{"id":"41ce16e2-0eec-4233-a1f6-c953c33c578f","arxiv_id":"2608.11610","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A finite-thickness superconducting film can support an acoustic plasmon whose velocity scales as sqrt(d) and sqrt(Tc-T), arising from the transverse mismatch of normal and superfluid charge profiles.","lead":"This paper explains a newly observed acoustic plasmon mode in thin superconducting films by showing that uneven distributions of normal and superfluid charge across the film thickness break local neutrality and produce a low-frequency mode that couples to light. If correct, it reconciles the measured mode with the Anderson-Higgs stiffness paradigm and suggests new uses in superconducting optoelectronics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The fitted structure-factor constant D0≈70 violates a geometric bound: for normalized transverse profiles on a film of thickness d, D0 cannot exceed about 2, so Eq. (18) cannot yield the claimed acoustic velocity.","rationale":"The reader's weakest assumption was that the transverse charge profiles are postulated rather than derived and that D0 is a fitted constant. My pass sharpens this into a concrete internal inconsistency: the same normalization condition used in Eq. (2) imposes an upper bound on D0 of about 2, while the paper requires D0≈70 to reach the experimental velocity. This is not a matter of unresolved microscopic details; it means the central mechanism, as formulated, cannot produce the claimed dispersion even with optimally chosen profiles. Because the paper's headline claim rests on this quantitative match, the verdict should move from CONDITIONAL to REJECT. I do not see an ad hominem issue; the concern is purely mathematical and would be settled by the proposed coefficient test. If the bound computation were somehow wrong, the verdict would need revisiting, but as stated the contradiction is decisive.","tokens_in":7628,"tokens_out":10423,"duration_ms":111950,"concrete_test":"For arbitrary nonnegative normalized profiles Aα(z) on [0,d], compute C(k) = (Mso + kaJss)/(Joo + ka) and evaluate the exact bound C(k)/k ≤ 2d + a for all small k, using Lαβ ≤ d. Then substitute this bound into Eq. (18) with the paper's parameters (d = 8 nm, k ≈ 80 cm^-1, their quoted N_s/N ≤ 1) and compare the resulting maximum acoustic velocity with the claimed s ≈ 1.4×10^9 cm/s and ωk(T=0) ≈ 113 GHz. If the velocity bound is exceeded by the paper's fit, the central acoustic branch cannot be produced by the proposed mechanism.","verdict_should_be":"REJECT","load_bearing_attack":"The paper's central quantitative result, Eq. (18), follows from the long-wavelength approximation in Eq. (17): (Mso + kaJss)/(Joo + ka) ≈ D0 kd, with D0 fitted to D0≈70 to reproduce the experimental velocity. This D0 is not a free parameter consistent with the model. For any nonnegative, normalized transverse charge distributions Aα(z) supported on a slab of width d, one has Jαβ(k) = 1 − k Lαβ + O(k^2), where Lαβ = ∫∫ Aα(z) Aβ(z′) |z − z′| dz dz′ satisfies 0 ≤ Lαβ ≤ d because |z−z′| ≤ d on the support. Therefore Mso = JssJoo − Jso^2 = k(2Lso − Lss − Loo) + O(k^2), so (Mso + kaJss)/(Joo + ka) = k(2Lso − Lss − Loo) + O(k^2), and the coefficient 2Lso − Lss − Loo is bounded above by 2d. Hence D0 ≤ 2 + O(a/d), not 70. With d = 8 nm, the fitted value D0 ≈ 70 requires a transverse charge separation of order 560 nm, thirty-five times the film thickness, which is impossible for the normalized profiles assumed in Eq. (2). The acoustic velocity in Eq. (18) is therefore not achievable with any physical choice of A_s, A_n, A_o consistent with the stated normalization; the quantitative agreement with experiment is enforced by an inadmissible fitting constant. This is a stronger, internal consistency failure than the reader's observation that the profiles are merely not derived: their normalization alone already caps the mode velocity far below the claimed value.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a microscopic theory for an acoustic plasmon mode in quasi-two-dimensional superconducting films. The authors model a superconducting film of finite thickness with transverse charge distribution functions for superconducting, normal, and deep-lying electrons, and derive a general dispersion relation from linearized kinetic theory or Mattis-Bardeen conductivities. In the long-wavelength limit they obtain an acoustic branch with linear dispersion, velocity scaling as the square root of the film thickness and of (Tc - T) near the critical temperature, and claim quantitative agreement with the experiments of Andreeva et al. [7]. The paper explicitly states that the microscopic origin of the transverse profiles is an open question, and the structure-factor constants D0 and S0 are fitted to the experimental mode velocity and damping.","tokens_in":8065,"tokens_out":8227,"duration_ms":84490,"significance":"If the proposed mechanism were sound, it would challenge the Anderson-Higgs stiffness picture and explain the recent observation of an EM-active subgap acoustic mode in superconducting films. The paper offers a concrete analytical framework and correctly identifies a physically interesting geometric mechanism (transverse charge separation). However, the central quantitative prediction rests on a fitted constant D0 that violates a fundamental geometric bound for normalized transverse profiles. The claimed agreement with experiment is therefore not supported by the model, and the significance of the result as stated cannot be established. The paper also honestly discloses the ad hoc nature of the transverse profiles, which further undercuts the predictive content.","major_comments":[{"comment":"The constant D0 ≈ 70 used to reproduce the experimental mode velocity is inadmissible for the normalized transverse profiles defined in the manuscript. Expanding J_αβ(k) = ∫∫ A_α(z) A_β(z') e^{-k|z-z'|} dz dz' for small k gives J_αβ = 1 - k L_αβ + O(k²) with L_αβ = ∫∫ A_α(z) A_β(z') |z-z'| dz dz'. Since the A_α are nonnegative and normalized to unity on a support of width d, one has 0 ≤ L_αβ ≤ d. Consequently (M_so + ka J_ss)/(J_oo + ka) = k(2L_so - L_ss - L_oo + a) + O(k²), so the coefficient D0 in Eq. (17) satisfies D0 ≤ 2 + a/d. With d = 8 nm and a ≈ 0.3 nm this yields D0 ≤ 2.04, not 70. Thus the acoustic velocity in Eq. (18) cannot be achieved with any physical choice of A_s, A_n, A_o consistent with the stated normalization. The quantitative agreement with experiment is enforced by an unphysical fitting constant.","section":"Collective mode, Eqs. (8), (17), (18)"},{"comment":"The transverse charge distribution functions A_s(z), A_n(z), and A_o(z) are postulated rather than derived, and the manuscript itself states that their microscopic origin remains an open question. Because the entire acoustic branch in Eq. (18) depends on the structure factor through D0, the central claim is conditional on an unmodeled geometric input. This would be acceptable if the resulting D0 were within the physical bound derived above, but it is not. The mechanism as formulated therefore does not explain the observed high-velocity acoustic mode; it only demonstrates that an acoustic dispersion would follow from an arbitrarily large, physically impossible structure factor.","section":"Model, Eq. (2) and final paragraph before Conclusions"},{"comment":"The claim of quantitative agreement with experiment is compromised by circularity: D0 is estimated from the experimental zero-temperature velocity and S0 is set by the measured damping at Tc. The subsequent temperature dependence is then carried by the superconducting density Ns(T) and normal density Nn(T), which are not independently fitted but also are not verified against the experimental data in the paper. Thus the agreement in Fig. 1 is not a predictive test of the theory but a consequence of fitting the two structure-factor constants. This circularity is compounded by the inadmissibility of D0 noted above.","section":"Collective mode and numerical analysis, Eq. (18) and following paragraph"}],"minor_comments":[{"comment":"The 'after algebraic simplification' step leading from Eq. (13) to Eq. (14) is not shown; a derivation or a supplementary note should be provided to allow verification of the approximations used.","section":"Collective mode, Eq. (13)"},{"comment":"The notation ω0 is k-dependent (ω0² = 2π e² k N/m), which is not stated explicitly. As written, Eq. (18) can be misread as implying ω_k ∝ sqrt(k); the authors should clarify that the k-dependence of ω0 cancels the sqrt(k) to yield a linear dispersion.","section":"Eqs. (13)-(18)"},{"comment":"The statement 'our theory predicts ω_k ∝ d' is only true at fixed k and with N = N_V d; please specify these conditions explicitly.","section":"Collective mode, sentence after Eq. (18)"},{"comment":"The damping Γ in Eq. (18) is independent of k because S0 is a constant, but the experimental data may exhibit k dependence; the paper does not discuss whether this is consistent with the observations.","section":"Numerical analysis and Fig. 1"},{"comment":"The dirty-limit entry T ≫ Δ for \tilde N_s/N is written as 'π Tτ_i Δ²/T²' in a way that is ambiguous; based on Eq. (20) it should presumably be π τ_i Δ²/(2T). Please correct the typesetting.","section":"Table I"},{"comment":"Reference [10] is 'private communications'; for a quantitative claim about experimental verification, an archival reference or a public data source should be provided.","section":"References"},{"comment":"There are several typographical and grammatical issues, including 'full a full cubic equation' in the paragraph after Eq. (14) and 'Beyond explanation recent measurements' in the Conclusions.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The stress-test concern about the geometric bound on D0 is correct and is the decisive issue: the model's central quantitative result requires a structure-factor coefficient that cannot arise from the normalized transverse profiles the paper itself defines. This is not a matter of presentation or a missing derivation; it is an internal inconsistency in the mechanism as formulated. The paper's own admission that the transverse profiles are unmodeled reinforces the conclusion that the claimed explanation of the experimental data is not supported. I see no straightforward fix within the current scope, hence a reject recommendation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, the core idea is genuinely interesting: a transverse mismatch between superfluid and normal charge profiles in a finite-thickness film could break local neutrality and produce an EM-active acoustic mode, which would explain the recent experiment challenging Anderson-Higgs stiffness. Second, the quantitative claim is built on a fitted constant D0≈70 that the model's own equations cannot accommodate. That is not a minor issue; it sinks the claimed agreement.\n\nWhat is actually new is the mechanism itself. The authors go beyond the standard Carlson-Goldman and Ohashi-Takada analyses by explicitly including transverse charge distributions, and they correctly identify that the resulting form factors can yield a linear dispersion. The derived scalings—√(Tc−T) near Tc and √d in thickness—are robust and do not depend on the fitted constants. The dirty-limit treatment via Mattis-Bardeen conductivities is standard and plausible, and the paper is careful to note that the microscopic origin of the profiles remains open.\n\nThe soft spot is load-bearing. For any nonnegative normalized profiles on a film of thickness d, the form factors satisfy Jαβ(k)=1−kLαβ+O(k²) with 0≤Lαβ≤d. Expanding the structure factor in (17) gives (Mso+kaJss)/(Joo+ka) = k(2Lso−Lss−Loo+a)+O(k²), so the dimensionless coefficient D0 = (2Lso−Lss−Loo+a)/d is bounded by about 2+a/d≈2.04. The paper fits D0≈70 to match the experimental velocity. That is a contradiction with the normalization of the very profiles the paper assumes. Equation (18) therefore cannot yield the claimed acoustic velocity with any physical A_s, A_n, A_o. The frequency and damping are not predictions; they are fits using an inadmissible constant, which the \"remarkable agreement\" rests on.\n\nThere are additional, lesser issues: the profiles themselves are postulated rather than derived, and S0 is likewise fitted to the damping. So the paper is better read as a proposal for a mechanism plus scaling laws, not as a quantitative microscopic explanation.\n\nI would still send this to peer review. The experimental puzzle is real and the mechanism is original; a serious referee should push the authors to confront the D0 bound, derive the profiles from a microscopic model, or recalibrate their claims. As it stands, the paper deserves engagement but not acceptance. If revised honestly, it could become a useful contribution.","headline":"Interesting mechanism for acoustic plasmons in superconducting films, but the fitted D0≈70 violates the model's own geometric bounds, so the quantitative agreement with experiment is not supported.","tokens_in":8494,"tokens_out":4755,"would_cite":false,"duration_ms":47947,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A superconducting film of finite thickness supports an acoustic plasmon whose velocity is set by thickness and by the superconducting density, breaking the usual stiffness of the plasma spectrum and coupling the mode to light.","keywords":["acoustic plasmon","superconducting film","plasma stiffness","Carlson-Goldman mode","transverse charge distribution","dirty-limit conductivity","subgap collective mode","electromagnetic coupling"],"falsifier":"Measure the acoustic mode in films of fixed areal density but different thickness: the theory requires $s \\propto \\sqrt{d}$, while a Carlson-Goldman-like mode is independent of $d$, and the mode must vanish in the strictly two-dimensional limit $d \\to 0$. A second check is to resolve the dispersion $\\omega_k = s k$ and compare the temperature dependence of the slope with the predicted $N_s(T)/N$; if the measured $s$ does not vanish as $\\sqrt{T_c - T}$ near $T_c$, the mechanism is not the one described.","tokens_in":7457,"feed_emoji":"⚡","tokens_out":13727,"duration_ms":129678,"temperature":0.7,"pith_summary":"The paper tries to establish that the plasma spectrum of a superconductor is not necessarily stiff: in a film of finite thickness, the superfluid and normal charge densities occupy different transverse profiles, and their spatial mismatch breaks local neutrality. Using the Poisson equation for these profiles together with kinetic-equation (clean) and dirty-limit conductivities, the authors derive an acoustic branch $\\omega_k = s k$ whose velocity is $s = \\omega_0 \\sqrt{(N_s/N)(M_{so}+k a J_{ss})/(J_{oo}+k a)}$, which for $a \\ll d$ reduces to $s = \\omega_0 \\sqrt{D_0 N_s d/N}$. This mode lives below the superconducting gap, carries a dipole moment, couples to external radiation, and its velocity scales as $\\sqrt{d}$ and as $\\sqrt{T_c - T}$ near $T_c$, matching recent optical and microwave observations on thin films. If correct, the result reconciles those observations with the basic stiffness constraints of superconductivity and separates the new mode cleanly from the quasi-neutral Carlson-Goldman mode.","feed_headline":"Acoustic plasmons in superconducting films track thickness","feed_subtitle":"A transverse mismatch between superfluid and normal charge gives the mode a dipole moment and makes it visible to light.","key_machinery":"The central object is the set of transverse charge distribution functions $A_\\alpha(z)$, normalized so $\\int A_\\alpha(z)\\,dz = 1$, and their geometric form factors $J_{\\alpha\\beta}(k) = \\int\\!\\int dz\\,dz'\\, A_\\alpha(z)\\,e^{-k|z-z'|}\\,A_\\beta(z')$. These enter a dispersion relation built from the 2D Coulomb potential and polarization operators $\\Pi_s$, $\\Pi_n$, and $\\Pi_0$; in the clean limit the response comes from the kinetic equation, and in the dirty low-frequency limit from the standard dirty-limit conductivities. The crucial simplification is that for $a \\ll d$ the combination $(M_{so} + k a J_{ss})/(J_{oo} + k a)$ becomes $D_0 k d$ with $D_0$ a fitted dimensionless constant, which turns what would be a gapped plasma oscillation into a linearly dispersing acoustic mode with velocity $s = \\omega_0 \\sqrt{D_0 N_s d/N}$. Damping is governed by the analogous structure factor $S(k) \\approx S_0$.","core_discovery":"The central claim is that a finite-thickness superconducting film supports a subgap collective mode with linear dispersion $\\omega_k = s k$, where the velocity $s$ is controlled by the superconducting density, the film thickness, and the temperature. The mechanism is geometric: the normalized transverse charge distributions $A_s(z)$, $A_n(z)$, and $A_o(z)$ for superfluid, normal, and deep-lying electrons are not identical, so the long-range Coulomb interaction couples to a net local charge in an oscillation that would be neutral in the Carlson-Goldman picture. Substituting the low-frequency polarization operators into the full dispersion relation and taking $a \\ll d$ gives $\\omega_k = \\omega_0 \\sqrt{D_0 N_s k d/N}$, i.e., an acoustic branch whose velocity scales as $\\sqrt{d}$ and, near $T_c$, as $\\sqrt{T_c - T}$. The normal component is overdamped and supplies a damping rate $\\Gamma = (\\omega_0^2 \\tau/2)(N_n/N) S_0$, which grows with temperature and shows a Hebel-Slichter-like peak. The paper states that the transverse charge mismatch is what gives the mode its dipole moment and strong coupling to external electromagnetic radiation.","pith_inferences":["An extension the authors leave implicit is that $D_0$ encodes the unmodeled transverse profiles; deliberately engineering the film interfaces or disorder (capping layers, asymmetric substrates, irradiation) should change the acoustic velocity, and sample-to-sample variation of $D_0$ would expose what the fitted constant is hiding.","By analogy, any two-component system whose charge carriers have different transverse confining wavefunctions could show the same mechanism—for example, coupled electron-hole bilayers or superconductor/normal-metal hybrids—so the paper's argument may generalize beyond superconducting films.","The polarization dependence of the optical coupling should carry direct information about $A_s(z)$ and $A_n(z)$; measuring the oscillator strength versus incidence angle could reconstruct the mismatch and test the assumed geometry without a specific microscopic model."],"forward_implications":["Near $T_c$, the acoustic velocity obeys $s \\propto \\sqrt{T_c - T}$ in both clean and dirty limits, with an extra suppression by $\\sqrt{\\tau T_c}$ in a disordered film, giving an optical handle on the condensate fraction.","Because the velocity scales as $\\sqrt{d}$ at fixed areal density, film thickness is a control parameter for the subgap resonance frequency, and strictly two-dimensional films should show no such mode.","If the bulk three-dimensional electron density is fixed instead, the frequency scales as $\\omega_k \\propto d$, a prediction the paper says is already checked for 5, 8, and 12 nm films.","The mode's damping contains a Hebel-Slichter-like coherence peak near $T_c$, so linewidth measurements can probe the same effect seen in nuclear magnetic resonance.","Unlike Carlson-Goldman modes, this acoustic plasmon has a dipole moment and couples to external radiation, so far-field optical and microwave spectroscopy can observe it directly."],"supporting_citations":[{"why":"Establishes the random-phase-approximation result that Coulomb interactions stiffen the plasma spectrum, the constraint this paper sets out to bypass.","marker":"[1]"},{"why":"Defines the Carlson-Goldman mode, the quasi-neutral acoustic collective mode with low velocity and no direct electromagnetic coupling that serves as the baseline comparison.","marker":"[3, 4]"},{"why":"Supplies the picture of out-of-phase oscillations between condensate and normal fluid that underlies the Carlson-Goldman picture the new mode must be distinguished from.","marker":"[5, 6]"},{"why":"Provides the experimental observation of the acoustic plasmon in thin films, including its temperature, damping, and thickness behavior.","marker":"[7]"},{"why":"Supplies the kinetic-equation description of transport in superconductors used for the clean-limit response.","marker":"[8]"},{"why":"Provides the dirty-limit conductivities from which the polarization operators are derived at low frequency.","marker":"[9]"},{"why":"Reported private communication cited for the experimental agreement with the predicted thickness scaling for 5, 8, and 12 nm films.","marker":"[10]"},{"why":"Explains the coherence peak used to interpret the peak in the calculated damping near the transition temperature.","marker":"[11]"},{"why":"Shows how transverse electron wave functions define the charge profiles in a normal film, the geometric input analogous to the assumed $A_\\alpha(z)$.","marker":"[12]"}],"fun_headline_variants":["Acoustic plasmons in films defy Anderson-Higgs stiffness","Film thickness unlocks acoustic plasmon beyond Carlson-Goldman","Superconducting films reveal acoustic mode with linear dispersion","Charge mismatch in films enables acoustic plasmon visibility","Acoustic mode in superconducting films: paradigm shifted"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the superfluid and normal charge distributions really are arranged differently across the film thickness; the paper assumes this mismatch rather than deriving it from a model, and the constant $D_0$ that fixes the velocity is fitted to the measured value.","fun_headline_variants_meta":{"raw":{"variants":["Acoustic plasmons in films defy Anderson-Higgs stiffness","Film thickness unlocks acoustic plasmon beyond Carlson-Goldman","Superconducting films reveal acoustic mode with linear dispersion","Charge mismatch in films enables acoustic plasmon visibility","Acoustic mode in superconducting films: paradigm shifted"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000182,"raw_usage":{"total_tokens":1310,"prompt_tokens":942,"completion_tokens":368,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":558,"completion_tokens_details":{"reasoning_tokens":293}},"tokens_in":558,"tokens_out":368,"duration_ms":4568,"temperature":1.0,"reasoning_tokens":293,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:33:15.415627+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the acoustic mode in films of fixed areal density but different thickness: the theory requires $s \\propto \\sqrt{d}$, while a Carlson-Goldman-like mode is independent of $d$, and the mode must vanish in the strictly two-dimensional limit $d \\to 0$. A second check is to resolve the dispersion $\\omega_k = s k$ and compare the temperature dependence of the slope with the predicted $N_s(T)/N$; if the measured $s$ does not vanish as $\\sqrt{T_c - T}$ near $T_c$, the mechanism is not the one described.","supporting_citations":[{"cited_title":"stiff- ness","cited_arxiv_id":null,"evidence_quote":"Establishes the random-phase-approximation result that Coulomb interactions stiffen the plasma spectrum, the constraint this paper sets out to bypass."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the experimental observation of the acoustic plasmon in thin films, including its temperature, damping, and thickness behavior."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the kinetic-equation description of transport in superconductors used for the clean-limit response."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the dirty-limit conductivities from which the polarization operators are derived at low frequency."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reported private communication cited for the experimental agreement with the predicted thickness scaling for 5, 8, and 12 nm films."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Explains the coherence peak used to interpret the peak in the calculated damping near the transition temperature."}],"review_version":1}