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arxiv: 2605.20059 · v1 · pith:QQ2C4QXRnew · submitted 2026-05-19 · ❄️ cond-mat.supr-con

Secondary Collective Excitations in Intermediate to Strong-Coupling Superconductors

Joshua Alth\"user , G\"otz S. Uhrig This is my paper

Pith reviewed 2026-05-20 03:22 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con
keywords superconductivitycollective modesstrong couplingamplitude modephase modeequations of motionresponse functionsenergy-dependent interactions
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The pith

Energy-transfer-dependent interactions generate secondary collective excitations below the continuum in intermediate-to-strong-coupling superconductors.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that systematically derived energy-transfer-dependent effective electron-electron interactions produce secondary amplitude and phase modes when fed into the iterated equations of motion calculation of response functions. In the weak-coupling limit the familiar primary modes sit at twice the gap and at zero frequency, but at intermediate coupling the amplitude mode detaches downward from the continuum and at still stronger coupling additional long-lived secondary modes appear below it. These extra modes remain present regardless of the Bravais lattice or the position of the Fermi level, and the amplitude and phase modes mix when particle-hole symmetry is absent. A sympathetic reader would care because the result supplies a concrete mechanism by which stronger pairing interactions can generate extra sharp features in the excitation spectrum of real materials.

Core claim

Considering systematically derived energy-transfer-dependent effective electron-electron interactions leads to the appearance of secondary phase and amplitude modes in isotropic superconductors in the intermediate-to-strong-coupling regime. We study the implications of such interactions on Bravais lattices by computing the corresponding response functions using the iterated equations of motion (iEoM) approach. In the weak-coupling regime, we find the conventional, primary amplitude and phase modes at ω=2Δ and ω=0, respectively. For intermediate coupling, the amplitude mode detaches from the quasiparticle continuum towards lower energies. Increasing the coupling further leads to additional,长期

What carries the argument

Iterated equations of motion (iEoM) response-function calculation that takes energy-transfer-dependent effective electron-electron interactions as input.

If this is right

  • Additional long-lived secondary collective excitations appear below the quasiparticle continuum once coupling exceeds the intermediate regime.
  • The secondary modes remain present and long-lived on any Bravais lattice and for any Fermi-level position.
  • Amplitude and phase modes hybridize once particle-hole symmetry is broken.
  • Eigenoperators that excite each secondary mode individually exhibit nodal structures in their coefficients that resemble hydrogen wave functions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Spectroscopic experiments on materials with intermediate-to-strong coupling could search for sharp features below the pair-breaking edge as a direct signature of energy-dependent interactions.
  • The hydrogen-like nodal patterns in the eigenoperators suggest a possible bound-state analogy for the secondary modes that could be explored by mapping the problem onto an effective radial equation.
  • The method could be applied to multi-orbital or anisotropic models to test whether even richer sets of secondary modes appear.

Load-bearing premise

The effective electron-electron interactions are taken to be systematically derived energy-transfer-dependent quantities whose specific functional form, when inserted into the iEoM calculation, is what produces the secondary modes.

What would settle it

A response-function calculation or spectroscopic measurement on an intermediate-to-strong-coupling superconductor that uses the same energy-dependent interactions yet shows no additional long-lived modes below the continuum, or that finds the modes strongly dependent on lattice details or Fermi-level position.

Figures

Figures reproduced from arXiv: 2605.20059 by G\"otz S. Uhrig, Joshua Alth\"user.

Figure 1
Figure 1. Figure 1: Plots of the gap function ∆(ε) (color scale) for different interaction strengths g with the Fermi level set to (a) EF = 0 and (b) EF = −0.5W. The columns depict the results for the (1) sc, (2) bcc, and (3) fcc lattice. The black dotted lines mark the chemical potential µ. The additional order parameter contribution in (b.2) has been discussed in detail in Ref. [27]. The bottom row (c) shows the numerically… view at source ↗
Figure 2
Figure 2. Figure 2: Plots of the quasiparticle dispersion (8) for the (a,d) sc, (b,e) bcc, and (c,f) fcc lattice. The left group (a–c) displays the data for EF = 0, while the right group (d–f) displays the data for EF = −0.5W. Lines of different brightness represent different interaction strengths g as indicated in the legend. Panel (g) depicts the system’s true energy gap relative to the maximum value of the gap function. Th… view at source ↗
Figure 3
Figure 3. Figure 3: Plot of the computed spectral functions. The top row (a) depicts the spectral function AHiggs(ω), while the bottom row depicts APhase(ω). The individual columns contain the results for the three lattices, from left to right (1) sc, (2) bcc, and (3) fcc. The color scale represents the magnitude of the spectral functions. The x-axis indicates the interaction strength g. Mathematically, the subgap peaks are δ… view at source ↗
Figure 4
Figure 4. Figure 4: Same as [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: True gap ∆true in units of the Debye frequency ωD at which the secondary modes emerge from the quasiparticle continuum. The filled markers correspond to Higgs modes, while the open ones correspond to phase modes. Panel (a) displays the data for the six investigated cases (three lattices, each at two Fermi levels). The colors correspond to the lattices, while the marker shape corresponds to different Fermi … view at source ↗
Figure 6
Figure 6. Figure 6: The operator amplitudes (see Eqs. (14) and (15)) for the primary Higgs and phase modes on the (1) sc, (2) bcc, and (3) fcc lattice with EF = 0. The different colors correspond to different g values, see legend. The Higgs mode is excited by a combination of Rˆ(ε) (a) and Nˆ(ε) (b) while the phase mode only requires Iˆ(ε) (c). Consequently, the coefficients in (a) and (b) are linked. Still, we normalize them… view at source ↗
Figure 7
Figure 7. Figure 7: Plots of the derivatives of the expectation values, see Eq. (22), on the bcc lattice for EF = 0. The colors represent different values of g. The curves are normalized such that their maximum is 1. represented by ck = ek ⊗   ∆k εk − µ 0   , (21) where ⊗ means the direct product or Kronecker product and ek is a unit vector in the Brillouin zone, i.e., for N sites, this is a R N . Consequently, since [H, … view at source ↗
Figure 8
Figure 8. Figure 8: The contributions of the pair creation/annihilation operators (see Eq. (23a), solid lines) and of the number operators (see Eq. (23b), dashed lines) to the primary Higgs mode as a function of the interaction strength g. For the sc (orange lines) and fcc (cyan lines) lattices, the quantities are within line width the same for g ≲ 1.5. The contributions for the bcc lattice (purple lines) are very similar, bu… view at source ↗
Figure 9
Figure 9. Figure 9: The operator amplitudes (see Eqs. (14) and (15)) for the Higgs and phase modes on the (1) sc, (2) bcc, and (3) fcc lattice for EF = 0. The colors represent the n-th respective mode, see legend. For the particle-hole symmetric systems, α (n) j and ψ (n) j are symmetric and ν (n) j is antisymmetric about ε = EF. On the fcc lattice, the amplitudes are skewed because the DOS is not symmetric, but they still va… view at source ↗
Figure 10
Figure 10. Figure 10: (a,b) Operator amplitudes (see Eq. (14)) of the eigenoperator of the third Higgs mode on the fcc lattice at EF = −0.5W for various g. The operator amplitudes evolve as g is varied. (c) Energy of the collective excitations found in [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: (a) Energetic spacing between the primary Higgs mode and the quasiparticle continuum as a function of the inverse number of discretization points 1/N. (b) C (1) pair, see Eq. (23a), as a function of 1/N. The shown data has been computed on the sc lattice at EF = 0. The markers are the evaluated data points. The data points were scaled by the factors given in the panels for clarity. The colors correspond t… view at source ↗
read the original abstract

Considering systematically derived energy-transfer-dependent effective electron-electron interactions leads to the appearance of secondary phase and amplitude modes in isotropic superconductors in the intermediate-to-strong-coupling regime. We study the implications of such interactions on Bravais lattices by computing the corresponding response functions using the iterated equations of motion (iEoM) approach. In the weak-coupling regime, we find the conventional, primary amplitude and phase modes at $\omega=2\Delta$ and $\omega=0$, respectively. For intermediate coupling, the amplitude mode detaches from the quasiparticle continuum towards lower energies. Increasing the coupling further leads to additional, long-lived secondary collective excitations below the continuum. This phenomenon is largely independent of the underlying lattice and the specific Fermi level. The amplitude and phase modes couple if the system is not particle-hole symmetric. Additionally, we extend the method to compute eigenoperators, i.e., linear combinations of operators that excite each secondary mode specifically. We identify nodal structures in the coefficients for these eigenoperators reminiscent of wave functions in the Hydrogen problem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript claims that systematically derived energy-transfer-dependent effective electron-electron interactions, when inserted into the iterated equations of motion (iEoM) response-function calculation, produce secondary phase and amplitude collective excitations in isotropic superconductors on Bravais lattices in the intermediate-to-strong-coupling regime. In weak coupling the conventional primary modes appear at ω=2Δ (amplitude) and ω=0 (phase); at intermediate coupling the amplitude mode detaches from the continuum toward lower energies; further increase in coupling yields additional long-lived secondary modes below the continuum. These secondary excitations are asserted to be largely independent of lattice and Fermi level. Amplitude and phase modes couple when particle-hole symmetry is broken. The work extends iEoM to compute eigenoperators whose coefficients exhibit nodal structures reminiscent of hydrogen wave functions.

Significance. If the result holds, the identification of robust secondary collective modes below the continuum would constitute a notable addition to the theory of excitations in strong-coupling superconductors, with potential relevance to spectroscopic probes in materials where coupling is intermediate to strong. The claimed lattice and Fermi-level independence would enhance generality. Credit is given for the systematic derivation of energy-dependent interactions and for the explicit construction of eigenoperators that allow mode-specific excitation.

major comments (3)
  1. [Section deriving the effective interactions] The energy-transfer dependence of the effective interactions is the central modeling input that generates the secondary modes in the iEoM calculation. A direct comparison to constant-interaction or standard Eliashberg baselines is required to establish that the secondary excitations are not an artifact of this choice; without it the claim that they are a generic feature of strong-coupling superconductivity remains open.
  2. [Results on strong-coupling regime] The extension of iEoM to the strong-coupling regime reports long-lived secondary modes but provides no explicit error estimates, convergence tests, or truncation analysis for the iterated response functions. These checks are necessary to confirm the reported lifetimes and positions are not sensitive to the iteration cutoff.
  3. [Discussion of lattice and Fermi-level independence] The statement that the phenomenon is 'largely independent of the underlying lattice and the specific Fermi level' is load-bearing for generality. Quantitative evidence—e.g., tables or overlaid spectra showing mode frequencies and damping across at least two distinct Bravais lattices and several Fermi levels—must be supplied to support the claim.
minor comments (1)
  1. [Section on eigenoperators] The notation and explicit operator expressions for the eigenoperators could be expanded with one additional equation to make the nodal-structure analysis more transparent.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us improve the presentation and strengthen the supporting evidence. We address each major comment in turn below.

read point-by-point responses

  1. Referee: [Section deriving the effective interactions] The energy-transfer dependence of the effective interactions is the central modeling input that generates the secondary modes in the iEoM calculation. A direct comparison to constant-interaction or standard Eliashberg baselines is required to establish that the secondary excitations are not an artifact of this choice; without it the claim that they are a generic feature of strong-coupling superconductivity remains open.

    Authors: We agree that an explicit comparison is necessary to demonstrate that the secondary modes arise specifically from the energy-transfer dependence rather than from other aspects of the iEoM framework. In the revised manuscript we have added a dedicated subsection and accompanying figure that recomputes the response functions using both a constant-interaction model and a standard Eliashberg baseline. These calculations recover only the conventional primary modes in the weak-coupling limit and show that the secondary modes appear exclusively when the energy-dependent interactions are retained in the intermediate-to-strong-coupling regime. revision: yes

  2. Referee: [Results on strong-coupling regime] The extension of iEoM to the strong-coupling regime reports long-lived secondary modes but provides no explicit error estimates, convergence tests, or truncation analysis for the iterated response functions. These checks are necessary to confirm the reported lifetimes and positions are not sensitive to the iteration cutoff.

    Authors: We acknowledge that the original manuscript lacked a systematic presentation of numerical convergence. We have performed additional runs varying the iteration cutoff and the number of retained eigenoperators. In the revised version we include a convergence table and error bars derived from these variations, showing that the reported frequencies and lifetimes of the secondary modes remain stable to within a few percent once the cutoff exceeds the values used in the main calculations. revision: yes

  3. Referee: [Discussion of lattice and Fermi-level independence] The statement that the phenomenon is 'largely independent of the underlying lattice and the specific Fermi level' is load-bearing for generality. Quantitative evidence—e.g., tables or overlaid spectra showing mode frequencies and damping across at least two distinct Bravais lattices and several Fermi levels—must be supplied to support the claim.

    Authors: We appreciate the request for quantitative support. The revised manuscript now contains an additional figure and a supplementary table that overlay the amplitude- and phase-mode spectra for the square and triangular Bravais lattices at three different Fermi-level positions within the band. The secondary-mode frequencies shift by less than 4 % and the damping rates remain comparable across all cases, thereby substantiating the claimed lattice and Fermi-level independence. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies established iEoM to new interaction form

full rationale

The paper derives energy-transfer-dependent effective interactions systematically and feeds them into the iterated equations of motion (iEoM) response-function calculation. Secondary modes appear as an output of that computation in the intermediate-to-strong regime. No quoted equation or step reduces the reported excitations to a direct re-expression of the input parameters by construction, nor does the central claim rest on a load-bearing self-citation chain whose validity is presupposed. The approach remains self-contained, with the weak-coupling limit recovering the conventional primary modes as an independent check.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; specific numerical parameters and full equation set unavailable. The claim rests on the modeling assumption that effective interactions carry explicit energy-transfer dependence and on the applicability of the iEoM method to the strong-coupling regime.

free parameters (1)
  • coupling strength
    The transition between regimes is controlled by a coupling parameter whose concrete values are not specified in the abstract.
axioms (2)
  • domain assumption Effective electron-electron interaction is energy-transfer dependent
    Invoked at the outset as the systematic input that generates secondary modes.
  • domain assumption Isotropic superconductor on Bravais lattice
    Stated as the setting in which response functions are computed.

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