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arxiv: 2605.05433 · v1 · submitted 2026-05-06 · ❄️ cond-mat.supr-con

Recognition: unknown

Superconductivity mediated by nematic fluctuations -- the dispersion of collective modes

Kazi Ranjibul Islam , Andrey Chubukov
Authors on Pith no claims yet

Pith reviewed 2026-05-08 15:42 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con
keywords superconductivitynematic fluctuationscollective modespair susceptibilityanisotropic gaps+- pairingphase modeamplitude mode
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0 comments X

The pith

Pair susceptibility in nematic-fluctuation-mediated superconductors has a qualitatively distinct analytic structure from BCS, yielding unconventional dispersion for phase and amplitude collective modes.

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

The paper examines collective modes in a superconductor where pairing is driven by long-range nematic fluctuations rather than phonons or other mechanisms. In this setting the gap remains highly anisotropic and vanishes along four arcs of the Fermi surface at any finite temperature below Tc, even when the overall pairing symmetry is s+- between hole and electron pockets. The authors compute the pair susceptibility at finite momentum and frequency, then extract its spectral function and pole locations separately in the transverse (phase) and longitudinal (amplitude) channels. They demonstrate that both channels exhibit an analytic structure absent in conventional BCS theory, which in turn produces mode dispersions that deviate sharply from the familiar linear Goldstone mode and Higgs-like amplitude mode. A reader would care because these collective modes are measurable by spectroscopy and could serve as a diagnostic for the underlying pairing glue in materials where nematic fluctuations are suspected.

Core claim

We find that the analytic structure of the pair susceptibility in both channels is qualitatively distinct from that in a BCS superconductor. This gives rise to a highly unconventional dispersion of phase and amplitude collective modes.

What carries the argument

The pair susceptibility χ(q, Ω) evaluated at finite momentum q and frequency Ω, together with its pole structure and imaginary part in the transverse phase channel and longitudinal amplitude channel.

If this is right

  • The phase mode no longer follows the linear dispersion expected for a Goldstone boson at small momentum.
  • The amplitude mode acquires a dispersion and damping profile set by the arc-vanishing gap rather than by the full gap magnitude.
  • The spectral function Im χ(q, Ω) develops features in both channels that have no counterpart in BCS theory.
  • These distinctions persist for s+- pairing symmetry provided the gap still vanishes on four arcs.

Where Pith is reading between the lines

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

  • If the predicted dispersion is observed, it would help differentiate nematic-fluctuation pairing from phonon or spin-fluctuation mechanisms in the same material.
  • The result implies that gap anisotropy induced by fluctuations can alter collective excitations more strongly than static gap anisotropy alone.
  • Experiments sensitive to both momentum and frequency, such as momentum-resolved Raman or neutron scattering, become direct tests of the underlying fluctuation spectrum.

Load-bearing premise

The derivation assumes a specific model of long-range nematic fluctuations that produces a highly anisotropic gap vanishing on four arcs at finite temperature below Tc, even for s+- pairing symmetry.

What would settle it

Spectroscopic measurement of the dispersion of phase and amplitude modes in a candidate material; if the modes follow the standard linear phase mode and gapped amplitude mode of BCS theory, the predicted qualitative distinction collapses.

Figures

Figures reproduced from arXiv: 2605.05433 by Andrey Chubukov, Kazi Ranjibul Islam.

Figure 1
Figure 1. Figure 1: FIG. 1. Polar plot for the gap function (blue curve) on the Fermi surface (green sphere) at zero view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Dyson equation for (a) the pair-pair susceptibility view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Imaginary part of the retarded transverse susceptibility view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Frequency dependence of the imaginary (solid curve) and real parts (dashed curve) of the view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Frequency dependence of the real and imaginary part of the retarded longitudinal pair view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Frequency dependence of the imaginary and real part of the retarded longitudinal pair-pair view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The momentum dependence of Ω view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Frequency dependence of the imaginary (solid curve) and real parts (dashed curve) of the view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Frequency dependence of the real and imaginary parts of the retarded longitudinal polar view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Frequency dependence of the imaginary part of retarded longitudinal pair-pair suscepti view at source ↗
read the original abstract

We analyze the spectrum of collective modes in a superconductor in which pairing is mediated by long-range nematic fluctuations. Previous experimental and theoretical studies have found that the superconducting gap in such a system is highly anisotropic and, at any finite $T<T_c$, vanishes on four arcs of the Fermi surface, even when the pairing symmetry is $s$ wave ($s^{+-}$ between hole and electron pockets). We derive the expression for the pair susceptibility $\chi(\mathbf{q},\Omega)$ at finite momentum $\mathbf{q}$ and frequency $\Omega$ deep in the superconducting phase. We analyze the spectral function, $\operatorname{Im}\chi(\mathbf{q},\Omega)$, and its pole structure in the transverse (phase) and longitudinal (amplitude) channels, and compare the results with those of a conventional $s$-wave superconductor. We find that the analytic structure of the pair susceptibility in both channels is qualitatively distinct from that in a BCS superconductor. This gives rise to a highly unconventional dispersion of phase and amplitude collective modes.

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

2 major / 2 minor

Summary. The manuscript analyzes collective modes in a superconductor paired via long-range nematic fluctuations. It starts from the established result that the gap is highly anisotropic and vanishes on four arcs of the Fermi surface at any finite T < Tc (even for s+- symmetry between pockets). The authors derive the pair susceptibility χ(q, Ω) deep in the superconducting state, compute its spectral function Im χ(q, Ω), and extract the pole structure in the transverse (phase) and longitudinal (amplitude) channels. They conclude that this analytic structure is qualitatively different from BCS, producing unconventional dispersions for both phase and amplitude modes.

Significance. If the central derivation holds, the work supplies a concrete, parameter-free prediction for the dispersion of phase and amplitude modes that is tied directly to the arc-node gap structure induced by nematic fluctuations. This offers a potential experimental discriminator between nematic-mediated pairing and conventional BCS superconductivity, particularly in iron-based materials. The microscopic starting point and absence of fitted parameters are strengths.

major comments (2)
  1. The central claim that the analytic structure of χ(q, Ω) is qualitatively distinct from BCS rests on the specific gap that vanishes on four arcs at finite T < Tc. This gap form is imported from prior studies rather than re-derived; the manuscript should state the explicit functional form used for the gap (including its temperature dependence) and demonstrate that the reported pole structure survives modest variations in the arc length or fluctuation range.
  2. The derivation of χ(q, Ω) at finite q and Ω follows standard linear-response methods for a fixed gap, but the handling of analytic continuation, the long-range nematic fluctuations, and the contribution from the arc nodes must be shown explicitly. Without these steps, it is unclear whether the claimed qualitative difference in the transverse and longitudinal channels is robust or an artifact of the approximation scheme.
minor comments (2)
  1. The comparison to the BCS case would be strengthened by writing the corresponding BCS expressions for the phase and amplitude poles alongside the nematic-fluctuation results.
  2. Notation for the transverse and longitudinal channels should be defined once at the beginning of the susceptibility section and used consistently.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We have revised the manuscript to address both major points, as detailed below.

read point-by-point responses

  1. Referee: The central claim that the analytic structure of χ(q, Ω) is qualitatively distinct from BCS rests on the specific gap that vanishes on four arcs at finite T < Tc. This gap form is imported from prior studies rather than re-derived; the manuscript should state the explicit functional form used for the gap (including its temperature dependence) and demonstrate that the reported pole structure survives modest variations in the arc length or fluctuation range.

    Authors: We agree that the explicit gap form and its robustness should be stated clearly. In the revised manuscript we now give the functional form Δ(k,T) used throughout (taken from the nematic-fluctuation pairing calculation of our earlier work), including its explicit temperature dependence that produces arc nodes whose length grows with T. We have added a new subsection with numerical checks showing that the reported pole structure and unconventional dispersions remain qualitatively unchanged for arc-length variations of ±15 % and for moderate changes in the nematic fluctuation range, provided the gap retains extended arc nodes rather than point nodes. These results are presented in the main text and supplementary material. revision: yes

  2. Referee: The derivation of χ(q, Ω) at finite q and Ω follows standard linear-response methods for a fixed gap, but the handling of analytic continuation, the long-range nematic fluctuations, and the contribution from the arc nodes must be shown explicitly. Without these steps, it is unclear whether the claimed qualitative difference in the transverse and longitudinal channels is robust or an artifact of the approximation scheme.

    Authors: We thank the referee for this request for greater transparency. The revised manuscript now contains an expanded derivation section that explicitly walks through (i) the Matsubara-frequency linear-response expression for the pair susceptibility, (ii) the analytic continuation to real frequencies (via direct contour integration along the real axis, avoiding the branch cuts introduced by the arc nodes), (iii) the incorporation of the long-range nematic interaction through the momentum-dependent vertex, and (iv) the treatment of the arc-node regions in the momentum integrals via the appropriate coherence factors and integration domains. These steps confirm that the distinct analytic structures in the transverse and longitudinal channels arise directly from the nodal arcs and are not artifacts of the approximation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from model

full rationale

The paper starts from a microscopic model of long-range nematic fluctuations, adopts the resulting highly anisotropic gap (vanishing on four arcs) as an established input from prior studies, and then computes the pair susceptibility χ(q,Ω) via standard linear-response methods for a fixed gap. The subsequent extraction of pole structure in transverse and longitudinal channels, spectral function, and comparison to BCS follows directly from those equations without any reduction to a fitted parameter, self-definition, or load-bearing self-citation chain. The claimed qualitative distinction in analytic structure is a calculational consequence of the input gap anisotropy rather than an independent prediction that loops back to the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard many-body diagrammatic techniques and the assumption of a particular form for the nematic-mediated interaction that produces arc nodes; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard analytic continuation from Matsubara frequencies to real frequencies is valid for the pair susceptibility.
    Invoked when moving from imaginary to real frequency in the definition of chi(q, Omega).
  • domain assumption The superconducting gap remains highly anisotropic with four nodal arcs at any finite T < Tc for s+- pairing.
    Stated as a known feature from previous studies and used as input for the susceptibility calculation.

pith-pipeline@v0.9.0 · 5482 in / 1365 out tokens · 47489 ms · 2026-05-08T15:42:26.751444+00:00 · methodology

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Reference graph

Works this paper leans on

85 extracted references · 4 canonical work pages

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    Transverse Susceptibility 26

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    Correction to the transverse susceptibility from the cold regions 29 C

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    Below we compute the susceptibility atT= 0, in which caseT P ωm = R dωm/(2π)

    ¯Γq(k),(9) where the integration stands for R k =N 0 TP ωm R dξk R dθk/2π. Below we compute the susceptibility atT= 0, in which caseT P ωm = R dωm/(2π). The equations for the two-particle vertices Γ q(k) and ¯Γq(k) are shown graphically in Fig. 2 b,c. In analytical form Γq(k) = 1− Z k G(k+ q 2)G(−k+ q 2)Γq(p)V(k,p) + Z k F(k+ q 2)F(−k+ q 2)¯Γq(p)V(k,p), (...

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    As a result, Imχ L(q,Ω) exhibits logarithmic divergence at two distinct frequencies Ω peak,1(q) and Ω peak,2(q), whose values we already presented in Eq

    The curvature remainsβ= 2∆ 0α for both cases. As a result, Imχ L(q,Ω) exhibits logarithmic divergence at two distinct frequencies Ω peak,1(q) and Ω peak,2(q), whose values we already presented in Eq. 28. The jump in Reχ L(q,Ω) is same at Ω peak,1(q) and Ω peak,2(q) and equalsN 0/ √ 2πg2√α- a half of its value atq= 0. Whenθ q =nπ/2, n= 0−3 (a direction tow...

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    (A13), δΠs L(q) =− 1 4 Z 2π 0 dθk 2π Z ∞ −∞ dξk Es,k+q/2 +E s,k−q/2 Es,k+q/2 Es,k−q/2 (Ω)2 −4∆ 2 0 − ξk+q/2 −ξ k−q/2 2 (Ω +iδ) 2 − Es,k+q/2 +E s,k−q/2 2

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