{"id":"ba71bd84-bf0a-47b0-bdd3-1410f713ca2d","arxiv_id":"2411.14782","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"In the Holstein model, the superconducting collective mode frequency evolves from about 2Δ_P in the BCS regime to the soft phonon in the crossover and to the superfluid stiffness in the BEC regime, with weight proportional to the pairing gap.","lead":"Using a non-perturbative quantum simulation of the Holstein model, this paper traces how the collective phase mode of a superconductor changes as pairing goes from weak BCS to strong BEC behavior. The mode's frequency shifts from twice the pairing gap to a soft lattice mode, then to the superfluid stiffness, giving an observable fingerprint of phase fluctuations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"BCS-regime branch of the central evolution rests on a single broad spectral feature; if that feature is continuum onset rather than a collective pole, the claimed BCS-to-BEC continuity lacks its anchor.","rationale":"The reader's weakest assumption was numerical convergence of the NRG parameters, which is a real concern. I partially agree: convergence checks are missing, and the extracted peak positions and weights could shift with Nph and Λ. However, I regard the more load-bearing issue as the sparse and ambiguous BCS-regime dataset: the entire 'gap mode near 2ΔP, increasing with coupling' branch rests on one broad shoulder at Eb=0.4, and the text itself calls it a shoulder. If that feature is the pair-breaking continuum edge rather than a collective pole, then the claimed evolution of a single collective mode from BCS to BEC lacks its starting point. The intermediate and BEC branches are better supported: multiple points show ω_col tracking the normal-state soft phonon ω_s, and in the BEC regime ω_col decreases monotonically with coupling in a way consistent with reduced superfluid stiffness. The W ∝ ΔP relation is presented across the full coupling range, but it is defined using an integral over the extracted peak region, so it does not independently certify the BCS feature as a collective pole. For these reasons, the concern does not overturn the qualitative BEC-region result or the paper's overall value, but it does justify keeping the verdict CONDITIONAL rather than ACCEPT. A small set of additional weak-coupling calculations, combined with a pole-versus-continuum fit, would settle whether the BCS anchor is real.","tokens_in":15097,"tokens_out":11446,"duration_ms":143431,"concrete_test":"Add DMFT-NRG calculations at Eb=0.2, 0.3, 0.5, and 0.6 using Nph=12, Λ=1.2, and z-averaging, and fit the low-energy superconducting phonon spectrum near 2ΔP with a pole-plus-continuum model (e.g., a Lorentzian plus a BCS pair-breaking threshold term). If the feature is a stable, underdamped pole whose position tracks 2ΔP across these additional points, the BCS anchor survives; if the feature is a broad continuum shoulder whose apparent position simply tracks the threshold, the BCS branch must be relabeled and the continuous-evolution claim should be restricted to the intermediate and BEC regimes.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that a single collective object, identified as the U(1) phase mode, evolves continuously from a gap mode near 2ΔP in the BCS regime to a low-energy mode tied to D_s in the BEC regime. The BCS anchor is supported by exactly one computed point: at Eb=0.4, the feature is described in the text as a 'small and broad peak or shoulder around ω=2ΔP'. No additional weak-coupling points (e.g., Eb=0.2, 0.3, 0.5, 0.6) are shown, and the claim that ω_col is 'proportional to 2ΔP' and increases with coupling is asserted from this single, barely resolved feature. A broad shoulder at 2ΔP in the superconducting density response can be the onset of the pair-breaking continuum rather than a genuine collective pole. Because the paper explicitly distinguishes the gap mode from the amplitude mode but does not separately establish that the gap-mode peak is a pole, the BCS leg of the crossover narrative is uncertain. This is compounded by the absence of NRG convergence checks for Λ=1.6, NS=1000, Nph=10; if the weak-coupling feature is an artifact of truncation or discretization, the claimed continuity between the BCS gap mode and the intermediate/BEC phase mode collapses. The intermediate- and BEC-regime branches are supported by more points and by the normal-state comparison, so the concern is localized but load-bearing for the full evolution picture.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Holstein model in infinite dimensions using DMFT with an NRG impurity solver, at zero temperature, half-filling, and antiadiabatic phonon frequency ω0=2.0 (in units of the half-bandwidth D). It computes phonon spectral functions and density-density correlation spectra in both the normal and superconducting states. The central claim is that the low-frequency collective mode in the superconducting state evolves continuously across the BCS-BEC crossover: at weak coupling it appears near 2ΔP (with a single computed point at Eb=0.4), in the intermediate regime it tracks the normal-state soft phonon frequency ωs when ωs<2ΔP, and in the BEC regime it decreases in proportion to the superfluid stiffness Ds. The integrated weight W of the mode is reported to be proportional to the pairing gap ΔP, which is interpreted as evidence that the mode originates from U(1) gauge-symmetry breaking. The phase-fluctuation interpretation is supported by an uncertainty-relation estimate based on lattice displacement fluctuations.","tokens_in":15421,"tokens_out":7705,"duration_ms":76473,"significance":"If correct, the result would establish a concrete scenario in which the phase mode of a strong-coupling superconductor appears at low energy, tied to Ds rather than to 2ΔP, with implications for BCS-BEC crossover materials such as magic-angle twisted graphene and iron chalcogenides, and for cold-atom analogues. The paper has clear methodological strengths: the key quantities (the phonon spectrum, the density-density correlation spectrum, ΔP, and Ds from Ref. 40) are computed independently within a non-perturbative DMFT-NRG framework, so the main comparison is not circular; the mode positions are read off directly from spectral functions and are falsifiable. However, the BCS leg of the evolution is anchored by a single broad spectral feature, no NRG convergence checks are reported, and the phase-fluctuation estimate is uncontrolled. These gaps are not fatal to the overall physical picture, but they must be addressed before the central continuity claim can be accepted.","major_comments":[{"comment":"The statement that ωcol is proportional to 2ΔP and increases with coupling in the BCS regime is supported by exactly one data point, Eb=0.4, which the text itself describes as a 'small and broad peak or shoulder around ω=2ΔP'. Such a feature can be the onset of the pair-breaking continuum rather than a genuine collective pole. The manuscript provides no additional weak-coupling points (e.g., Eb=0.2, 0.3, 0.5, 0.6), no pole/residue analysis, and no comparison with the expected continuum threshold. Because this single point anchors the claimed continuity from the BCS gap mode to the intermediate/BEC phase mode, the BCS branch of the central evolution is not yet established.","section":"Phonon and density-density correlation spectra; Fig. 3(a)"},{"comment":"The NRG parameters Λ=1.6, NS=1000, Nph=10 are stated but no convergence checks with respect to these truncations are provided. The low-frequency features in Figs. 1 and 2 are broad and resolution-dependent, and both the extracted ωcol and the integrated weight W depend on spectral resolution and on the phonon Hilbert-space truncation Nph. In particular, the existence and position of the weak-coupling shoulder at Eb=0.4, and the quantitative values in the crossover and BEC regimes, could be numerical artifacts. The authors should report tests of ωcol and W for different Λ (e.g., 1.5 and 2.0), different NS, and different Nph, or else give an explicit numerical uncertainty estimate.","section":"Model and methods"},{"comment":"The weight is defined as W = ∫_0^{2ωcol} ρsc_ph(ω), so the upper integration limit is itself proportional to the extracted mode frequency. Since ωcol varies strongly across the crossover, the integration range changes with coupling, and the reported proportionality W ∝ ΔP could partly be an artifact of this coupling-dependent cutoff. The authors should show that the proportionality is robust to alternative cutoffs (e.g., a fixed frequency window, or an analysis based on a fitted peak profile) and, ideally, that the weight is separated from the normal-state phonon contribution.","section":"Phonon and density-density correlation spectra; Fig. 3(b)"},{"comment":"The quantitative link between the collective-mode frequency and phase fluctuations rests on the relations Δn = (ω0/2g)Δx and Δθ ∼ 1/Δn. At the operator level, the coupling g x(n−1) relates the displacement operator to the density operator, but a mean-field relation between expectation values does not determine the fluctuation Δn; moreover, Δθ is not a well-defined observable in the BEC regime where the claimed phase fluctuations are large. Therefore the statement that the decrease of ωcol 'originates from' increasing phase fluctuations is an interpretation rather than a demonstrated result. The authors should either compute a direct number-phase uncertainty measure within the NRG ground states or explicitly soften the causal claim.","section":"Expectation values and the phase fluctuation; Fig. 4(b)"}],"minor_comments":[{"comment":"The text reads 'we take the Plank constant ℏ as 1'; this should be 'Planck's constant' and 'we set ℏ=1'.","section":"Introduction"},{"comment":"The caption says 'Fig. 1(b) zooms in on panel (b)'; it should refer to Fig. 2(b) zooming into the low-frequency region of Fig. 1(b).","section":"Fig. 2 caption"},{"comment":"The statement that 'the findings are also valid for adiabatic phonons (not shown in this paper)' is unverifiable as written; it should either be supported by the data or removed.","section":"Conclusion"},{"comment":"The labels 'SM I', 'SM II', etc. in the main text do not match the SM section labels 'S1', 'S2', etc.; please make the cross-referencing consistent.","section":"Supplemental Material"},{"comment":"The text refers to the weak-coupling feature as a 'gap mode' and notes that it 'may be too tiny to observe experimentally', but later states that 'the gap mode becomes less prominent' with increasing coupling; the relationship between these statements is unclear and should be clarified.","section":"Phonon and density-density correlation spectra"}],"recommendation":"major_revision","confidential_remarks":"The manuscript reports a potentially interesting and largely non-circular numerical study of collective modes in the Holstein model across the BCS-BEC crossover. The main weakness is not the physics idea but the evidence base for part of the central claim: the BCS anchor is one broad shoulder, and the numerical truncation dependence is not characterized. I would be willing to look at a revision that adds more weak-coupling data points, convergence tests, and a more careful treatment of the mode weight and the phase-fluctuation estimate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version. This paper computes the phonon spectra of the superconducting Holstein model across the BCS-BEC crossover using DMFT-NRG, and finds a collective mode whose peak tracks the normal-state soft phonon in the crossover region and the superfluid stiffness in the BEC regime, with a weight proportional to the pairing gap. That is a genuinely new calculation for a model with explicit phonon dynamics — prior RPA work on the attractive Hubbard model could not capture the phonon softening and retardation. The overall picture is plausible and the comparison between phonon and density-density correlation spectra is consistent.\n\nThe main soft spot is the BCS anchor. The claim that ω_col sits near 2ΔP and increases with coupling rests on a single broad shoulder at Eb=0.4, which could be the pair-breaking continuum onset rather than a collective pole. The crossover- and BEC-regime branches have multiple points and a clean normal-state reference, but the BCS leg needs at least two or three more weak-coupling points and some analysis showing the feature is a pole. The stress-test note is right to flag this as load-bearing.\n\nSecond, there are no NRG convergence checks. Λ=1.6, NS=1000, Nph=10 are stated, but nothing shows the peak positions and weights are stable against these choices. A short convergence test would resolve this and would be expected in a letter.\n\nThe D_s/5 factor in the BEC comparison is unexplained. The trend is likely correct, but a factor of 5 without derivation reads as a fit. The adiabatic-phonon claim in the conclusion is explicitly \"not shown,\" so it should be treated as an aside, not evidence. The phase-fluctuation estimate via Δn = ω0Δx/(2g) is a rough mean-field argument, fine as supporting context but not the main evidence.\n\nWhat the paper does well: it is a non-trivial numerical study of a model with explicit bosons, the two independent spectra agree, and the W ∝ ΔP observation is a nice handle on U(1) breaking. The self-citation for ΔP and D_s is legitimate since those come from their earlier DMFT-NRG study of the same model.\n\nI would send this to a serious referee. The result is novel for the Holstein model and the weaknesses are fixable, but the evolutionary picture needs more BCS-regime data and a convergence check before it is fully convincing. The CONDITIONAL verdict is fair.","headline":"A plausible first DMFT-NRG study of the collective mode across the BCS-BEC crossover in the Holstein model, but the BCS-regime branch and NRG convergence need strengthening.","tokens_in":15961,"tokens_out":3037,"would_cite":true,"duration_ms":30876,"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 Holstein superconductor, the collective phase mode tracks 2Δ_P, then the soft phonon, then superfluid stiffness across the BCS-BEC crossover.","keywords":["BCS-BEC crossover","Holstein model","collective modes","phonon spectra","superfluid stiffness","phase fluctuations","dynamical mean-field theory","numerical renormalization group"],"falsifier":"Recompute the superconducting phonon spectrum at $E_b = 2.2$ and $2.6$ with progressively larger phonon cutoffs ($N_{ph} = 12, 15, 20$) and finer NRG discretizations ($\\Lambda = 1.2, 1.4$); if the extracted $\\omega_{col}$ and $W$ shift by more than the quoted trends, the claimed BCS-to-BEC evolution is a numerical artifact rather than a physical result.","tokens_in":14834,"feed_emoji":"⚛️","tokens_out":8762,"duration_ms":77339,"temperature":0.7,"pith_summary":"This paper establishes a three-stage fate for the collective phase mode of a superconductor as the electron-phonon coupling is tuned through the BCS-BEC crossover in the Holstein model. In the weak-coupling BCS regime the mode sits near twice the pairing gap, $2\\Delta_P$, and rises with coupling; in the intermediate crossover regime it tracks the normal state's soft phonon mode; and in the strong-coupling BEC regime it falls toward zero with the superfluid stiffness, reflecting growing phase fluctuations of local pairs. The paper further argues that the mode's integrated spectral weight is proportional to the pairing gap across the whole crossover, identifying the mode as the U(1) phase (Goldstone) mode. The reason this matters is that it predicts a low-energy, observable phase mode in strong-coupling superconductors, well below the plasma frequency, and connects the mode frequency to superfluid stiffness rather than to the pairing gap alone.","feed_headline":"Collective mode tracks superfluid stiffness at strong coupling","feed_subtitle":"As local pairs form, the phase mode falls to low energy, within reach of scattering and Raman probes.","key_machinery":"The load-bearing object is the phonon spectral function $\\rho_{\\mathrm{ph}}(\\omega)=-\\mathrm{Im}\\,D(\\omega)/\\pi$, obtained from the Dyson equation $D^{-1}(\\omega)=D_0^{-1}(\\omega)-\\Pi(\\omega)$ with self-energy $\\Pi(\\omega)=g\\langle\\langle n_f,x\\rangle\\rangle_\\omega/\\langle\\langle x,x\\rangle\\rangle_\\omega$; the collective mode is extracted as the lowest-frequency zero of $d\\rho_{\\mathrm{ph}}(\\omega)/d\\omega$ in the superconducting state. The argument then proceeds by comparing $\\omega_{col}$ to three independently computed energy scales: twice the pairing gap $2\\Delta_P$, the normal-state soft-phonon frequency $\\omega_s$ set by lattice instability, and the superfluid stiffness $D_s$. The mode's weight $W$ is integrated up to $2\\omega_{col}$ and compared with $\\Delta_P$. This comparison structure, supported by dynamical mean-field spectra of the phonon and density-density correlation functions, is what turns the raw peaks into the claimed BCS-to-BEC evolution.","core_discovery":"On the author's own terms, the central claim is that in the superconducting state of the Holstein model the phonon spectrum develops a sharp low-frequency collective peak whose position $\\omega_{col}$ encodes which side of the BCS-BEC crossover the system is on. For $E_b < 0.6$ the peak sits near $2\\Delta_P$ and increases with coupling; for intermediate coupling, once the normal-state soft phonon $\\omega_s$ drops below $2\\Delta_P$, the peak follows $\\omega_s$ and decreases; and for $E_b > E_{c2}$, where the normal state is a bipolaron insulator, the peak scales as roughly $D_s/5$ and vanishes in the infinite-coupling limit as a massless Goldstone mode. Throughout, the mode's weight $W = \\int_0^{2\\omega_{col}} \\rho_{\\mathrm{ph}}^{sc}(\\omega)\\,d\\omega$ is proportional to $\\Delta_P$, which the authors read as evidence that the mode originates from U(1) gauge-symmetry breaking at every coupling strength.","pith_inferences":["An editorial extension: if $\\omega_{col} \\propto D_s$ in the BEC regime, the ratio $\\omega_{col}/D_s$ (reported as roughly $1/5$) should be universal across different phonon frequencies and lattice geometries within the Holstein model; recomputing this ratio for adiabatic phonons such as $\\omega_0 = 0.05$ would test whether the constant survives the antiadiabatic limit.","The weight proportionality $W \\propto \\Delta_P$ implies a sum-rule-like statement that the collective mode exhausts a fixed fraction of the pairing-gap spectral weight; extracting that fraction from the present data and comparing it with attractive-Hubbard random-phase-approximation results would connect this letter to earlier collective-mode studies.","The normal-state first-order metal-insulator transition with hysteresis between $E_{c1}$ and $E_{c2}$ suggests that, if a superconducting dome sits over this coexistence region, sharp sample-dependent softening of $\\omega_{col}$ near $E_{c1}$ should be observable; the paper does not discuss sample-to-sample variation."],"forward_implications":["A strong-coupling superconductor should show a collective mode well below both $2\\Delta_P$ and the plasma frequency, so the phase mode can be probed by inelastic X-ray scattering, Raman spectroscopy, or electron energy loss spectroscopy rather than only by terahertz nonlinear optics.","The mode frequency in the BEC regime is tied to the superfluid stiffness $D_s$, so measurements of the low-energy mode across a pressure- or doping-tuned crossover should track the inverse penetration depth squared, not the pairing gap.","Because $W \\propto \\Delta_P$ throughout the crossover, the integrated weight of the phonon collective peak offers a spectroscopic measure of the pairing gap that is independent of the normal-state soft-phonon background.","The mode's onset in the crossover regime coincides with the normal-state lattice instability around $E_{c1}$, so the soft-phonon energy scale inherited from the normal state controls the collective mode whenever $\\omega_s < 2\\Delta_P$.","The same physics is claimed to persist for adiabatic phonons, where the mode frequency varies non-monotonically across the crossover while $W \\propto \\Delta_P$ remains universal, suggesting applicability beyond the antiadiabatic phonon regime."],"supporting_citations":[{"why":"Supplies the pairing gap $\\Delta_P$, superfluid stiffness $D_s$, and critical temperature data that the collective-mode behavior is compared against.","marker":"[40]"},{"why":"Provides the Lanczos-basis numerical renormalization-group transformation used in the impurity solver of the dynamical mean-field self-consistency.","marker":"[37]"},{"why":"Supplies the dynamical mean-field theory framework for solving the infinite-dimensional lattice model.","marker":"[43]"},{"why":"Supplies the numerical renormalization-group technique used to compute the low-frequency phonon and density correlation spectra.","marker":"[44]"},{"why":"Derives the effective retarded electron-electron interaction $U_{\\mathrm{eff}}(\\omega)$ from the Holstein electron-phonon coupling, framing the comparison with the attractive Hubbard model.","marker":"[41]"},{"why":"Establishes the gap formation and soft phonon mode in the Holstein model that the crossover regime's $\\omega_s$ is compared with.","marker":"[46]"},{"why":"Supplies the polaron-crossover and bipolaronic metal-insulator transition physics that defines the normal-state instability region between $E_{c1}$ and $E_{c2}$.","marker":"[47]"},{"why":"Shows that the amplitude mode does not couple to charge in linear response, which motivates reading the low-energy phonon peak as the phase mode.","marker":"[24, 25]"}],"fun_headline_variants":["Phonon collective mode maps BCS-BEC crossover in Holstein superconductor","Collective mode weight tied to pairing gap across all couplings","From 2Δ to soft phonon: collective mode tracks crossover","Superfluid stiffness reduction mirrors collective mode softening in BEC"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole picture rests on the assumption that the calculated spectra resolve the low-frequency collective peak accurately with the numerical settings used (a phonon-state cutoff of ten and an NRG discretization of 1.6), and the paper does not show how the peak moves when those settings are tightened.","fun_headline_variants_meta":{"raw":{"variants":["Phonon collective mode maps BCS-BEC crossover in Holstein superconductor","Collective mode weight tied to pairing gap across all couplings","From 2Δ to soft phonon: collective mode tracks crossover","Superfluid stiffness reduction mirrors collective mode softening in BEC"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001394,"raw_usage":{"total_tokens":5658,"prompt_tokens":982,"completion_tokens":4676,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":598,"completion_tokens_details":{"reasoning_tokens":4610}},"tokens_in":598,"tokens_out":4676,"duration_ms":73028,"temperature":1.0,"reasoning_tokens":4610,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:54:04.499511+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the superconducting phonon spectrum at $E_b = 2.2$ and $2.6$ with progressively larger phonon cutoffs ($N_{ph} = 12, 15, 20$) and finer NRG discretizations ($\\Lambda = 1.2, 1.4$); if the extracted $\\omega_{col}$ and $W$ shift by more than the quoted trends, the claimed BCS-to-BEC evolution is a numerical artifact rather than a physical result.","supporting_citations":[{"cited_title":"Potential-energy-driven (bcs) to kinetic-energy-driven (bec) pairing in the two-dimensional attractive hubbard model: Cellular dynamical mean-field theory,","cited_arxiv_id":null,"evidence_quote":"Supplies the pairing gap $\\Delta_P$, superfluid stiffness $D_s$, and critical temperature data that the collective-mode behavior is compared against."},{"cited_title":"Low-temperature properties of the infinite-dimensional attractive hubbard model,","cited_arxiv_id":null,"evidence_quote":"Provides the Lanczos-basis numerical renormalization-group transformation used in the impurity solver of the dynamical mean-field self-consistency."},{"cited_title":"I: Derivation of effective electron-electron interaction from electron-phonon cou- pling","cited_arxiv_id":null,"evidence_quote":"Supplies the dynamical mean-field theory framework for solving the infinite-dimensional lattice model."},{"cited_title":"Dynamical mean-field theory of strongly correlated fermion systems and the limit of infi- nite dimensions,","cited_arxiv_id":null,"evidence_quote":"Supplies the numerical renormalization-group technique used to compute the low-frequency phonon and density correlation spectra."},{"cited_title":"Dynamical effects on superconductivity in a bcs-bec crossover: Dynamical mean field theory and numerical renormalization group study of the holstein model in infinite dimensions,","cited_arxiv_id":null,"evidence_quote":"Derives the effective retarded electron-electron interaction $U_{\\mathrm{eff}}(\\omega)$ from the Holstein electron-phonon coupling, framing the comparison with the attractive Hubbard model."},{"cited_title":"Renormalization-group approach to the anderson model of dilute magnetic alloys. i. static properties for the sym- metric case,","cited_arxiv_id":null,"evidence_quote":"Establishes the gap formation and soft phonon mode in the Holstein model that the crossover regime's $\\omega_s$ is compared with."},{"cited_title":"Gap formation and soft phonon mode in the holstein model,","cited_arxiv_id":null,"evidence_quote":"Supplies the polaron-crossover and bipolaronic metal-insulator transition physics that defines the normal-state instability region between $E_{c1}$ and $E_{c2}$."}],"review_version":1}