{"id":"76281c64-72fc-473f-9702-ff2b413f712c","arxiv_id":"2501.11936","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"75As NMR on Ba0.234K0.766Fe2As2 shows a pseudogap below T* of about 12 K above Tc and no spin-fluctuation enhancement at T_c^Z2, indicating the time-reversal-symmetry-broken state is nonmagnetic.","lead":"Nuclear magnetic resonance measurements on the iron-based superconductor Ba1-xKxFe2As2 at x=0.77 reveal a drop in the electronic spin-lattice relaxation rate starting near 12 K, well above the superconducting transition, which the authors interpret as a pseudogap formed by preformed electron pairs.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The absence-of-spin-magnetism claim rests on 8 T data, yet at 8 T the spontaneous Nernst signal defining T_c^Z2 is no longer resolvable and the authors' own GL analysis allows field-induced restoration of TRS; the BTRS phase may not be present at the field of measurement.","rationale":"The most load-bearing assumption in the paper is that the 8 T NMR and µSR data are informative about the BTRS phase. The paper itself acknowledges that at 8 T the Nernst signal identifying T_c^Z2 is no longer resolvable, so the persistence of the quadrupling phase at that field is an extrapolation rather than an observation. The authors' own Ginzburg–Landau simulations (Appendix F) explicitly show that external field can restore time-reversal symmetry and eliminate the quartic phase, making the extrapolation fragile rather than conservative. The reader's weakest_assumption identified exactly this 8 T limitation, and the evidence of disagreement is essentially the same. Since the reader already set the verdict to CONDITIONAL on this basis, the present stress-test does not move that verdict. The concrete test of measuring at 5 T, where the Nernst signal is still visible, would directly resolve whether the nonmagnetic conclusion holds where the BTRS phase is actually present. I did not find a more serious independent flaw: the pseudogap interpretation, while theory-laden, is supported by the monotonic decrease of 1/T1T and consistency across probes, and the linewidth/Knight-shift arguments would be persuasive if the field-mismatch concern were addressed. No issues with fraudulent or deceptive presentation arose; the limitation is openly stated but not adequately weighted in the conclusions.","tokens_in":26702,"tokens_out":6580,"duration_ms":68055,"concrete_test":"Perform 75As NMR spin-lattice relaxation and linewidth measurements at 5 T, where the spontaneous Nernst signal is still resolvable in Fig. 1(b), on the same x=0.77 crystal, and look for any anomaly at the measured T_c^Z2(5 T). If 1/T1T and linewidth remain featureless across T_c^Z2 at 5 T, the non-magnetism conclusion is corroborated; if an anomaly appears, the 8 T-based argument fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's second central claim—that 1/T1T, NMR linewidth, and Knight shift 'prove the absence of a magnetic transition at T_c^Z2'—is based entirely on 75As NMR and µSR measurements at 8 T (Figs. 2–4, Secs. III–IV). However, Section II states that the spontaneous Nernst signal defining T_c^Z2 'cannot be resolved at 8 T and beyond', and explicitly: 'we cannot determine whether the quadrupling state is completely suppressed, or if a small quadrupling phase remains at that field with the Nernst signal becoming too small to be resolved in the measurements.' The authors assume via the dashed extrapolation in Fig. 1(d) that T_c^Z2 remains above Tc at 8 T, but this is not a measurement. Their own Ginzburg–Landau analysis (Appendix F, Figs. 17–18) demonstrates that external fields restore TRS by suppressing the subdominant component and/or via the kinetic phase-locking term, concluding that for the parameters examined 'weaker fields eliminate the quartic phase'. Therefore, if the quartic BTRS phase is absent at 8 T, the absence of NMR anomalies near the extrapolated T_c^Z2 carries no information about the zero-field BTRS state, and the conclusion that the spontaneous fields detected by zero-field µSR are nonmagnetic is not established by the present data.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper reports 75As NMR and transverse-field µSR measurements on Ba1−xKxFe2As2 with x ≈ 0.77, with the aim of probing the electronic spectral properties of the time-reversal-symmetry-breaking (BTRS) state that sets in above Tc. The authors identify a decrease of the spin-lattice relaxation rate 1/T1T below a pseudogap temperature T* that lies above the extrapolated BTRS temperature T_c^Z2 and above Tc, and they associate this with the formation of preformed electron pairs that precede the proposed electron quadrupling condensate. They also find no NMR line broadening and no enhancement of 1/T1T near the extrapolated T_c^Z2 at 8 T, which they interpret as excluding spin magnetism and as evidence that the spontaneous fields seen by zero-field µSR arise from persistent real-space currents rather than from spin order. The paper includes detailed sample characterization, transport, Nernst, and specific-heat data, and a Ginzburg-Landau analysis of field-induced suppression of the BTRS phase.","tokens_in":26992,"tokens_out":5704,"duration_ms":58069,"significance":"If the conclusions hold, this would be the first direct spectroscopic evidence for a pseudogap tied to the proposed fermion quadrupling state above Tc in this material, and it would strengthen the case for nonmagnetic, current-based origin of the spontaneous fields. The paper combines several complementary probes (NMR, µSR, transport, specific heat, STM) and is unusually candid about limitations, explicitly stating in Section II that the quadrupling state cannot be confirmed at the 8 T measurement field and in Appendix F that the mean-field GL analysis is not valid close to Tc. The comparison with literature NMR data (Hirano et al.) and the detailed sample characterization are assets. However, the central absence-of-spin-magnetism conclusion depends on an extrapolation of the BTRS phase boundary to 8 T, and the primary pseudogap evidence in Fig. 4 is presented without visible error bars, so the strength of the claims currently exceeds what the data justify.","major_comments":[{"comment":"The claim that the NMR data \"prove the absence of a magnetic transition at T_c^Z2\" (abstract and Section IV) is not supported by the data shown, because all NMR and µSR measurements were taken at 8 T. Section II states that the spontaneous Nernst signal \"cannot be resolved at 8 T and beyond\" and that the authors \"cannot determine whether the quadrupling state is completely suppressed, or if a small quadrupling phase remains.\" The dashed extrapolation of T_c^Z2 in Fig. 1(d) is not a measurement, and the GL analysis in Appendix F explicitly shows that external fields can restore TRS, with the text concluding that for the parameters examined \"weaker fields eliminate the quartic phase\" (Section F.3). Consequently, the absence of an NMR anomaly near the extrapolated T_c^Z2 at 8 T does not constrain the zero-field BTRS state. This affects the second central conclusion of the paper and must be addressed, either by adding lower-field NMR data where the Nernst signal is resolvable or by substantially tempering the claims in the abstract and Section IV.","section":"Section II and IV"},{"comment":"The primary evidence for the pseudogap is the decrease of 1/T1T below T*(NMR) shown in Fig. 4, but no error bars are visible in this figure. The text states that 1/T1T is \"constant within error bars\" above T* and that the decrease below T* is monotonous, yet without visible uncertainties the reader cannot assess the significance of the trend, the precise location of the kink, or the claimed absence of an enhancement near T_c^Z2. Please add error bars (or state explicitly how the scatter defines the uncertainty and why the trend is statistically significant) so that the central pseudogap claim can be independently evaluated.","section":"Fig. 4 and Section IV"}],"minor_comments":[{"comment":"The doping level is given as x = 0.776(1) in the first paragraph of Section IV and in Fig. 4, but as x = 0.766(1) in Section II, the Methods, and Fig. 2. Please correct the typo.","section":"Section IV"},{"comment":"In Fig. 2(b)-(d) the vertical dotted lines are labeled Tc and T*, but the text discusses crossing T_c^Z2; please also mark T_c^Z2 (or explain why it is not marked, given that the measurements are at 8 T).","section":"Fig. 2 and Section III"},{"comment":"The sentence \"we compare this to the quadrupole frequencies measured by Hirano et al. .\" has a missing citation number and an extra period; please fix the reference formatting.","section":"Appendix B.2"},{"comment":"The phrase \"The theoretical analysis ... demonstrates the mechanism of suppression of the quadrupling state when the external magnetic field exceeds a certain value\" is stronger than what the finite-element simulations support, since they are performed for specific parameter sets; suggest \"is consistent with\" or \"illustrates\".","section":"Section II"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the scope of the journal and the NMR/µSR dataset is valuable, but the logical gap between the 8 T measurement field and the zero-field BTRS state is a load-bearing issue for the absence-of-spin-magnetism claim. If the authors can provide lower-field NMR data (or clearly reframe the conclusion as conditional on the extrapolated phase boundary), the paper may become publishable. The \"prove\" language in the abstract and Section IV should be softened regardless, and the error bars in Fig. 4 should be shown."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the 75As NMR pseudogap observation is the real news here, and it looks solid; the paper overreaches when it uses the same 8 T data to “prove” the absence of spin magnetism in the zero-field BTRS state, because the spontaneous Nernst signal that defines that state is already unresolvable at 8 T. The authors admit as much in Section II, then proceed as if the extrapolated T_c^Z2 is a real transition at that field. That is a logical gap, not a dishonest one.\n\nWhat is genuinely new: the 1/T1T drop starting around 12 K, well above Tc = 6.8 K at 8 T, together with the Knight shift and µSR data, provides the first direct spectral evidence for a pseudogap-like reduction of low-energy excitations in this “magic doping” range. The sample characterization is thorough: x-ray, quadrupole frequency, RF penetration, and the comparison with Hirano’s systematic 1/T1T data give confidence that they are looking at the right material. The specific heat and transport data are consistent. This part of the paper deserves careful reading.\n\nThe soft spots: first, the 8 T problem. Their own GL simulations (Appendix F) show that external fields can suppress the quartic phase, and they explicitly say they cannot determine whether the quadrupling state remains at 8 T. So the absence of a 1/T1T anomaly or line broadening near the extrapolated T_c^Z2 does not by itself rule out a magnetic transition in the zero-field BTRS state. The claim that the spontaneous fields are nonmagnetic would need zero-field or low-field NMR or another probe that catches the actual transition. Second, no error bars are visible in the key 1/T1T plot, and the two-Gaussian decomposition adds a layer of uncertainty to the linewidth analysis, though not to T1. Third, the abstract says “prove” – that is too strong.\n\nWho is this for: people working on multiband superconductivity, TRSB, and iron pnictides. The pseudogap data will be a useful anchor for theory. A serious referee should engage with it, but the nonmagnetic conclusion needs to be reworded and supported by data at fields where the BTRS phase actually exists, or at least the caveat needs to be front and center. I would accept it for review, with the expectation of major revision.","headline":"The pseudogap NMR data are a genuinely new and useful result; the absence-of-spin-magnetism claim is oversold because the 8 T measurements may not be in the BTRS phase at all.","tokens_in":27684,"tokens_out":4051,"would_cite":true,"duration_ms":40810,"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":"High-field NMR shows the time-reversal-symmetry-breaking state in Ba1−xKxFe2As2 at magic doping is preceded by a pseudogap and is free of spin magnetism.","keywords":["time-reversal symmetry breaking","electron quadrupling","pseudogap","nuclear magnetic resonance","muon spin rotation","iron-based superconductor","spin-lattice relaxation","Knight shift"],"falsifier":"Measure 75As NMR 1/T1T and linewidth on the same doping at fields of about 1–3 T, where the spontaneous Nernst signal clearly marks $T_c^{{Z2}}$; a peak or step in 1/T1T or line broadening at $T_c^{{Z2}}$ would contradict the claim that the transition is non-magnetic, whereas continued monotonic behavior would confirm it.","tokens_in":26448,"feed_emoji":"🧲","tokens_out":5935,"duration_ms":55529,"temperature":0.7,"pith_summary":"This paper argues that the time-reversal-symmetry-breaking (BTRS) state observed above the superconducting transition in Ba1−xKxFe2As2 near x≈0.77 is an electronic condensate, not a magnetic phase. From 75As nuclear magnetic resonance and muon-spin-rotation Knight-shift data, it establishes two connected results: a pseudogap opens at a temperature T* well above the BTRS transition, indicating preformed bound electron pairs, and the spin-lattice relaxation rate shows no enhancement or line broadening through the transition, ruling out spin magnetism and proximity to a magnetic instability. The claim matters because it narrows the origin of the observed spontaneous fields to persistent real-space currents and connects the pseudogap to the predicted precursor of a four-electron (quadrupling) condensate.","feed_headline":"NMR rules out spin magnetism in the electron-quadrupling state","feed_subtitle":"Measurements locate the pseudogap precursor of four-electron condensation and rule out spin order as its cause.","key_machinery":"The load-bearing observable is the 75As nuclear spin-lattice relaxation rate 1/T1T, which is proportional to the q-averaged low-energy dynamic spin susceptibility; a magnetic instability would show up as an enhancement near the transition, and none is observed. This is paired with the NMR linewidth (a measure of the distribution of local fields) and the NMR and muon Knight shifts (measures of the static spin susceptibility). The interpretive machinery is the electron-quadrupling scenario, in which the order parameter is fourth order in fermionic fields, ⟨ΔaΔb*⟩≠0 while the individual pairing fields ⟨Δa⟩=0, so BTRS occurs through interband relative-phase locking without spin order, and the pseudogap at T* marks the precursor formation of non-condensed Cooper pairs.","core_discovery":"The paper's central claim is that the broken-time-reversal-symmetry state in Ba1−xKxFe2As2 near x≈0.77 is a non-magnetic electronic condensate with a distinct spectral signature: a pseudogap that develops at T* well above $T_c^{{Z2}}$, the temperature where the Z2 symmetry is spontaneously broken. The 75As spin-lattice relaxation rate 1/T1T begins to decrease below T* and keeps decreasing through $T_c^{{Z2}}$ and Tc, with no sign of critical slowing of spin fluctuations, and the NMR linewidth shows no broadening across either transition. Together with NMR and muon-spin-rotation Knight shifts that show no Curie-Weiss behavior, the authors conclude that the spontaneous magnetic fields detected by zero-field muon-spin rotation do not come from spin order or proximity to a magnetic instability, but from persistent real-space currents associated with interband phase locking of the multicomponent order parameter. The pseudogap is interpreted as the fingerprint of non-condensed Cooper pairs that form prior to the fermion-quadrupling ordering, consistent with theory and with specific-heat, transport, and muon-shift data.","pith_inferences":["A testable extension would be to measure 1/T1T across a range of potassium dopings: if the pseudogap is the universal precursor of quadrupling, its separation from Tc should systematically widen as the magic doping is approached from either side.","A direct check of the paper's field-extrapolation assumption would be NMR at fields of about 1–3 T, where the spontaneous Nernst signal clearly marks T_c^{Z2}; a feature in 1/T1T or the linewidth there would challenge the non-magnetic interpretation, whereas continued monotonic behavior would strengthen it.","The same probe combination could distinguish spin-current from orbital-current loop-order candidates in other time-reversal-symmetry-breaking superconductors by looking for the absence of low-energy spin fluctuations at the ordering transition.","Because the Knight shift is suppressed below T*, the underlying gapped density of states appears spin-singlet-like; fitting the temperature dependence with a Yosida-type form could quantify the gap and test whether it matches the specific-heat anomaly scale."],"forward_implications":["The time-reversal-symmetry-breaking state above Tc in this compound is a fermion-quadrupling condensate rather than a magnetic phase.","The absence of low-energy spin-fluctuation enhancement rules out proximity to a spin-density-wave or other magnetic instability at the magic doping level.","The pseudogap T* is a genuine electronic spectral feature observable by NMR, setting the energy scale for pair formation before quadrupling ordering.","The spontaneous Nernst signal and the spontaneous magnetic fields detected in zero-field muon-spin rotation share a common origin in persistent currents from interband phase locking.","Other iron-based superconductors with reported pseudogaps and BTRS superconductivity may host quartic states at appropriate doping levels, and NMR pseudogap signatures offer a way to search for them."],"supporting_citations":[{"why":"Established the BTRS state above Tc via the spontaneous Nernst effect and defined T*, the state that this paper's NMR data probe.","marker":"[4]"},{"why":"Provided calorimetric evidence for two phase transitions, fixing the pseudogap onset and the BTRS transition scales used here.","marker":"[5]"},{"why":"Reported muon-spin-rotation evidence for BTRS in the superconducting state and the phase diagram at magic doping, supplying the sample context.","marker":"[9]"},{"why":"Supplied the s+is pairing model from which the non-magnetic BTRS scenario and its external-field suppression are analyzed.","marker":"[17]"},{"why":"Theory showing that fluctuations can drive BTRS above Tc, the mechanism that motivates the pseudogap-before-quadrupling picture.","marker":"[19]"},{"why":"Theory of phase transitions and the anomalous normal state in BTRS superconductors, providing the basis for precursor bound-state formation.","marker":"[20]"},{"why":"Effective model of the resistive electron quadrupling state and its magnetic properties, supporting persistent currents as the source of spontaneous fields.","marker":"[23]"},{"why":"Provided the transverse-field muon-spin-rotation Knight-shift analysis method used for the muon data in this paper.","marker":"[31]"},{"why":"Benchmark 75As NMR relaxation data on Ba1−xKxFe2As2 across doping levels, against which this sample's 1/T1T behavior is compared.","marker":"[35]"}],"fun_headline_variants":["NMR finds pseudogap, no spin magnetism in electron quadrupling","Four-electron condensate: pseudogap shows, magnetism doesn't","Non-magnetic pseudogap state precedes electron quadrupling","Pseudogap signals four-electron condensate minus spin order"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that the ordered time-reversal-breaking state still exists, at least in a small residual form, at the 8 T field where the NMR and muon-spin-rotation data were recorded, even though the spontaneous Nernst signal that defines $T_c^{{Z2}}$ could not be resolved at that field.","fun_headline_variants_meta":{"raw":{"variants":["NMR finds pseudogap, no spin magnetism in electron quadrupling","Four-electron condensate: pseudogap shows, magnetism doesn't","Non-magnetic pseudogap state precedes electron quadrupling","Pseudogap signals four-electron condensate minus spin order"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000908,"raw_usage":{"total_tokens":4034,"prompt_tokens":1208,"completion_tokens":2826,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":824,"completion_tokens_details":{"reasoning_tokens":2754}},"tokens_in":824,"tokens_out":2826,"duration_ms":21087,"temperature":1.0,"reasoning_tokens":2754,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T17:41:50.444723+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure 75As NMR 1/T1T and linewidth on the same doping at fields of about 1–3 T, where the spontaneous Nernst signal clearly marks $T_c^{{Z2}}$; a peak or step in 1/T1T or line broadening at $T_c^{{Z2}}$ would contradict the claim that the transition is non-magnetic, whereas continued monotonic behavior would confirm it.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Established the BTRS state above Tc via the spontaneous Nernst effect and defined T*, the state that this paper's NMR data probe."},{"cited_title":"For this, we monitored changes of the complex radio-frequency (RF) reflection coefficient S11 at the NMR circuit, using a vector net- work analyzer","cited_arxiv_id":null,"evidence_quote":"Provided calorimetric evidence for two phase transitions, fixing the pseudogap onset and the BTRS transition scales used here."},{"cited_title":"(b), Histogram showing the height distribution of the As-lattice points in (b)","cited_arxiv_id":null,"evidence_quote":"Reported muon-spin-rotation evidence for BTRS in the superconducting state and the phase diagram at magic doping, supplying the sample context."},{"cited_title":"Shipulin, N","cited_arxiv_id":null,"evidence_quote":"Theory showing that fluctuations can drive BTRS above Tc, the mechanism that motivates the pseudogap-before-quadrupling picture."},{"cited_title":"Grinenko, R","cited_arxiv_id":null,"evidence_quote":"Effective model of the resistive electron quadrupling state and its magnetic properties, supporting persistent currents as the source of spontaneous fields."},{"cited_title":"Maiti and A","cited_arxiv_id":null,"evidence_quote":"Provided the transverse-field muon-spin-rotation Knight-shift analysis method used for the muon data in this paper."},{"cited_title":"Maccari and E","cited_arxiv_id":null,"evidence_quote":"Benchmark 75As NMR relaxation data on Ba1−xKxFe2As2 across doping levels, against which this sample's 1/T1T behavior is compared."}],"review_version":1}