdict remains CONDITIONAL; no change from the reader's assessment is required.","tokens_in":14197,"tokens_out":8819,"duration_ms":82537,"concrete_test":"Perform a time-dependent Bogoliubov–de Gennes simulation on a 2D lattice with a non-separable d-wave pairing interaction that includes a small off-diagonal momentum scattering component (e.g., V_{k,k'} = -V γ_k γ_k' - δV (cos k_x - cos k_y)(cos k_y' - cos k_x') or a next-nearest-neighbor term). Quench the pairing strength by δλ/λ = 0.05 and extract the long-time envelope of Δ(t) up to tΔ0 ~ 10^3. If the envelope remains ~1/t^2 with frequency 2√2Δ0, the approximation is validated for weak quenches; if it becomes exponential or a different power law, the central claim is restricted to the separable model. Alternatively, compute the q=0 pair susceptibility including the off-diagonal self-energy (à la Refs. 56,57) and check whether the singularity at Ω = 2√2Δ0 changes from a power-law edge to a pole.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central quantitative claim—Eq. (47): Δ(t) ≈ Δ0[1 + A cos(2ω_SH t + π/4)/(tΔ0)^2] with ω_SH = √2Δ0—is derived entirely from Hamiltonian (1), which keeps only the separable (diagonal-in-momentum) BCS pairing term. As the authors state in Sec. II, it is 'not a priori clear if this approximation is justified here also given the existence of the nodal quasiparticles. Specifically, one would expect that due to the excitation of the nodal quasiparticles, the longitudinal (Schmid-Higgs) mode will decay exponentially fast with time.' The later justification in Sec. V—'which is justified as long as we are interested in the dynamics at q=0'—is an assertion, not a derivation. The q=0 limit does not remove the nodal quasiparticle continuum, nor does it suppress scattering between different Fermi-surface directions. If such off-diagonal momentum processes are not negligible, the long-time behavior is exponential, and both Eq. (47) and the susceptibility-based 1/t^2 claim fail. Because the quasiclassical calculation and the Anderson-pseudospin numerics both use the same truncated Hamiltonian (1), their mutual agreement does not test this assumption; it only confirms internal consistency. The s-wave pair-breaking analogy (Refs. 56,57) invoked in Sec. II is not directly applicable: in the s-wave case the density of states has a square-root edge, whereas d-wave nodal quasiparticles have a linear density of states, so the power law could be qualitatively different if this term were included.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the longitudinal Schmid-Higgs mode in a d-wave superconductor after a weak quench of the pairing interaction. The authors use a quasiclassical Eilenberger/Keldysh approach to derive the zero-momentum pairing susceptibility and find, via numerical Fourier transform, that the mode oscillates at frequency 2√2 Δ0 (twice the anti-nodal gap) and decays as ~1/t^2. They then independently simulate the Anderson pseudospin dynamics and extract the same frequency and decay law, summarized in Eq. (47)–(48). The central quantitative claim is Δ(t) ≈ Δ0[1 + A cos(2ω_SH t + π/4)/(tΔ0)^2] with ω_SH = √2 Δ0. The model Hamiltonian (1) deliberately neglects off-diagonal-in-momentum pairing terms, an approximation the authors acknowledge may be problematic for nodal d-wave superconductors.","tokens_in":14582,"tokens_out":3063,"duration_ms":29088,"significance":"If correct, the paper provides a concrete and non-trivial prediction: in a clean d-wave superconductor the Schmid-Higgs mode frequency is set by the anti-nodal gap and its amplitude decays as 1/t^2, distinctly faster than the 1/√t law in s-wave superconductors. The use of two independent computational routes—quasiclassical susceptibility and classical pseudospin dynamics—is a strength: their mutual agreement checks internal consistency of the formalism. The authors also explicitly discuss the key approximation and its potential limitations, which is commendable. However, the central prediction is conditional on the validity of neglecting off-diagonal pairing terms at q=0, and the 1/t^2 exponent is extracted from numerical fits rather than derived analytically. These caveats limit the strength of the claim.","major_comments":[{"comment":"The neglect of off-diagonal-in-momentum pairing terms is load-bearing. The authors themselves state in Sec. II that nodal quasiparticles would be expected to make the Schmid-Higgs mode decay exponentially, and the justification in Sec. V ('which is justified as long as we are interested in the dynamics at q=0') is an assertion, not a derivation. The q=0 limit does not remove the nodal quasiparticle continuum, nor does it obviously suppress scattering between different Fermi-surface directions. Both calculations use the same truncated Hamiltonian, so their agreement does not test this assumption. Please either provide a controlled estimate of the off-diagonal terms at q=0 (e.g., a diagrammatic or RG argument, or a numerical test with a more complete kernel) or explicitly weaken the central claim to apply only within the truncated model.","section":"Sec. II, Hamiltonian (1), and Sec. V"},{"comment":"The asymptotic decay exponent α≈2 is determined solely from numerical fits ('From our numerical analysis it follows...'), with no error bars, no fit residuals, and no analytic asymptotic derivation from χ_SH(Ω). Given that the 1/t^2 law is a central claim, it should be backed by an analytic asymptotic analysis near the relevant threshold (e.g., saddle-point or van Hove analysis of the angular integral) rather than read off a log–log fit. Please provide the analytic exponent, or at minimum a quantitative fitting procedure with confidence intervals and a demonstration that the result is robust to the fit window.","section":"Sec. III, Fig. 2, and Eq. (47)"},{"comment":"The frequency 2√2 Δ0 is extracted numerically (peak in Im χ_SH and FFT) without identifying the analytic origin. The susceptibility expression (38) is a two-dimensional integral; the peak frequency and the oscillation frequency should be tied to a specific singularity or stationary-phase contribution. Without such an identification, the claim that the frequency is 'determined by the anti-nodal gap' remains a numerical observation. It would strengthen the paper to show analytically where 2√2 Δ0 emerges from the angular average of γ_n^2 in Eq. (38).","section":"Fig. 1 and Fig. 2, Eq. (38)"}],"minor_comments":[{"comment":"Typos: 'staisfies' in Eq. (2) line; 'the the' in the first introductory paragraph; 'paring' should be 'pairing' in Sec. IV or figure captions. Please proofread.","section":"General"},{"comment":"The notation Y(n) appears in Eq. (36) before it is explained; later text says 'after setting Y(n)=1' but does not define Y(n) for the d-wave case. Presumably Y(n)=γ_n, but this should be stated explicitly.","section":"Eq. (36)"},{"comment":"The discussion of finite-momentum dynamics is speculative and leans on Ref. [71], which is unpublished. Either provide some preliminary results or mark the statement as an outlook without citing an unavailable work as support.","section":"Sec. V"},{"comment":"The inset axis labels ('60 80 100 t Δ0') are unclear; please format as tΔ0 with a clear axis. The main panel x-axis should also specify units (tΔ0).","section":"Fig. 3"},{"comment":"The claim that the difference from Refs. [44,47] stems from the normalized versus unnormalized form factor would be more convincing if a brief numerical check or a comment on the angular average were included. As written it is a plausible but unverified attribution.","section":"Introduction, Refs. [44,47]"}],"recommendation":"major_revision","confidential_remarks":"The paper is internally consistent and the two calculations agree, but the central claim rests on an approximation that the authors themselves flag as potentially unjustified for nodal d-wave superconductors. The requested revisions—an analytic derivation or at least rigorous fit for the 1/t^2 exponent, an analytic justification of the 2√2 Δ0 frequency, and a more quantitative treatment of the off-diagonal term concern—are within the scope of a revision and would substantially raise the paper's impact. Without these, the result is a model-calculation claim of limited robustness."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper computes the q=0 Schmid-Higgs mode in a d-wave superconductor using Eilenberger/Keldysh theory and Anderson pseudospin dynamics. The headline result: after a weak quench, the order parameter oscillates at frequency 2√2 Δ₀ = 2Δ_an (the anti-nodal gap) and the amplitude decays as 1/t^2. The two methods agree, which is the strongest evidence that the calculation is internally consistent.\n\nWhat's actually new is the explicit long-time 1/t^2 decay law and the identification of the frequency with the anti-nodal gap in the normalized form-factor convention. That frequency statement goes beyond Refs. 44 and 47, though the difference may partly be a normalization artifact—once the gap convention is converted, the physical claim may be consistent with those earlier works.\n\nThree soft spots, in proportion. First, the 1/t^2 exponent is read off from numerical fits (Fig. 3 inset and the susceptibility FFT) with no analytic asymptotic analysis and no error bars. Since that is the load-bearing quantitative claim, it deserves a derivation or at least a careful asymptotic evaluation. Second, the Hamiltonian neglects off-diagonal-in-momentum pairing terms, and the authors are upfront that this is not a priori justified for a nodal d-wave state. Their appeal to q=0 does not obviously suppress scattering between Fermi-surface directions. However, the susceptibility calculation already includes the gapless quasiparticle continuum, so the decay law is not simply an artifact of ignoring the nodes; it is a genuine prediction of the model. Whether including the omitted terms would turn the decay exponential is plausible but unproven—the stress-test speculation is fair but not demonstrated. Third, no code or data accompanies the fits, so the numerics can't be independently checked without reimplementation.\n\nThis is a useful paper for people working on collective modes in unconventional superconductors and on THz pump-probe experiments in cuprates. The derivation is careful, the limitations are acknowledged, and the internal consistency is real. It deserves a proper referee, with the request that the authors supply an analytic derivation of the 1/t^2 law and a more nuanced comparison with the earlier d-wave Higgs mode literature.","headline":"A solid, internally consistent model calculation of the d-wave Higgs mode frequency and 1/t^2 decay, but the decay law is read off numerics and the key approximation is acknowledged as uncertain.","tokens_in":15083,"tokens_out":3598,"would_cite":true,"duration_ms":35767,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["67.85.De","34.90.+q","74.40.Gh"],"model":"deepseek-v4-flash","headline":"This paper claims that after a weak quench, the amplitude (Schmid-Higgs) mode of a d-wave superconductor oscillates at a frequency set by the anti-nodal gap, 2√2Δ0, and its amplitude decays as 1/t^2.","keywords":["Schmid-Higgs mode","d-wave superconductor","amplitude mode","pairing susceptibility","Anderson pseudospins","quench dynamics","anti-nodal gap","Eilenberger equation"],"falsifier":"Measure the time-dependent order parameter (e.g., by time-resolved ARPES or THz pump-probe) in a clean d-wave superconductor after a weak pump: if the dominant oscillation frequency is not 2√2 Δ0 or the envelope decays faster than a power law (exponentially), the central claim is wrong. Alternatively, include the off-diagonal pairing terms in a numerical solution of the pseudospin or time-dependent BCS equations; if the 1/t^2 tail disappears, the simplification is not justified.","tokens_in":14066,"feed_emoji":"⚛️","tokens_out":8141,"duration_ms":58617,"temperature":0.7,"pith_summary":"This paper aims to pin down the frequency and decay law of the longitudinal amplitude (Schmid-Higgs) mode in a d-wave superconductor after a weak sudden perturbation. Using the quasiclassical Eilenberger formalism, the authors compute the zero-momentum pairing susceptibility and find that the mode oscillates at 2√2Δ0, i.e., twice the pairing amplitude along the anti-nodal direction, and that the oscillation amplitude falls off as 1/t^2. They confirm this by direct numerical solution of the Anderson pseudospin equations of motion. If correct, the result distinguishes d-wave superconductors from conventional s-wave ones, where the mode frequency is 2Δ0 and the decay is 1/√t, and gives a concrete target for time-resolved experiments.","feed_headline":"d-wave Higgs mode oscillates at twice anti-nodal gap; decays as 1/t^2","feed_subtitle":"The result gives a concrete prediction for pump-probe experiments on cuprate superconductors.","key_machinery":"The argument rests on two equivalent descriptions of the same mean-field dynamics: the Eilenberger equation for the quasiclassical propagator (used to derive the Schmid-Higgs pair susceptibility χ_SH(Ω) from the self-consistency condition), and the classical equations of motion for Anderson pseudospins S_k, which are solved numerically for a weak quench. The load-bearing identity is the relation between the mode frequency and the anti-nodal gap, ω_SH = √2 Δ0 = Δ_an, which follows from the angular averages over the normalized d-wave form factor γ_n.","core_discovery":"The central claim is that after a weak quench the order parameter of a d-wave superconductor evolves as Δ(t) ≈ Δ0 [1 + A cos(2ω_SH t + π/4)/(tΔ0)^2] with ω_SH = √2 Δ0 ≡ Δ_an, the anti-nodal gap. The same frequency, 2√2Δ0, and the same 1/t^2 decay are found in the pairing susceptibility computed from the Eilenberger equation. The authors interpret the factor √2 as a consequence of using the normalized d-wave form factor γ_n = √2(n_x^2 - n_y^2); with the unnormalized cos2φ form factor, previous work obtained a different mode energy.","pith_inferences":["The paper's own caveat: the mean-field Hamiltonian drops off-diagonal-in-momentum pairing terms. At finite momentum, or when nodal quasiparticle scattering is included, the 1/t^2 tail may give way to exponential decay, which would make the mode much harder to observe; the paper's regime of validity is q=0 and the collisionless limit.","The discrepancy with earlier works is attributed to the normalization of the d-wave form factor. An independent re-derivation using the unnormalized form factor could verify this explanation and settle the numerical value of the mode frequency.","The same susceptibility machinery could be applied at finite momentum q; the paper notes that nodal vs anti-nodal directions may behave differently, so a spatially resolved calculation could reveal anisotropic decay of the mode.","For experiments, the prediction suggests that the observable oscillation window is short (since 1/t^2 decays quickly), so ultra-short-pulse setups would be needed to see the mode."],"forward_implications":["In a d-wave superconductor, the amplitude mode frequency is 2√2 Δ0 (2Δ_an), not 2Δ0 as in s-wave superconductors.","The oscillation amplitude decays as 1/t^2 at long times, much faster than the 1/√t law of the s-wave case.","The peak in the imaginary part of the Schmid-Higgs susceptibility is broad and sits above 2Δ0, so the mode is not a sharp resonance in d-wave systems.","Time-resolved pump-probe experiments on clean d-wave materials (e.g., cuprates) should see an oscillatory component at 2√2 Δ0 with a 1/t^2 envelope."],"fun_headline_variants":["d-wave Higgs mode: frequency 2Δ_an, decay 1/t^2","d-wave SC Higgs mode: oscillations at 2×anti-nodal gap, 1/t^2 decay","New d-wave Higgs mode law: 2Δ_an frequency, 1/t^2 amplitude","d-wave Higgs mode predicted: 2Δ_an, 1/t^2 decay in pump-probe"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing assumption is that Cooper-pair scattering between different Fermi-surface directions (off-diagonal-in-momentum pairing terms) can be neglected at zero momentum; if nodal quasiparticles couple to the amplitude mode at q=0, the 1/t^2 law is replaced by exponential decay.","fun_headline_variants_meta":{"raw":{"variants":["d-wave Higgs mode: frequency 2Δ_an, decay 1/t^2","d-wave SC Higgs mode: oscillations at 2×anti-nodal gap, 1/t^2 decay","New d-wave Higgs mode law: 2Δ_an frequency, 1/t^2 amplitude","d-wave Higgs mode predicted: 2Δ_an, 1/t^2 decay in pump-probe"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000267,"raw_usage":{"total_tokens":1442,"prompt_tokens":726,"completion_tokens":716,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":470,"completion_tokens_details":{"reasoning_tokens":611}},"tokens_in":470,"tokens_out":716,"duration_ms":6432,"temperature":1.0,"reasoning_tokens":611,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T23:50:33.340863+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the time-dependent order parameter (e.g., by time-resolved ARPES or THz pump-probe) in a clean d-wave superconductor after a weak pump: if the dominant oscillation frequency is not 2√2 Δ0 or the envelope decays faster than a power law (exponentially), the central claim is wrong. Alternatively, include the off-diagonal pairing terms in a numerical solution of the pseudospin or time-dependent BCS equations; if the 1/t^2 tail disappears, the simplification is not justified.","supporting_citations":[],"review_version":1}