{"id":"862f3dc8-44bb-4b5c-a3a5-17f8eface5c8","arxiv_id":"2507.14466","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"Terahertz third-harmonic spectroscopy reveals a sub-gap collective mode in superconducting FeSe, attributed to a Bardasis-Schrieffer mode of an s+d-wave order parameter.","lead":"Using terahertz light pulses, researchers observed a new collective vibration inside the superconducting state of the iron-based superconductor FeSe, at an energy well below the superconducting gap. The result supports the idea that FeSe's superconducting pairing mixes s-wave and d-wave components, and shows that terahertz nonlinear spectroscopy can reveal such hidden modes.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper's own Supplementary Note 7 shows Higgs and charge-density-fluctuation terms can produce a resonance near the anisotropic gap minimum, so the observed low-energy phase jumps do not uniquely require a Bardasis-Schrieffer mode until that alternative is excluded on the same footing.","rationale":"The paper presents a carefully executed terahertz THG experiment: the NbN reference validates the phase-jump method, the I3ω ∝ Iω^3 scaling supports a third-order nonlinear process, and the substrate multiple-reflection analysis in Supplementary Note 4 shows the measured phase is not an artifact of the substrate. The observation itself of low-temperature phase jumps at 0.1 and 0.2 THz, and their absence at 0.3 and 0.5 THz, is credible and likely stands. The central theoretical attribution to a Bardasis-Schrieffer mode is less secure. The main text presents only the BS-mode calculation in Fig. 3c, while the competing Higgs/CDF calculation is relegated to Supplementary Note 7, where it is admitted that for a nodeless anisotropic gap both terms produce a resonance and phase jump near the gap minimum—precisely the spectral region of the observed resonances. The authors argue this alternative fails because no resonance appears at the gap maximum, but that exclusion is not quantified: the model is single-pocket, impurity and paramagnetic corrections are stated to be beyond scope, and the high-frequency probes are two discrete frequencies, neither reaching the hole gap maximum. The reader's concern about gap values taken from bulk FeSe is partially addressed by the same-film THz conductivity in Supplementary Note 3, which supports maximum gaps of roughly 4.5–8 meV, but it does not directly determine the minimum gap. If the same-film 2Δmin is comparable to or below the two-photon energies at the phase jumps, the 'substantially below the gap' framing weakens, and the Higgs/CDF alternative becomes more plausible. These are addressable concerns rather than fatal flaws, so the CONDITIONAL verdict remains appropriate.","tokens_in":25217,"tokens_out":6615,"duration_ms":84527,"concrete_test":"Compute the full THG intensity and phase in a single model that includes Higgs, CDF, and BS contributions using the same hole-pocket parameters listed in the paper (r ≈ 0.5, V0 ≈ 3Vl, c0x = 0.5, c0y = 0.2, c1x = c1y = −0.001) and the gap forms of Fig. S10, then overlay the normalized I3ω(T) and phase data for all four incident frequencies. If the Higgs+CDF-only calculation reproduces the observed phase jumps at 0.1 and 0.2 THz and the absence at 0.3 and 0.5 THz within noise, the BS-mode assignment is not required. In parallel, extract the minimum superconducting gap of the same 65-nm FeSe/CaF2 film from a two-gap BCS fit to the measured superfluid density in Fig.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the exclusion of conventional low-energy Higgs and charge-density-fluctuation (CDF) responses. The paper itself, in Supplementary Note 7 and Fig. S10, calculates that for an anisotropic s+d-like gap, both the Higgs mode and CDF give a THG resonance and phase jump in the vicinity of the gap minimum—exactly the low-energy region where the 0.1 and 0.2 THz phase jumps are observed. The only stated reason for preferring the Bardasis-Schrieffer interpretation is the absence of a high-energy resonance at the gap maximum, but that exclusion is not quantitative: the calculation is single-pocket, the impurity and paramagnetic contributions are acknowledged as beyond scope, and the high-frequency probes are only two discrete frequencies (2ω = 2.4 meV at 0.3 THz and 4.1 meV at 0.5 THz), neither of which reaches the hole-pocket maximum 2Δh(0) = 7.0 meV. Moreover, the BS-mode calculation in Fig. 3c is presented separately and is not compared with the Higgs/CDF curves on the same footing, so the relative weight of the alternatives is not established. The claim that the mode is 'substantially below the superconducting gap' also depends on the minimum gap 2Δmin ≈ 0.8 meV taken from bulk STS [S4]; if the same-film minimum gap is close to or below the two-photon energies at which phase jumps occur, the sub-gap character of the resonance is not independently demonstrated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":25480,"tokens_out":8503,"duration_ms":92631,"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":[{"comment":"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.","section":"Main text, 'Theoretical analysis'; Supplementary Note 3"},{"comment":"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.","section":"Supplementary Note 7 and Fig. S10"},{"comment":"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.","section":"Supplementary Note 8, Table S1; main text Fig. 2e-f"},{"comment":"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.","section":"Supplementary Note 9 and Table S1"}],"minor_comments":[{"comment":"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.","section":"Main text, 'THz-THG experiments'"},{"comment":"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.","section":"Fig. 2d caption"},{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Supplementary Note 7"},{"comment":"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.","section":"Fig. 1e caption"}],"recommendation":"major_revision","confidential_remarks":"The manuscript reports a striking experimental observation that is likely to attract attention, and the experimental quality appears good. However, the central attribution to a Bardasis-Schrieffer mode depends on a chain of assumptions: same-film gap values, resonance-condition mapping, and quantitative exclusion of Higgs/CDF responses. These are fixable within the scope of a revision, but they are load-bearing rather than cosmetic, so I recommend major revision rather than accept or minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, what to know: this is a careful THz third-harmonic experiment on a FeSe thin film that finds a low-energy resonance (phase jump and peak in THG at 0.1 and 0.2 THz) and validates the phase-jump method using a NbN reference. The observation itself is new and interesting. The claim that it is a Bardasis-Schrieffer mode is plausible, but it is not as solid as the title suggests. The paper's own Supplementary Note 7 shows that Higgs and charge-density-fluctuation (CDF) contributions can also produce a resonance near the gap minimum for an anisotropic s+d gap. The only stated reason to prefer BS is the absence of a high-energy resonance at the gap maxima, but the highest drive frequency used, 0.5 THz, corresponds to 2ω=4.1 meV, which is below the estimated electron gap 2Δe=4.6 meV and far below the hole gap 2Δh=7.0 meV. So the gap maximum region is not actually probed. That, plus the fact that the gap values are extrapolated from bulk STS via Tc scaling rather than measured on the same film, leaves the sub-gap identification with real uncertainty.\n\nWhat the paper does well: the sample characterization is thorough (XRD, optical conductivity, superfluid density), the multiple-reflection correction for the substrate is carefully treated, and the polarization dependence, while averaged over twinned domains, is consistent with the C2 symmetry expected for a BS mode. The theory is not fitted to the THG data; parameters come from previous literature and the mode position is a consistency check rather than a posteriori fit. That is honest and avoids the worst kind of circularity.\n\nThe soft spots are, in order: (1) same-film gap measurement is missing, so 'substantially below the gap' is an extrapolation; (2) the Higgs/CDF alternatives are not excluded on the same footing—the calculated curves for the BS mode and for Higgs/CDF are not directly compared in one figure, and the single-pocket model omits the electron pockets; (3) the high-frequency drive does not reach the gap maxima, so the absence of a resonance there is not demonstrated. These are addressable. The experiment is likely correct; the interpretation is the open question.\n\nWho this is for: superconductivity and THz nonlinear spectroscopy researchers. It deserves a serious referee: the experimental observation is important if it holds, and the paper engages honestly with the alternatives. I would send it to peer review with a request that referees push on the gap measurement and the quantitative comparison of the candidate modes.","headline":"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.","tokens_in":26125,"tokens_out":4429,"would_cite":true,"duration_ms":51063,"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":"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.","keywords":["iron-based superconductors","FeSe","electronic nematicity","Bardasis-Schrieffer mode","terahertz third-harmonic generation","collective modes","multicomponent order parameter","superconducting gap"],"falsifier":"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.","tokens_in":24940,"feed_emoji":"⚡","tokens_out":7194,"duration_ms":535523,"temperature":0.7,"pith_summary":"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.","feed_headline":"Sub-gap resonance fingerprints s+d pairing in FeSe","feed_subtitle":"A sub-gap terahertz resonance reveals the mixed s+d order parameter of nematic FeSe.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Bardasis and Schrieffer's original prediction of an exciton-like sub-gap collective mode from a subdominant pairing channel.","marker":"[24]"},{"why":"Establishes the terahertz third-harmonic generation method and the two-photon Higgs-mode resonance criterion used as the reference protocol.","marker":"[30]"},{"why":"Supplies the bulk FeSe STS superconducting gap values that the film's gaps are scaled from.","marker":"[12]"},{"why":"Provides the nematic-order-based pairing model whose two-channel s/d interaction structure the paper's gap equation follows.","marker":"[17]"},{"why":"Predicted that the Bardasis-Schrieffer mode produces a terahertz third-harmonic resonance near $2\\omega = \\omega_m$, the theoretical basis for the THG signature.","marker":"[36]"},{"why":"Supplies the effective two-component gap-equation framework and the s+d/Leggett-mode language for FeSe.","marker":"[44]"},{"why":"Gives the ARPES-measured anisotropic gap and BCS-like temperature dependence assumed for the film's gap estimates.","marker":"[13]"}],"fun_headline_variants":["Nematic FeSe hosts Bardasis-Schrieffer mode","Sub-gap resonance exposes s+d pairing in FeSe","Collective mode reveals two-channel pairing in FeSe","Bardasis-Schrieffer mode seen in nematic FeSe"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Nematic FeSe hosts Bardasis-Schrieffer mode","Sub-gap resonance exposes s+d pairing in FeSe","Collective mode reveals two-channel pairing in FeSe","Bardasis-Schrieffer mode seen in nematic FeSe"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000809,"raw_usage":{"total_tokens":3577,"prompt_tokens":998,"completion_tokens":2579,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":614,"completion_tokens_details":{"reasoning_tokens":2511}},"tokens_in":614,"tokens_out":2579,"duration_ms":20560,"temperature":1.0,"reasoning_tokens":2511,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:55:33.230402+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":", author Hanaguri, T","cited_arxiv_id":null,"evidence_quote":"Bardasis and Schrieffer's original prediction of an exciton-like sub-gap collective mode from a subdominant pairing channel."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the terahertz third-harmonic generation method and the two-photon Higgs-mode resonance criterion used as the reference protocol."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the bulk FeSe STS superconducting gap values that the film's gaps are scaled from."},{"cited_title":", author Sigrist, M","cited_arxiv_id":null,"evidence_quote":"Provides the nematic-order-based pairing model whose two-channel s/d interaction structure the paper's gap equation follows."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Predicted that the Bardasis-Schrieffer mode produces a terahertz third-harmonic resonance near $2\\omega = \\omega_m$, the theoretical basis for the THG signature."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the effective two-component gap-equation framework and the s+d/Leggett-mode language for FeSe."},{"cited_title":"& author Hirschfeld, P","cited_arxiv_id":null,"evidence_quote":"Gives the ARPES-measured anisotropic gap and BCS-like temperature dependence assumed for the film's gap estimates."}],"review_version":1}