{"id":"a435e172-41d3-491e-83f9-94f666465847","arxiv_id":"2601.09782","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In d-wave superconductors, the amplitude mode's peak frequency depends on momentum direction but its 1/t² decay does not; the phase mode softens at finite temperature due to nodal quasiparticles.","lead":"This theory paper computes how collective waves of paired electrons move in d-wave superconductors with electric repulsion, using two complementary methods. It predicts that one wave's speed depends on direction while its damping does not, and that another wave softens at finite temperature in a way unique to d-wave materials.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Finite-T transverse 'much larger decay rate' is asserted but never computed; only the pole position (dispersion) is derived. This is the load-bearing unsupported claim.","rationale":"The paper's two headline results are (i) finite-T softening plus enhanced decay of the transverse mode and (ii) q-independent 1/t² decay of the longitudinal mode. The first is the more load-bearing because it is stated in the abstract, introduction, and conclusions, yet no damping calculation appears in Section III or the appendices. The finite-T calculation only determines the real combination ζ(T) that sets the pole position; the determinant condition (39) fixes a real frequency, not a width. The claimed 'larger decay rate' is attributed to nodal-quasiparticle screening, but screening affects the real part of the density response; the width requires Im χ_T, which is not computed. I therefore do not see correlations as the deepest issue — the authors explicitly scope them out in Sec. V — although their admission that correlations 'may change the exponent' is a legitimate caveat. The reader already flagged the missing damping calculation as a shortcoming, so my agreement is partial. A concrete computation of Im χ_T at finite T would settle the matter; until then, the existing internal evidence supports keeping the verdict CONDITIONAL rather than moving it.","tokens_in":33256,"tokens_out":15308,"duration_ms":142059,"concrete_test":"Evaluate the full retarded transverse susceptibility χ_T(q,Ω) at finite T by analytic continuation of the diagrammatic expression (55) (or via the quasiclassical χ_AB^{-1}(q,Ω), Eq. 34) for v_F q = 0.125Δ and T/Δ = 0.01, 0.03, 0.05, 0.07, 0.09, matching Fig. 3. Compute Im χ_T(q,Ω) around the mode frequency and extract the half-width at half maximum (or fit a Lorentzian to the spectral function). Compare with the same calculation for an s-wave superconductor at identical T and q. If Γ_d-wave(T) is not significantly larger than Γ_s-wave(T) (or if the damping does not grow with T in the claimed manner), the abstract's 'much larger decay rate' statement is not supported by the model.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing gap is the finite-T transverse-mode damping claim. The Abstract, Introduction, and Conclusions state that at finite T the transverse mode is 'softer and has a much larger decay rate' due to partial Coulomb screening by nodal quasiparticles. However, no decay rate is calculated. Section III derives the real part of the inverse transverse susceptibility (Eq. 34) and the finite-T polarization bubble Π_T (Eq. 55), and uses them only to extract the pole position: ζ(T) in Eqs. (37)-(40) enters the determinant (39), which fixes a real mode frequency (e.g., Ω^(2D)(q) ∝ √[q ζ(T)]), not the pole width. The width would come from Im χ_T(q,Ω) (or from Im Π_T, Im Π_GF, Im Π_0), which is neither evaluated nor plotted. The 'partial screening' argument concerns the real part of the density response and does not by itself establish an enhanced damping rate. Since this enhanced decay is one of the paper's two headline results and is invoked for CG-mode experiments near T_c (Sec. V), the claim is currently unsupported by the calculation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies collective excitations of a clean d-wave superconductor with long-range Coulomb interaction using two complementary methods: quasiclassical Keldysh-Nambu theory and diagrammatic ladder/bubble resummation. The transverse (phase/plasma) mode is found to have the same dispersion as in an s-wave superconductor at T=0, while at finite T its velocity is reduced by a factor set by ζ(T), which decreases linearly with T because of nodal quasiparticles. The longitudinal (amplitude) mode is analyzed at zero temperature; it is a resonance in the continuum whose peak frequency depends on the direction of q for q in the nodal versus antinodal directions, while the time-domain decay is claimed to be 1/t² independent of |q| and direction. The two formalisms are shown to agree for the dispersion relations, and the gap equation eliminates the coupling constant. The paper also discusses possible connections to recent pump-probe and Raman experiments on cuprates.","tokens_in":33501,"tokens_out":3624,"duration_ms":45319,"significance":"If the central claims hold, the paper provides concrete, falsifiable predictions for the momentum-resolved collective-mode spectrum of a d-wave superconductor, especially the direction-dependent amplitude-mode dispersion and the softening of the Carlson-Goldman mode. The manuscript has clear strengths: the two independent derivations agree; the dispersion formulas are derived in detail, including the low-temperature asymptotic ζ(T) in Eqs. (37)-(38); and the results are parameter-free in the sense that the pairing coupling is eliminated via the gap equation. The formal machinery is therefore largely sound. However, two headline claims — the finite-T transverse-mode damping and the q-independent longitudinal decay — are asserted rather than derived, and the paper itself concedes in Sec. V that correlations may change the predicted power-law decay, limiting the direct experimental transferability.","major_comments":[{"comment":"The central claim that the transverse mode at finite T has a 'much larger decay rate' is not supported by the calculation. The text computes only real parts of the relevant functions: Re χ_AB in Eq. (34) and Fig. 3, and the dispersion from the determinant of Eq. (39), leading to Eq. (40). No imaginary part of χ_T, Π_T, Π_GF, or Π_0 is evaluated, plotted, or used to obtain a pole width. A broader peak in Re χ_AB is not a quantitative damping rate. Since this enhanced damping is a headline result and motivates the CG-mode discussion in Sec. V, it must either be derived by computing Im χ_T at finite T or the claim must be removed/weakened.","section":"Abstract, §III.B.2, §V"},{"comment":"The claim that the longitudinal mode decays as 1/t² independent of |q| and direction is based on Fig. 7 and a sentence in Sec. IV.C, but no analytic derivation or quantitative fit is provided. The abstract states 'the decay rate of this mode does not depend on momentum,' yet the width of Im χ_L is never extracted or compared. Given that this is the other headline result, the authors should provide either an analytic argument (e.g., from the ω→0 behavior of Im χ_L) or a careful numerical demonstration, including the extraction of the exponent.","section":"§IV.C, Conclusions"},{"comment":"The discussion concedes that electronic correlations 'may change the exponent in the power law decay' relative to the clean parabolic-band result. This is an honest limitation, but it undercuts the direct experimental application to cuprates stated in the same section. The formal result for the clean model is still valuable; however, the manuscript should more carefully separate the model prediction from the experimental expectation, especially in the Abstract and Conclusions, where the predictions are stated without this caveat.","section":"§V"}],"minor_comments":[{"comment":"The phrase 'the longitudinal mode in a d-wave superconductor (CG mode at T≤T_c)' appears to be a typo: the Carlson-Goldman mode is a transverse phase mode, not a longitudinal mode.","section":"Introduction, p. 4"},{"comment":"In the right (s-wave) panel, the legend entry 'v_F q = Δ' appears to be missing a numerical value; several entries are incomplete. Please check all legend labels.","section":"Fig. 4"},{"comment":"The expressions contain log(W²/ν) and log(W²/|ν|) with ν later defined as ν=2ν/Δ_max. Please clarify the dimensionless normalization of the argument throughout, since logarithms of dimensionful quantities are ambiguous.","section":"Eq. (76)"},{"comment":"The finite-T calculation approximates Δ(T)≈Δ(0) while keeping thermal factors from nodal quasiparticles. This is reasonable for T≪Δ, but the approximation should be stated more explicitly with the expected size of the correction.","section":"§III.B.2"},{"comment":"The data availability statement ('not publicly available upon publication because it is not technically feasible') is unusual; the paper contains no data files, so it might be clearer to state that no experimental data were generated.","section":"§VIII"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is technically careful and the two-method cross-check is a genuine strength. The main barrier is the unsupported finite-T damping claim, which appears in the Abstract and Conclusions and is used in the experimental discussion. If the authors can compute Im χ_T at finite T and either verify or retract the 'much larger decay rate' claim, the paper could become acceptable. The q-independence of the longitudinal decay also needs firmer support. I do not see a load-bearing error in the dispersion derivations themselves."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth your time. This paper works out finite-q dispersions for both transverse (phase/plasma) and longitudinal (amplitude) modes in a clean d-wave superconductor with Coulomb interaction, using two independent formalisms that agree. That is the real substance: at T=0 the transverse mode matches s-wave, at finite T it softens linearly in T through the nodal quasiparticle screening captured in zeta(T); the longitudinal mode is a resonant continuum feature whose peak position depends on the direction of q relative to the nodes, while the time-domain decay stays 1/t^2 across momenta. The derivation is careful, the two methods cross-check each other, and the main dispersion formulas are reproduced rather than quoted. The 1/t^2 decay for the longitudinal mode is shown numerically, not proven analytically, but the numerics are consistent and the s-wave analogue is established, so I would call that a minor gap.\n\nThe genuine soft spot is the abstract's claim of a 'much larger decay rate' for the finite-T transverse mode. No damping rate is computed anywhere. The paper derives the pole position from the real part of the susceptibility and the screening argument concerns the real part of the density response. The width would come from Im chi_T, which is never evaluated. So the quantitative claim about enhanced damping is unsupported. The softening itself is derived and is fine.\n\nThe other caveat is the model: clean, single-band, parabolic dispersion, with correlations and disorder neglected. The authors say this in Sec. V, explicitly noting correlations may change the 1/t^2 exponent. That is honest, but it means the experimental implications for cuprates are suggestive rather than predictive. The formal result stands on its own.\n\nWho benefits: people working on collective modes in unconventional superconductors, especially those connecting to Raman and THz experiments. The citation list covers the prior q=0 and T=0 work; Ref. [87] is a same-group preprint but it is used as a q=0 benchmark and the present paper goes beyond it, so I do not treat that as a problem. No code or data shipped, but the derivations are detailed enough to check.\n\nMy recommendation: send it to peer review. A referee should push for the damping calculation or for a softened claim, and for a statement about how the 1/t^2 result might be affected by correlations. But the core is sound and the predictions are specific enough to be useful. I would cite it once the damping claim is either computed or qualified.","headline":"Solid, cross-checked derivation of finite-q collective-mode dispersions in a clean d-wave superconductor; the headline finite-T damping claim is asserted, not computed, and the clean-model caveat is real but openly acknowledged.","tokens_in":33983,"tokens_out":634,"would_cite":true,"duration_ms":9590,"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":"In a clean d-wave superconductor with Coulomb interaction, the transverse phase mode matches the s-wave plasma mode at zero temperature but softens and broadens at finite temperature, while the longitudinal amplitude mode is a direction-dep","keywords":["d-wave superconductivity","collective modes","plasma mode","amplitude (Higgs) mode","Carlson-Goldman mode","Coulomb screening","nodal quasiparticles","pair susceptibility"],"falsifier":"Measure the time-domain decay of the coherent amplitude-mode oscillations in a clean d-wave superconductor (e.g., via THz pump-probe); if the envelope decays as t^{-α} with α≠2, or if α depends on the pump wavevector direction, the central claim is falsified. Alternatively, measure the Carlson-Goldman mode frequency near Tc; if its softening with T does not follow the linear-in-T reduction of ζ(T) relative to an s-wave reference, the finite-T screening mechanism is wrong.","tokens_in":33150,"feed_emoji":"🌊","tokens_out":7621,"duration_ms":68998,"temperature":0.7,"pith_summary":"This paper aims to establish how the two collective excitations of a d-wave superconductor—the phase (transverse) and amplitude (longitudinal) modes—behave when long-range Coulomb interaction is included. The authors show that at zero temperature the transverse mode is exactly the s-wave plasma mode, but at finite temperature it becomes softer and much broader because nodal quasiparticles partially screen the Coulomb field. The longitudinal mode, a resonance inside the quasiparticle continuum, has a peak frequency that depends on the direction of the wavevector relative to the d-wave nodes, while its decay in time is 1/t² independent of both the magnitude and direction of momentum. A sympathetic reader would care because these are concrete, momentum-resolved signatures that distinguish d-wave from s-wave superconductors and can be compared with pump-probe and Raman experiments on cuprates.","feed_headline":"Plasma mode softens in d-wave superconductors; amplitude decay 1/t²","feed_subtitle":"Thermal nodal quasiparticles screen Coulomb, softening the phase mode while amplitude decay remains 1/t².","key_machinery":"The load-bearing object is the pair-pair susceptibility χ(q,Ω), decomposed into transverse and longitudinal polarization bubbles. The transverse bubble is dressed by the screened Coulomb interaction via a bubble insert Π_GF²/(V_q^{-1}-2Π_0); the longitudinal bubble is not. The d-wave form factor γ(θ)=√2 cos 2θ makes the gap vanish at nodal points, so thermally excited nodal quasiparticles sit at low energy and partially screen the Coulomb potential at finite T. Two independent methods—quasiclassical Eilenberger equations in Keldysh-Nambu space and diagrammatic ladder sums—yield the same mode dispersions.","core_discovery":"The central claim is that in a clean d-wave superconductor the pair susceptibility separates into a Coulomb-dressed transverse (phase) part and a longitudinal (amplitude) part untouched by Coulomb. At T=0 the phase mode is the same plasma mode as in s-wave: ω ∝ √q in 2D and ω = √(4πne²/m) in 3D. At finite T it softens and broadens because nodal quasiparticles partially screen the Coulomb potential; the effective velocity falls linearly with T. The amplitude mode is a resonance at 2Δ_max at q=0; at finite q its peak frequency depends on momentum direction, while its time-domain decay is 1/t² independent of momentum magnitude and direction.","pith_inferences":["If electron correlations renormalize nodal quasiparticle damping, the 1/t² decay is likely to become a different exponent; the paper itself notes this, so measuring the exponent in cuprates could quantify correlation strength.","The directional dependence of the amplitude-mode resonance suggests that angle-resolved experimental probes, such as directional THz pump-probe or momentum-resolved Raman, should observe different oscillation periods for excitations along nodal versus antinodal directions.","The finite-T plasma-mode softening could be tested by near-field or transmission experiments that tune T/Tc and look for the linear-in-T reduction of the mode velocity predicted here.","Since pair-breaking disorder suppresses d-wave superconductivity, extending the calculation to include disorder may replace the clean 1/t² decay with a faster decay, offering another experimental handle."],"forward_implications":["The Carlson-Goldman/plasma mode in a d-wave superconductor near Tc should appear at lower frequency and with broader width than in s-wave, providing a direct finite-T signature.","The longitudinal (amplitude) susceptibility is nonzero at all frequencies due to nodal quasiparticles, unlike s-wave where it vanishes below 2Δ; pump-probe oscillations should show a momentum-direction-dependent period.","The decay of the amplitude mode's oscillations in the time domain is 1/t² independent of both magnitude and direction of momentum, a universal signature in clean d-wave superconductors.","At zero temperature the plasma-mode dispersion matches s-wave, so any deviation in experiment at low T would indicate physics beyond the clean single-band model."],"fun_headline_variants":["Nodal quasiparticles screen Coulomb, softening plasma mode in d-wave SC","Plasma mode softens in d-wave SC due to nodal quasiparticle screening","Amplitude decay 1/t² independent of momentum in d-wave SC","Direction-dependent longitudinal mode in d-wave superconductors","Plasma mode softens with T, amplitude decay stays 1/t² in d-wave SC"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The calculation assumes a clean, single-band, parabolic-band d-wave superconductor with only BCS pairing plus Coulomb interaction; if electron correlations or disorder change the screening or the decay exponent, as the paper itself concedes, the quantitative predictions would not transfer to real cuprates.","fun_headline_variants_meta":{"raw":{"variants":["Nodal quasiparticles screen Coulomb, softening plasma mode in d-wave SC","Plasma mode softens in d-wave SC due to nodal quasiparticle screening","Amplitude decay 1/t² independent of momentum in d-wave SC","Direction-dependent longitudinal mode in d-wave superconductors","Plasma mode softens with T, amplitude decay stays 1/t² in d-wave SC"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000809,"raw_usage":{"total_tokens":3364,"prompt_tokens":696,"completion_tokens":2668,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":440,"completion_tokens_details":{"reasoning_tokens":2568}},"tokens_in":440,"tokens_out":2668,"duration_ms":19206,"temperature":1.0,"reasoning_tokens":2568,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T10:29:49.358942+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the time-domain decay of the coherent amplitude-mode oscillations in a clean d-wave superconductor (e.g., via THz pump-probe); if the envelope decays as t^{-α} with α≠2, or if α depends on the pump wavevector direction, the central claim is falsified. Alternatively, measure the Carlson-Goldman mode frequency near Tc; if its softening with T does not follow the linear-in-T reduction of ζ(T) relative to an s-wave reference, the finite-T screening mechanism is wrong.","supporting_citations":[],"review_version":1}