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REVIEW 4 major objections 5 minor 55 references

Collective mode across the BCS-BEC crossover in Holstein model

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read In a Holstein superconductor, the collective phase mode tracks 2Δ_P, then the soft phonon, then superfluid stiffness across the BCS-BEC crossover.

desk verdict 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. read the letter →

arxiv 2411.14782 v1 pith:YMRNBXTA submitted 2024-11-22 cond-mat.supr-con cond-mat.str-el

Tae-Ho Park , Han-Yong Choi This is my paper · ORCID
classification cond-mat.supr-concond-mat.str-el
keywords BCS-BECcrossoverHolsteinmodelcollectivemodesphononspectrasuperfluidstiffnessphasefluctuationsdynamicalmean-fieldtheorynumericalrenormalizationgroup
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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

4 major / 5 minor

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.

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 (4)
  1. [Phonon and density-density correlation spectra; Fig. 3(a)] 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.
  2. [Model and methods] 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.
  3. [Phonon and density-density correlation spectra; Fig. 3(b)] 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.
  4. [Expectation values and the phase fluctuation; Fig. 4(b)] 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.
minor comments (5)
  1. [Introduction] The text reads 'we take the Plank constant ℏ as 1'; this should be 'Planck's constant' and 'we set ℏ=1'.
  2. [Fig. 2 caption] 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).
  3. [Conclusion] 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.
  4. [Supplemental Material] 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.
  5. [Phonon and density-density correlation spectra] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the collective-mode frequency, gap, stiffness, and soft-phonon frequency are independently computed quantities.

full rationale

The paper's central comparison is between the peak position ω_col extracted from the DMFT-NRG superconducting phonon spectrum and three independently obtained quantities: 2Δ_P and D_s (taken from the authors' prior Ref. 40) and ω_s (extracted from the normal-state phonon spectrum). None of these quantities is defined in terms of ω_col, and ω_col is not fitted to reproduce them. The extraction rule dρ/dω = 0 at the lowest peak is applied directly to the spectra, while Δ_P is obtained from the gap condition Δ_P = ReΔ(ω = Δ_P) and D_s from the Kubo formula; neither involves the phonon spectral peak. The collective-mode weight W = ∫_0^{2ω_col} ρ_sc_ph(ω) is a spectral integral whose claimed proportionality to Δ_P is a numerical observation, not an identity imposed by the integration limit. The use of Ref. 40 for Δ_P and D_s is a self-citation, but it is independent computational evidence from a prior NRG-DMFT study with stated formulas (SM III) and does not assume the target ω_col evolution. The BCS-regime anchor rests on a single broad shoulder at E_b = 0.4, and no NRG convergence checks are provided; these are correctness and robustness concerns, not circularity, because the quantities compared are not mutually constructed. No load-bearing step reduces by construction to its own input.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The ledger shows that the paper's load-bearing extras are numerical truncation parameters, a scaling factor for display, and a rough uncertainty-principle estimate for phase fluctuations. No new physical entities are introduced. The model itself (Holstein) and the method (DMFT-NRG) are established, and the main physics claim is an interpretation of independently computed spectral features.

free parameters (5)
  • NRG discretization parameter Λ = 1.6
    Chosen by hand for the numerical renormalization group calculation; the paper gives no convergence study showing that the collective mode peak position and weight are stable as Λ varies.
  • Number of kept NRG states NS = 1000
    Chosen by hand; no convergence checks are shown, so the low-frequency spectra could depend on this truncation.
  • Phonon Hilbert space truncation Nph = 10
    The phonon states are truncated to Nph=10; with strong coupling and soft phonons, this could affect the collective mode position and weight, but no convergence test is reported.
  • Ds scaling factor = 5
    The paper plots D_s/5 to show ω_col scaling with superfluid stiffness in the BEC regime (Fig. 3a). The factor 5 is not derived; it is a non-derived scale factor introduced to make the curves overlap.
  • Weight integration cutoff = 2ω_col
    The collective mode weight is defined as W = ∫_0^{2ω_col} ρ_ph(ω)dω; the upper cutoff is chosen ad hoc and depends on the peak position, so W is not a model-independent measure.
assumptions (5)
  • domain assumption The DMFT solution in the superconducting state is unique and independent of initial configuration.
    Stated in the main text before Fig. 3: 'in the superconducting state, a unique solution is obtained from the DMFT calculation regardless of the initial condition.' This justifies comparing results across Eb without hysteresis effects.
  • domain assumption The effective electron-electron interaction U_eff(ω) = 2g²ω0/(ω²-ω0²) after integrating out phonons, and Eb = 2g²/ω0 as the coupling parameter.
    Derived in SM I and used throughout to characterize the coupling strength; this maps the Holstein model to a retarded attractive interaction.
  • domain assumption The low-frequency peak in the superconducting phonon spectrum is identified as the U(1) phase mode.
    The paper argues that the peak is the collective mode from symmetry breaking, with the evidence being the proportionality of its weight to the pairing gap (Fig. 3b). This identification is not proven from first principles and could in principle mix with phonon softening.
  • ad hoc to paper The phase fluctuation Δθ is estimated through the uncertainty relation Δn·Δθ ~ 1, with Δn derived from the mean-field relation ⟨n-1⟩ = (ω0/g)⟨x⟩/2.
    Used in the 'Expectation values and the phase fluctuation' section. This is a rough estimate: the uncertainty relation is used as an equality, and the relation between density and displacement is confirmed only numerically in SM IV.
  • standard math Kramers-Kronig relations are valid for obtaining the real parts of the correlators.
    Used in SM S2 to construct the phonon self-energy and spectral functions from the imaginary parts obtained in NRG.

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Cite this review

Pith. "Pith review of Collective mode across the BCS-BEC crossover in Holstein model." pith.science (2026). https://pith.science/paper/YMRNBXTA

@misc{pith2026241114782,
  author       = {Pith},
  title        = {Pith review of: Collective mode across the BCS-BEC crossover in Holstein model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YMRNBXTA}},
  note         = {Machine review of arXiv:2411.14782}
}
abstract

We investigate the emergence of the collective mode in the phonon spectra of the superconducting state within the Holstein model by varying the electron-phonon coupling. Using dynamical mean field theory (DMFT) combined with the numerical renormalization group (NRG) technique, we calculate the phonon spectra. In the superconducting state with a pairing gap ($\Delta_P$), the peak position of the collective mode ($\omega_{col}$) evolves from the Bardeen-Cooper-Schrieffer (BCS) regime, manifesting near $2\Delta_P$ and increasing with coupling, to the Bose-Einstein condensation (BEC) regime, where $\omega_{col}$ decreases with increasing coupling. The decrease of $\omega_{col}$ matches well with the reduction of superfluid stiffness, which originates from the increasing phase fluctuations of local pairs with coupling strength. In the crossover regime with intermediate coupling, $\omega_{col}$ aligns with the soft phonon mode ($\omega_s$) of the normal state and decreases with increasing coupling when $\omega_s < 2\Delta_P$. Additionally, comparing the collective mode weight to $\Delta_P$ suggests that the collective mode predominantly stems from U(1) gauge symmetry breaking across all coupling strengths.

Figures

Figures reproduced from arXiv: 2411.14782 by the authors.

Figure 1
Figure 1. FIG. 1. Comparison of phonon spectra of the normal state [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Comparison of the density-density correlation spec [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. (a) Square of the lattice displacement fluctuation [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

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