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A new collective mode in an iron-based superconductor with electronic nematicity

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Terahertz nonlinear spectroscopy on strained FeSe thin films finds a collective-mode resonance well below the superconducting gap, which the paper attributes to a Bardasis-Schrieffer mode between s-wave and d-wave pairing channels.

desk verdict Careful THz-THG experiment finds a low-energy resonance in FeSe, but the Bardasis-Schrieffer assignment is not unique until gap values on the same film and Higgs/CDF alternatives are addressed. read the letter →

arxiv 2507.14466 v1 pith:2GPATHDJ submitted 2025-07-19 cond-mat.supr-con cond-mat.str-el

Haruki Matsumoto , Silvia Neri , Tomoki Kobayashi , Atsutaka Maeda , Dirk Manske , Ryo Shimano This is my paper · ORCID
classification cond-mat.supr-concond-mat.str-el
keywords iron-basedsuperconductorsFeSeelectronicnematicityBardasis-Schrieffermodeterahertzthird-harmonicgenerationcollectivemodesmulticomponentorderparametersuperconductinggap
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

The paper sets out to identify the symmetry and structure of the superconducting order parameter in FeSe, an iron-based superconductor that becomes superconducting inside an electronic nematic phase. Using terahertz third-harmonic generation on a strained FeSe thin film, it finds a low-energy collective-mode resonance at 0.1 and 0.2 THz that lies clearly below the estimated superconducting gap. The authors argue this mode is not the amplitude Higgs mode and not a quasiparticle charge-density fluctuation; instead, their theory attributes it to a Bardasis-Schrieffer mode, the relative phase oscillation between the s+d-wave-like ground state and a subleading pairing channel. If correct, the observation fingerprints a multicomponent s+d order parameter that becomes active because the nematic phase breaks the fourfold lattice symmetry. The resonance matters because collective modes provide one of the few direct experimental windows onto the internal structure of a superconducting order parameter.

What carries the argument

The argument runs through an effective linearized gap equation on the hole pocket, written as a $2\times 2$ matrix coupling the s-wave component $\Delta_1$ and d-wave component $\Delta_2\cos(2\phi)$, with off-diagonal mixing produced by the nematic distortion of the Fermi pockets. Diagonalizing the interaction matrix gives the ground-state form factor $f_0(k)=1+r\cos(2\phi)$, a sign-preserving s-wave-dominated gap, and a subleading form factor $f_1(k)=\cos(2\phi)-r'$. The Bardasis-Schrieffer mode is the relative-phase oscillation between these two pairing channels, and it appears in the theory as a sharp resonance just below the gap minimum whose third-harmonic current reproduces the measured resonance position, the absence of a resonance at the gap maximum, and the near-isotropic polarization after averaging over twinned domains.

What would settle it

Measure the superconducting gaps directly on the same FeSe film used for the THG experiment, for example by terahertz optical conductivity or scanning tunneling spectroscopy. If the minimum gap $2\Delta_{\min}$ falls to or below about 1.65 meV, the resonances at 0.1 and 0.2 THz would no longer be sub-gap, and the Bardasis-Schrieffer assignment would lose its footing; equally, observing a third-harmonic phase jump at the gap maximum would contradict the paper's central claim.

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Extended reading notes

Core claim

The central discovery is a collective-mode resonance in terahertz third-harmonic generation from a strained FeSe film that appears substantially below the superconducting gap energy and is distinct from the amplitude Higgs mode. A phase jump and intensity peak in the third-harmonic response occur only at driving frequencies \omega = 0.1 and 0.2 THz, corresponding to resonance energies below $2\hbar\omega = 2.4$ meV, whereas the estimated superconducting gaps are $2\Delta_h(0) = 7.0$ meV and $2\Delta_e(0) = 4.6$ meV. Theoretical modeling with an effective two-component gap equation for the hole pocket shows that nematicity mixes s-wave and d-wave pairing channels, with a sign-preserving s-wave-dominated ground state and a subleading d-wave-dominated channel. The observed resonance is then identified as the Bardasis-Schrieffer mode: a collective fluctuation of the relative phase between these two channels, which the paper notes can also be viewed as an intraband Leggett mode. The result corroborates multicomponent pairing in FeSe activated by the lower space-group symmetry of the electronic nematic phase.

Load-bearing premise

The sub-gap claim depends on the strained FeSe film's actual minimum superconducting gap being larger than the two-photon energies at which the resonance appears, about 1.65 meV for 0.2 THz; those gap values are not measured on the same film but scaled from bulk FeSe STS data by the critical temperature and given BCS temperature dependence.

Editorial extensions

If this is right

  • If the interpretation is right, FeSe's superconducting state in the nematic phase carries a multicomponent s+d order parameter rather than a single-component gap.
  • The THz third-harmonic phase-jump criterion used here can identify sub-gap Bardasis-Schrieffer or Leggett modes in other superconductors where the gap cannot be measured on the same film.
  • The Higgs and charge-density-fluctuation contributions alone cannot explain the low-energy resonance or the absence of a resonance at the gap maximum, and the Bardasis-Schrieffer calculation reproduces both.
  • The observed mode can be regarded as an intraband Leggett mode, so it speaks to phase fluctuations inside a single pocket rather than only to interband oscillations.

Reading between the lines

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

  • Our inference: a detwinned, single-domain FeSe sample should show the predicted C2 anisotropy of the third-harmonic signal instead of the near-isotropic response seen here, providing a cleaner test of the s+d assignment.
  • Our inference: if the mode is a Bardasis-Schrieffer bound state near the gap minimum, its energy should track the minimum gap, not the maximum gap, across strain, doping, and pressure; this is testable with the same technique on films with different critical temperatures.
  • Our inference: the same two-channel gap equation should predict mode softening as the subleading interaction approaches the leading one, so materials tuned toward a pairing-channel degeneracy should show the resonance shift downward and strengthen.
  • Our inference: measuring the superconducting gap on the very same film used for the THG experiment, rather than scaling bulk STS values by the critical temperature, is the shortest route to confirming the sub-gap placement.
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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 reports terahertz third-harmonic generation (THG) measurements on a strained FeSe thin film and observes low-temperature phase jumps and THG intensity peaks for incident frequencies of 0.1 and 0.2 THz, which are absent at 0.3 and 0.5 THz. Interpreting the phase jumps as a collective-mode resonance, the authors locate the mode near 0.8–1.65 meV, well below their estimated gap energies of 2Δh(0)=7.0 meV and 2Δe(0)=4.6 meV, and therefore exclude the amplitude Higgs mode. Comparing with a two-channel gap model with s+d-like pairing on the hole pocket, they attribute the mode to a Bardasis-Schrieffer relative phase mode between the dominant and subleading pairing channels, supported by a polarization-dependence analysis and a twin-domain average. The paper presents a new experimental observation of a low-energy THG resonance in FeSe and proposes a specific collective-mode interpretation.

Significance. If the interpretation is correct, this is the first observation of a Bardasis-Schrieffer type mode in an iron-based superconductor with electronic nematicity and provides evidence for a multicomponent s+d superconducting order parameter in FeSe. The experimental methods are careful: the NbN reference establishes the phase-jump criterion, substrate multiple-reflection effects are analyzed in Supplementary Note 4, and the polarization measurements include appropriate controls. The theory is not fitted to the data but is presented as a consistency check using parameters from previous literature. The significance is therefore high if the sub-gap nature and the exclusion of conventional Higgs/charge-density-fluctuation responses can be made quantitative.

major comments (4)
  1. [Main text, 'Theoretical analysis'; Supplementary Note 3] The central claim that the mode lies substantially below the superconducting gap is not quantitatively secured, because the gap values 2Δh(0)=7.0 meV and 2Δe(0)=4.6 meV are extrapolated from bulk FeSe STS data by assuming Δ∝Tc, and the minimum gap 2Δmin≈0.8 meV used to place the mode inside the gap is taken from bulk FeSe, not measured on the same film. The optical conductivity in Fig. S3 shows only a broad missing-weight edge near 8 meV and a downturn near 4.5 meV, neither of which determines the minimum gap. If the actual minimum gap on the film were ≤1.65 meV (the two-photon energy at 0.2 THz), the observed resonances at 2ω=0.83 and 1.65 meV would be at or above the gap, and the sub-gap collective-mode interpretation would not follow. Please provide a same-film lower bound on the minimum gap, or explicitly analyze how the conclusion depends on this quantity.
  2. [Supplementary Note 7 and Fig. S10] The exclusion of conventional Higgs and charge-density-fluctuation (CDF) contributions is incomplete. Supplementary Note 7 and Fig. S10 show that for a nodeless anisotropic gap of the form 1+0.65 cos2θ, both the Higgs mode and CDF produce a THG resonance and phase jump near the gap minimum, which is precisely the region where the observed 0.1 and 0.2 THz phase jumps occur. The authors reject these mechanisms because no resonance is seen at the gap maxima, but the two highest probe frequencies only reach 2ω=2.48 meV (0.3 THz) and 4.13 meV (0.5 THz), both below the hole-pocket maximum 2Δh(0)=7.0 meV and only near the electron-pocket maximum 2Δe(0)=4.6 meV. Thus the absence of a high-energy resonance is not established by the data. Moreover, the relative weight of Higgs/CDF versus Bardasis-Schrieffer contributions is not computed on the same footing: the BS curve in Fig. 3c is not overlaid with the S10 curves, and Supplementary Note 7 explicitly states that the impurity and paramagnetic coupling terms needed for such a comparison are beyond the scope of the work.
  3. [Supplementary Note 8, Table S1; main text Fig. 2e-f] There is an internal tension between the measured resonance frequencies and the theoretical mode energy. Table S1 and Supplementary Note 8 give ωBS≈0.9(2Δmin)≈0.72 meV, and the THG resonance condition quoted from ref. 36 is 2ω=ωBS. This condition can account for a phase jump at ω=0.1 THz (2ω=0.83 meV) but not for the observed 0.2 THz phase jump (2ω=1.65 meV), which would require the mode energy to reach at least 1.65 meV at low temperature. Please specify the resonance condition explicitly and show that both phase-jump temperatures are consistent with a single BCS-like mode dispersion. The magenta curve in Fig. 2d is not defined in the text or caption, which makes this consistency check difficult to evaluate.
  4. [Supplementary Note 9 and Table S1] The theoretical identification is only a qualitative consistency check, not a quantitative fit. All interaction parameters in Table S1 (Vs, Vd, W, W', Φh, Φe, r, r') are adopted from refs. [S17] and earlier literature, no sensitivity or uncertainty analysis is given, and Supplementary Note 9 reduces the calculation to a single hole pocket even though Eq. (1) is derived from a three-pocket model. The comparison with experiment is made by eye rather than by a quantitative metric. The authors should either confront the model with the measured temperature/frequency dependence of the THG signal or clearly state that the agreement is limited to the qualitative position of the resonance.
minor comments (5)
  1. [Main text, 'THz-THG experiments'] The sentence 'a collective mode exists in the low energy region at least below 2ℏω=2.4 meV' is confusing, because phase jumps are observed only for ω=0.1 and 0.2 THz, not for 0.3 THz; please clarify how this upper bound is inferred from the data.
  2. [Fig. 2d caption] The magenta curve is not defined in the main text or the caption. Please state explicitly what quantity it represents (e.g., the assumed temperature dependence of the collective-mode energy) and how it was obtained.
  3. [Throughout] There are several typographical errors that should be corrected: 'admixiture' in the main text, 'correspnding' in the Discussion, 'grater' in the Theoretical analysis section, 'T able S1' in the Table S1 header, and 'auperconductors' in ref. 47.
  4. [Supplementary Note 7] The statement that the low-energy resonance 'cannot be ascribed to the Higgs mode or to CDF' is based only on the absence of a phase jump at 0.3 and 0.5 THz; please phrase this as an upper-bound argument rather than a demonstrated exclusion, given the sparse frequency sampling.
  5. [Fig. 1e caption] The description '0.61 THz-FFT low pass and high pass filters' is not self-explanatory; please specify the filter cutoff frequencies and the filtering procedure used to extract the FH and TH components.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: BS-mode assignment is a consistency check with externally sourced model parameters, and the remaining weaknesses are evidentiary limitations rather than inputs renamed as predictions.

full rationale

The paper's derivation chain does not reduce to its own inputs. The observed low-energy THG phase jumps and intensity resonances are experimental facts; the theoretical BS-mode response in Fig. 3c is computed from a two-channel gap equation whose interaction parameters (Us=Ud=U, Phi_h=0.3, Phi_e=-0.1) are adopted from external references [17/S10, 44/S17], not fitted to the 0.1/0.2 THz data. The mode position omega_BS ~ 0.9(2Delta_min) is an output of those parameters (g_l ~ g_0/3), and the paper explicitly frames the comparison as consistency rather than a parameter-free prediction. The NbN reference measurement independently demonstrates the phase-jump criterion, so the use of refs. [21,30,37,38] is not load-bearing circular self-citation. Supplementary Note 7 and Fig. S10 explicitly calculate Higgs and charge-density-fluctuation contributions and acknowledge that they can also resonate near the gap minimum; the paper's preference for BS over these alternatives rests on the asserted absence of a high-energy resonance, but the highest two-photon energy used (4.1 meV at 0.5 THz) does not reach 2Delta_h=7.0 meV, and the calculations are single-pocket with impurity/paramagnetic effects noted as beyond scope. These are evidentiary limitations that weaken the discrimination between mechanisms; they are not equation-level circularity, fitted parameters renamed as predictions, or self-citation chains that force the conclusion. No specific reduction of the claimed prediction to the input has been identified.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The theoretical support for the BS-mode assignment rests on a set of interaction and band parameters taken from earlier literature, plus the extrapolated gap energies for the film. The mode calculation is a consistency check with those inputs, not a parameter-free first-principles prediction. No new particles or forces are introduced.

free parameters (7)
  • gap anisotropy parameter r = 0.5
    Appears in the ground-state form factor f0(k)=1+r cos2φ; obtained from the interaction parameters in Table S1, not measured on this sample.
  • subleading form-factor parameter r' = -0.5
    Appears in f1(k)=cos2φ-r'; taken from diagonalization of Eq. (S17) with chosen interactions.
  • pairing interaction couplings Vs, Vd, W, W' = 2, 1, 0.6, 0.7 (in units of U^2)
    Table S1; these values control the BS-mode energy, gl≈g0/3, and hence the predicted mode position near the gap minimum.
  • nematic order parameters Φh, Φe = 0.3, -0.1
    Adopted from ref. [S17] to model the nematic band distortions.
  • hole pocket effective mass ratio = mx=2.37 my
    Sets the elliptical Fermi surface used in the light-coupling calculation.
  • band curvature coefficients c0, c1 = c0x=0.5, c0y=0.2, c1x=c1y=-0.001
    Used in the expansion of the band dispersion in Eq. (S26) for the THG calculation.
  • film superconducting gap values = 2Δh(0)=7.0 meV, 2Δe(0)=4.6 meV
    Scaled from bulk STS values by Tc; not directly measured on the strained film, though optical conductivity suggests a similar scale.
assumptions (6)
  • domain assumption The superconducting gap on the hole pocket can be parametrized as Δh=Δ1+Δ2 cos2θh with s-wave and d-wave components mixed by nematicity.
    Used to construct the two-channel gap equation Eq. (1); relies on the pairing picture of refs. [17,44].
  • domain assumption The nematic order in the film is strong enough that the ground state is an s+d state rather than a time-reversal-symmetry-broken s+eiαd state.
    The identification with the BS mode and the rejection of the Leggett/TRSB interpretation depends on this; the paper argues from literature but does not measure the relative phase.
  • domain assumption The THG response is dominated by the superconducting collective mode, and the temperature-dependent transmission correction is negligible below Tc.
    Supplementary Note 4 shows the substrate multiple-reflection correction changes the amplitude by less than 4% and phase by less than 5e-3 π rad below 20 K, which justifies analyzing the raw transmitted field.
  • domain assumption A single-hole-pocket model captures the light-matter coupling of the BS mode, despite FeSe being multiband.
    Supplementary Note 9 explicitly retains only the hole pocket for the current calculation; the authors note the full multiband equations are coupled, so this is a simplification.
  • domain assumption The BCS temperature dependence with Tc scaling of the bulk gap applies to the strained thin film.
    Used to draw the magenta mode-energy curve in Fig. 2d and to conclude the mode lies below 2Δe; no direct gap measurement on this film.
  • standard math Standard BCS mean-field and Anderson pseudospin formalism with linearized Bloch equations describes THG from the BS mode.
    This is the framework of refs. [S8, S18, S19]; no novel mathematical assumptions.

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Pith. "Pith review of A new collective mode in an iron-based superconductor with electronic nematicity." pith.science (2026). https://pith.science/paper/2GPATHDJ

@misc{pith2026250714466,
  author       = {Pith},
  title        = {Pith review of: A new collective mode in an iron-based superconductor with electronic nematicity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2GPATHDJ}},
  note         = {Machine review of arXiv:2507.14466}
}
read the original abstract

Elucidation of the symmetry and structure of order parameter(OP) is a fundamental subject in the study of superconductors. Recently, a growing number of superconducting materials have been identified that suggest additional spontaneous symmetry breakings besides the primal breaking of U(1) gauge symmetry, including time-reversal, chiral, and rotational symmetries. Observation of collective modes in those exotic superconductors is particularly important, as they provide the fingerprints of the superconducting OP. Here we investigate the collective modes in an iron-based superconductor, FeSe, a striking example of superconductivity emergent in an electronic nematic phase where the rotational symmetry of electronic degree of freedom is spontaneously broken. By using terahertz nonlinear spectroscopy technique, we discovered a collective mode resonance located substantially below the superconducting gap energy, distinct from the amplitude Higgs mode. Comparison with theoretical calculations demonstrates that the observed mode is attributed to a collective fluctuation between the s+d-wave-like ground state and the subleading pairing channel, which corresponds to the so-called Bardasis-Schrieffer mode but also resembles an intraband Leggett mode. Our result corroborates the multicomponent pairing channels in FeSe activated in the lower space group symmetry in the electronic nematic phase.

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    write newline

    " write newline "" before.all 'output.state := FUNCTION add.period duplicate empty 'skip "." * add.blank if FUNCTION if.digit duplicate "0" = swap duplicate "1" = swap duplicate "2" = swap duplicate "3" = swap duplicate "4" = swap duplicate "5" = swap duplicate "6" = swap dupl...

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    write newline

    " write newline "" before.all 'output.state := FUNCTION output.doi doi empty skip "doi:" doi * "" * output if FUNCTION format.archive archivePrefix empty "" archivePrefix ":" * if FUNCTION format.primaryClass primaryClass empty "" " [" primaryClass * "] " * if FUNCTION format....

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    write newline

    " write newline "" before.all 'output.state := FUNCTION string.to.integer 't := t text.length 'k := #1 'char.num := t char.num #1 substring 's := s is.num s "." = or char.num k = not and char.num #1 + 'char.num := while char.num #1 - 'char.num := t #1 char.num substring FUNCTI...

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    write newline

    " write newline "" before.all 'output.state := FUNCTION string.to.integer 't := t text.length 'k := #1 'char.num := t char.num #1 substring 's := s is.num s "." = or char.num k = not and char.num #1 + 'char.num := while char.num #1 - 'char.num := t #1 char.num substring FUNCTI...

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    ENTRY address archive author booktitle chapter edition editor eprint howpublished institution journal key keywords month note number organization pages publisher school series title type url doi volume year archivePrefix primaryClass eid adsurl adsnote version label INTEGERS o...

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    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

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    Available from:

    ENTRY address assignee author booktitle chapter cartographer day edition editor howpublished institution inventor journal key keywords month note number organization pages part publisher school series title type volume word year eprint doi url lastchecked updated archive archi...

  69. [77]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.