{"id":"e24fc93e-b53d-4b7a-9029-ad8082062eb6","arxiv_id":"2608.07238","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Phase fluctuations alone produce a magnetic pseudogap in 1/T1T with onset set by ξ(T) ≈ ξ_BCS, and an s-wave vertex correction generates a normal-state coherence peak governed by ξ(T) ≈ ℓ.","lead":"A theory paper shows that the slow suppression of the NMR relaxation rate seen above the superconducting transition in two-dimensional superconductors can come purely from phase fluctuations, with no competing order. The same calculation predicts a coherence peak just above the transition in s-wave systems, analogous to the Hebel-Slichter peak in conventional superconductors.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The s-wave coherent-enhancement scale ξ(T_coh)/ℓ=2.22 comes from a one-vertex truncation that is uncontrolled near Tc; an exact Gaussian-disorder calculation could shift or remove it.","rationale":"The reader's weakest_assumption correctly flags the Gaussian correlation form, temperature-independent Δ0, and leading-order Born self-energy. I sharpen this into a more specific correctness risk: the one-vertex truncation for the s-wave coherent enhancement is uncontrolled exactly where the effect is claimed to be strongest, near Tc. The self-energy on the Fermi surface is not a small perturbation in the quoted regime, and the vertex divergence signals that higher-order diagrams are not parametrically suppressed. The d-wave vertex cancellation and the BCS limit at ξ→∞ are internally consistent, and the recent preprint [60] gives qualitative numerical support for a normal-state s-wave Hebel-Slichter-like peak in an XY-type model, but it does not test the specific ratio ξ(T_coh)/ℓ = 2.22. The exact Gaussian-disorder calculation I propose is a finite computational task that would directly determine whether the leading-order vertex controls 1/T1T in this regime. This concern does not justify rejection: the existence of a magnetic pseudogap and of a coherent enhancement in some phase-fluctuation model is plausible and partly supported. It does, however, mean the quantitative central claim should be conditioned on a controlled calculation or an explicit demonstration that higher-order diagrams are negligible.","tokens_in":18180,"tokens_out":20179,"duration_ms":231406,"concrete_test":"Perform an exact disorder-average calculation of 1/T1T for the same Gaussian random pairing model: generate static complex pairing configurations Δ(r) with correlation exp(−|r−r′|²/2ξ²), diagonalize the BdG Hamiltonian on a finite lattice for each realization, compute the spin susceptibility from the eigenstates, and average over many realizations. Scan the same parameter range as Fig. 7 (πξ_BCS/ℓ from 0.05 to 0.4 and β|Δ0| = 5), then extract ξ(T_coh)/ℓ from the derivative maximum of the averaged 1/T1T. If the exact result deviates from 2.22(3) by more than the quoted fit error, the leading-order vertex is not the controlling contribution and that quantitative scale should not be presented as a prediction.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The quantitative s-wave prediction is the least secure part of the argument. The spin response in Eqs. (9)-(10) is computed with one explicit pairing correlation beyond the Dyson-resummed Green's function. Near Tc, however, the expansion is not controlled. On the Fermi surface the Born self-energy Eq. (3) gives Σ(k_F, 0) = −i|Δ0|²/(2Γ) = −i|Δ0|²ξ/v_F, which for the quoted parameters (Δ0/E_F = 1/4 and k_Fξ ≫ 1) is not small compared with E_F once k_Fξ exceeds order 10. The leading vertex diagram, containing four Green's functions, diverges faster than the bubble as Γ_k → 0, and there is no small parameter suppressing the neglected multi-vertex diagrams of the same Gaussian model, whose additional powers of |Δ0|²ξ/v_F are of order unity in the regime of interest. Therefore the derivative maximum that defines ξ(T_coh), and hence the headline numerical ratio ξ(T_coh)/ℓ = 2.22(3), may be an artifact of the one-vertex truncation rather than a robust property of phase-fluctuating superconductors. The magnetic-pseudogap scale is less exposed because it inherits the single-particle charge pseudogap, but the coherent-enhancement scale is exactly the part of the central claim that lacks independent support beyond the leading-order calculation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the NMR spin-lattice relaxation rate 1/T1T in two-dimensional superconductors with strong phase fluctuations. Building on the authors' previous Gaussian-disorder description of static superconducting phase fluctuations, they evaluate the spin susceptibility in perturbation theory, keeping the bubble diagram and the leading vertex correction. For both s-wave and d-wave pairing, they find a magnetic pseudogap in the normal state whose onset is set by the competition between the BKT correlation length ξ(T) and the BCS coherence length ξ_BCS (Eq. 16). For s-wave pairing, the vertex correction leads to a normal-state coherent enhancement of 1/T1T, regularized by a background scattering rate Γ0; the associated coherence scale is claimed to satisfy ξ(T_coh)/ℓ = 2.22(3) with ℓ = v_F/Γ0 (Eq. 17, Fig. 7d). The vertex correction is shown to vanish for d-wave in the large-ξ limit. The paper argues that the full temperature dependence of 1/T1T can be understood as an interplay of the three length scales ξ(T), ξ_BCS, and ℓ.","tokens_in":18576,"tokens_out":12919,"duration_ms":116851,"significance":"The proposed mechanism is physically appealing and, if correct, would unify the charge pseudogap, magnetic pseudogap, and Hebel-Slichter-like coherence effects in quasi-2D superconductors without invoking competing orders. The paper's main strength is that it makes specific, falsifiable predictions: the magnetic pseudogap scale is tied to ξ_BCS, and the coherent enhancement scale is linear in ℓ. The analytic demonstration of the d-wave suppression of the vertex correction is elegant. The authors also point to a recent independent Monte Carlo study (Ref. [60]) that finds a normal-state coherence peak in s-wave BKT superconductors, which supports the qualitative conclusion. The numerical integrations are carried out with a documented adaptive Monte Carlo method. However, the quantitative s-wave prediction rests on a one-loop truncation whose validity near the transition is not established.","major_comments":[{"comment":"The headline ratio ξ(T_coh)/ℓ = 2.22(3) is extracted from the leading-order vertex correction, but the perturbative expansion in |Δ0|² is not controlled in the regime where the coherence scale is determined. From Eq. (3), the on-shell self-energy is Σ(k_F,0) ≈ −i |Δ0|² ξ/v_F, so |Σ|/E_F ≈ (1/2)(Δ0/E_F)² k_F ξ (for m=1). With the parameters of Fig. 7 (E_F=2, Δ0/E_F=1/4, Γ0/Δ0=0.05) and the claimed scaling ξ(T_coh)=2.22ℓ, one finds k_F ξ ~ 350 and |Σ|/E_F ~ 10, i.e., the self-energy is no longer a small correction. The higher-order vertex diagrams of the same Gaussian model carry additional factors of |Δ0|² ξ/v_F and are therefore of order one in this regime; no small parameter justifies their neglect. The regularization by Γ0 makes the leading vertex finite, but it does not by itself make the one-loop truncation reliable. Thus the derivative maximum that defines ξ(T_coh), and the fitted ratio 2.22(3), may be an artifact of the truncation. The authors should either compute the exact disorder average for the Gaussian model (feasible in principle since the model is quadratic) to test the one-loop result, or provide an explicit estimate of the omitted diagrams and restrict the quantitative claim to a regime where the expansion parameter |Δ0|² ξ/v_F ≪ 1.","section":"III B, Eqs. (14a)-(14b), Fig. 7"},{"comment":"The paper does not provide any estimate of the size of the omitted higher-order self-energy and vertex diagrams. The only stated smallness condition is that Δ0 ≪ E_F (Sec. II A), but the actual expansion parameter for the disorder average is |Δ0|² ξ/v_F, which diverges as T → T_c. Even if the Dyson resummation of the leading self-energy is accepted, the consistency of the approximation requires a conservation law or a small dimensionless parameter; neither is demonstrated. The analogy with the Hebel-Slichter peak is suggestive, but in BCS theory the coherence peak is obtained from a nonperturbative anomalous Green's function, whereas here the normal-state vertex is computed only to leading order. I would like the authors to state clearly the range of ξ (or T) for which the perturbative treatment is valid, and to provide a numerical estimate of the first neglected diagram (e.g., the two-vertex diagram) in that range.","section":"II A, Eq. (3) and Sec. III B"},{"comment":"The magnetic pseudogap scale is identified with the maximum of the derivative of (1/T1T)_bd with respect to log(ξ_BCS/ξ). This is a reasonable operational definition, but the resulting coefficient ξ(T_mPG)/ξ_BCS ≈ 0.5–0.7 is obtained for a single Gaussian correlation function and for fixed Γ0. The paper would be strengthened by showing that this coefficient is insensitive to the shape of g(x) and to the regularization Γ0, or by stating explicitly that only the parametric scaling (Eq. 16) is claimed to be universal. Without such a check, the 'quantitative' hierarchy in Fig. 5 may be partly a model artifact.","section":"III A, Eq. (16), Fig. 5"}],"minor_comments":[{"comment":"The symbol ξ is used both for the BKT correlation length and for the single-particle dispersion ξ_k in Eq. (3). This is confusing; consider using ε_k or ζ_k for the dispersion.","section":"II A, Eq. (3)"},{"comment":"The statement that the temporal fluctuations are integrated out uses β ≪ ξ/v_F; this condition breaks down well above T_c where ξ becomes short. The model is therefore best justified only in a window near T_c, which conflicts with the use of the same formalism to describe the high-temperature Fermi-liquid limit.","section":"II A"},{"comment":"The comparison with the recent Monte Carlo study (Ref. [60]) is only qualitative. A side-by-side plot of the normalized 1/T1T from the two approaches, or a table of the extracted ξ(T_mPG) and ξ(T_coh) values, would substantially increase confidence in the leading-order vertex result.","section":"III B"},{"comment":"The linear regression in Fig. 7(d) is performed over four points at each β|Δ0|; the reported uncertainty 2.22(3) is the statistical fit error. It would be useful to state the number of independent Monte Carlo samples used in the vegas integration and the criteria for convergence, as the numerical error may be correlated across points.","section":"III B, Fig. 7(d)"},{"comment":"The phrase 'quantitatively in a unified picture' overstates the certainty given the uncontrolled truncation; suggest softening to 'semi-quantitatively' or adding a caveat about the perturbative regime.","section":"Introduction, final paragraph"}],"recommendation":"major_revision","confidential_remarks":"The paper is well written and addresses a topical problem. The qualitative predictions are attractive, and the independent Monte Carlo preprint offers support. My main reservation is the uncontrolled vertex expansion; this is a technical issue that could be resolved with an exact Gaussian-disorder calculation or a careful estimate of higher-order terms. I recommend major revision, not rejection, because the magnetic-pseudogap part is likely robust and the coherent-enhancement part has concurrent numerical support. I would ask the authors to either supply the missing control or clearly downgrade the quantitative status of ξ(T_coh)/ℓ."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper should be refereed, not desk-rejected. It gives a genuine analytic calculation of 1/T1T in phase-fluctuating 2D superconductors, including the leading vertex correction, and it is careful about what its model can and cannot do. The magnetic-pseudogap scale ξ(TmPG)~ξ_BCS is a natural corollary of the authors' earlier charge-pseudogap work, and the d-wave result that the vertex vanishes identically in the weak-fluctuating regime is a clean, verifiable statement. The s-wave case is where I get nervous.\n\nThe bubble contribution to 1/T1T is just the local-DOS product, so the pseudogap scale is inherited from the single-particle physics that the group already established. That part is robust, and the vertex correction is shown to be numerically negligible at that scale, which is reassuring. The new quantitative claim lives in the s-wave vertex correction: the coherence scale ξ(Tcoh)=2.22ℓ, Eq. (17) and Fig. 7(d). The problem is that this is a leading-order vertex diagram in an expansion whose small parameter is not small near Tc. On the Fermi surface the Born self-energy is |Δ0|²ξ/vF, which for their parameters (Δ0/EF=1/4, kFξ>>1) is of order unity once kFξ reaches ~10. There is no obvious suppression of the multi-vertex diagrams that have been dropped. So the number 2.22(3) could easily shift, or the peak could smear, once higher-order vertices are included. The qualitative existence of an s-wave normal-state coherence peak is supported by the numerical preprint [60], so I would not call the phenomenon an artifact — but the scaling ratio is a truncation-dependent prediction, not a robust consequence of the model.\n\nOther soft spots are minor and mostly acknowledged: the Gaussian correlation and temperature-independent Δ0 are modeling choices, the regularization Γ0 is physically motivated but phenomenological, and the whole calculation is static. None of these undermine the central claim about the magnetic pseudogap, which the authors appropriately frame as a corollary.\n\nWho should read it: anyone working on preformed pairs, pseudogaps, and NMR in quasi-2D superconductors. It is a useful and honest paper, and it deserves a serious referee. The referee should push the authors on the s-wave vertex convergence — ideally an estimate of the next-order contribution, a comparison with an exact Gaussian-disorder calculation, or at least a quantitative flag on the 2.22 prediction.","headline":"A clean, honest extension of the phase-fluctuation program to NMR; the s-wave vertex-correction scale 2.22ℓ is a truncation-dependent prediction that needs a closer look, but the magnetic-pseudogap result is robust and the paper deserves refereeing.","tokens_in":19018,"tokens_out":3900,"would_cite":true,"duration_ms":42045,"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":"Pure phase fluctuations, with no competing order, produce the magnetic pseudogap in two-dimensional superconductors and, for s-wave pairing, a normal-state coherence peak in $1/T_1T$.","keywords":["magnetic pseudogap","phase fluctuations","BKT transition","spin-lattice relaxation rate","NMR relaxation","Hebel-Slichter peak","vertex correction","two-dimensional superconductors"],"falsifier":"Run a sign-problem-free quantum Monte Carlo simulation of the attractive-U Hubbard model in two dimensions and compute the NMR relaxation rate $1/T_1T$, or perform NMR on a quasi-2D $s$-wave superconductor with tunable disorder. The central claim would be falsified if the normal-state $1/T_1T$ does not begin to drop when the BKT correlation length reaches $\\xi_{\\mathrm{BCS}}$, if an $s$-wave system with small $\\Gamma_0$ shows no normal-state coherence peak just above $T_c$, if a $d$-wave system shows such a peak, or if the onset of the $s$-wave peak obeys a ratio $\\xi(T_{\\mathrm{coh}})/\\ell$ far from $2.22(3)$ while the Gaussian correlator is in force.","tokens_in":17978,"feed_emoji":"🧲","tokens_out":13063,"duration_ms":106909,"temperature":0.7,"pith_summary":"This paper argues that the magnetic pseudogap observed in normal-state NMR relaxation — the smooth drop of $1/T_1T$ as temperature falls toward $T_c$ — can be generated entirely by superconducting phase fluctuations in two dimensions, without any competing order like antiferromagnetic spin fluctuations. Using a model with static Gaussian phase fluctuations of a uniform pairing amplitude $\\Delta_0$, the authors compute the spin susceptibility from the bubble graph and the leading vertex correction, and show that the bubble contribution already produces a magnetic pseudogap whose onset scale $T_{\\mathrm{mPG}}$ is set by $\\xi(T_{\\mathrm{mPG}}) \\sim \\xi_{\\mathrm{BCS}}$, where $\\xi(T)$ is the BKT correlation length and $\\xi_{\\mathrm{BCS}}=v_F/\\pi\\Delta_0$ is the BCS coherence length. For $s$-wave pairing the vertex correction diverges at $T_c$ because of the sharp coherence peak at the gap edge; once regularized by a scattering length $\\ell=v_F/\\Gamma_0$, it drives a coherent enhancement of $1/T_1T$ just above $T_c$, with scale $\\xi(T_{\\mathrm{coh}}) \\approx 2.22\\,\\ell$. For $d$-wave pairing the vertex correction is suppressed, so no such normal-state peak appears. If correct, the paper provides a unified scale hierarchy — $\\xi(T)$, $\\xi_{\\mathrm{BCS}}$, and $\\ell$ — that quantitatively organizes the full temperature evolution of $1/T_1T$ in quasi-2D superconductors, turning NMR measurements into a probe of these lengths.","feed_headline":"Phase fluctuations alone can create the magnetic pseudogap","feed_subtitle":"BKT, Cooper-pair, and scattering lengths organize the whole 1/T1T curve in 2D superconductors.","key_machinery":"The load-bearing machinery is the phase-fluctuating spectral function generated by the disorder-averaged pairing correlation. The two central identities are the Gaussian correlator $\\langle\\Delta(r)\\Delta^*(r')\\rangle = |\\Delta_0|^2 \\exp(-|r-r'|^2/2\\xi^2)$ and the self-energy $\\Sigma(k,\\omega) = |\\Delta_k|^2/(\\omega+\\xi_k+2i\\Gamma_k)$ with pair-breaking rate $\\Gamma_k = v_k/2\\xi$; this self-energy turns the BKT correlation length $\\xi(T)$ into a scale for the smearing of the spectral peaks, so that all single-particle pseudogap physics is expressed as a competition between $\\xi(T)$ and the BCS coherence length $\\xi_{\\mathrm{BCS}} = v_F/\\pi\\Delta_0$. On top of this, the Kubo formula $1/T_1T = (2|A_{\\mathrm{hf}}|^2/V)\\sum_q \\lim_{\\omega\\to 0^+} \\mathrm{Im}\\,\\chi^{-+}(q,\\omega)/\\omega$ converts the spectral functions into the spin response, with the bubble term (Eq. 13) carrying the magnetic pseudogap and the leading vertex term (Eqs. 14a, 14b) carrying the $s$-wave coherence peak whose divergence is regularized by the scattering length $\\ell = v_F/\\Gamma_0$. The emergence of all temperature scales from length ratios is what the authors call the hierarchy of scales.","core_discovery":"The central discovery, stated on the paper's own terms, is that a magnetic pseudogap — a smooth suppression of $1/T_1T$ beginning well above $T_c$ — is an unavoidable consequence of phase-only superconducting fluctuations in two dimensions, and that the same fluctuation physics reproduces the charge pseudogap and (for $d$-wave pairing) Fermi arcs without invoking any competing order. The calculation is carried out in a quadratic Hamiltonian with static, spatially fluctuating pairing $\\Delta(r)$ whose correlation is $\\langle\\Delta(r)\\Delta^*(r')\\rangle = |\\Delta_0|^2 e^{-|r-r'|^2/2\\xi^2}$, and the resulting Born self-energy $\\Sigma(k,\\omega)=|\\Delta_k|^2/(\\omega+\\xi_k+2i\\Gamma_k)$, with $\\Gamma_k=v_k/2\\xi$, supplies a pair-breaking rate controlled by the BKT correlation length. In the weak-fluctuating regime $k_F\\xi \\gg 1$, the bubble contribution to $1/T_1T$ drops rapidly when $\\xi(T)$ becomes comparable to $\\xi_{\\mathrm{BCS}}$, defining the magnetic-pseudogap scale $T_{\\mathrm{mPG}}$; numerically $\\xi(T_{\\mathrm{mPG}})/\\xi_{\\mathrm{BCS}}$ lies between about 0.5 and 0.7 and is essentially independent of $\\ell$. The leading vertex correction is exactly zero for $d$-wave pairing in this regime (the nodal form factor kills the angular integral), but for $s$-wave pairing it diverges at $T_c$ because the spectral function develops an infinitely sharp coherence peak at the gap edge. A finite background scattering rate $\\Gamma_0$, equivalently a length $\\ell=v_F/\\Gamma_0$, regularizes this divergence and turns it into a normal-state coherent enhancement of $1/T_1T$, with the numerical scaling $\\xi(T_{\\mathrm{coh}})/\\ell = 2.22(3)$; the authors identify this effect as the normal-state analogue of the Hebel-Slichter peak. The paper further extracts the dip scale $T_m$ and shows that the full $1/T_1T$ curve is organized by the competition among $\\xi(T)$, $\\xi_{\\mathrm{BCS}}$, and $\\ell$.","pith_inferences":["The same hierarchy of $\\xi(T)$, $\\xi_{\\mathrm{BCS}}$, and $\\ell$ likely controls other two-particle probes, such as the optical conductivity or Raman response, so the paper's picture could be tested without NMR; a normal-state coherence peak in $s$-wave systems might show up there too.","The Gaussian assumption for $\\langle\\Delta\\Delta^*\\rangle$ is an input, not a derivation; if real BKT fluctuations produce a different short-distance correlator, the numerical ratios 0.5–0.7 and 2.22(3) may change even though the qualitative competition of lengths survives.","A clean experimental test would be a quasi-2D $s$-wave superconductor with weak disorder: the theory predicts a double feature in $1/T_1T$ — first a pseudogap drop, then a rise just above $T_c$ — which is unusual and easy to look for.","Including spin fluctuations (which the paper explicitly sets aside) might suppress the $s$-wave coherence peak, as it does for the BCS Hebel-Slichter peak, so the predicted normal-state peak is most likely to be seen in systems with weak magnetic correlations."],"forward_implications":["In any quasi-2D superconductor with preformed pairs, $1/T_1T$ should start falling well above $T_c$ even with no spin gap, and the fall begins when the BKT correlation length $\\xi(T)$ is comparable to the BCS coherence length $\\xi_{\\mathrm{BCS}}$.","The magnetic-pseudogap temperature $T_{\\mathrm{mPG}}$ is a distinct energy scale, separate from both $T_c$ and $\\Delta_{\\mathrm{SC}}$, and can be predicted from the known BKT correlation length and the weak-coupling value $\\xi_{\\mathrm{BCS}}=v_F/\\pi\\Delta_0$.","For $s$-wave pairing, a coherent enhancement of $1/T_1T$ in the normal state just above $T_c$ should appear when the background scattering is weak enough ($\\ell > \\ell^*$), with its onset scale fixed by $\\xi(T_{\\mathrm{coh}}) \\approx 2.22\\,\\ell$; this is the phase-fluctuation analogue of the Hebel-Slichter peak.","For $d$-wave pairing, the vertex correction is negligible, so no normal-state coherence peak is expected; the absence of such a peak in nodal superconductors is consistent with the theory.","Measuring $\\xi(T_{\\mathrm{mPG}})$ and $\\xi(T_{\\mathrm{coh}})$ from NMR data would provide a direct extraction of $\\xi_{\\mathrm{BCS}}$ and $\\ell$, making $1/T_1T$ a quantitative probe of the coherence and scattering hierarchy."],"supporting_citations":[{"why":"Supplies the phase-fluctuating self-energy and the charge-pseudogap criterion that the magnetic-pseudogap analysis directly extends.","marker":"[24]"},{"why":"Provides the Gaussian pairing correlation and the d-wave Fermi-arc framework from which the d-wave vertex-cancellation is inherited.","marker":"[27]"},{"why":"Gives the BKT scaling form for the correlation length $\\xi(T)$ used to convert length ratios into temperature scales.","marker":"[16, 17]"},{"why":"Defines the BCS coherence length $\\xi_{\\mathrm{BCS}} = v_F/\\pi\\Delta_0$ that sets the magnetic-pseudogap scale.","marker":"[50]"},{"why":"The Hebel-Slichter coherence-peak phenomenon in BCS theory that the s-wave normal-state vertex divergence is compared with.","marker":"[29, 30]"},{"why":"Basis for the $1/T_1T \\sim N(0)^2$ relation and for regularizing the coherence-peak divergence with a background scattering rate.","marker":"[31]"},{"why":"Recent Monte Carlo study of BKT s-wave and d-wave superconductors cited as numerical support for the normal-state coherence peak and its d-wave absence.","marker":"[60]"},{"why":"Experimental NMR data on a layered FeSe superconductor with BKT behavior and pseudogap, used as a possible observation of the predicted magnetic pseudogap.","marker":"[10]"}],"fun_headline_variants":["Phase-only fluctuations yield magnetic pseudogap without competing order","Magnetic pseudogap scale set by BKT vs BCS coherence lengths in 2D","1/T1T curve explained by interplay of BKT, BCS, and scattering lengths","s-wave phase fluctuations give normal-state Hebel-Slichter-like 1/T1T peak"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The calculation assumes the pairing amplitude keeps a fixed size while its phase wanders with a Gaussian bell-shaped spatial correlation of width $\\xi$, that only the leading-order self-energy is kept, and that the weak-fluctuation regime $k_F\\xi \\gg 1$ holds; if the actual fluctuations are non-Gaussian, if the amplitude itself shrinks with temperature, or if the scattering rate $\\Gamma_0$ depends on temperature, the numerical ratios $\\xi(T_{\\mathrm{mPG}})/\\xi_{\\mathrm{BCS}}\\approx 0.5\\text{--}0.7$ and $\\xi(T_{\\mathrm{coh}})/\\ell = 2.22(3)$ could shift or smear.","fun_headline_variants_meta":{"raw":{"variants":["Phase-only fluctuations yield magnetic pseudogap without competing order","Magnetic pseudogap scale set by BKT vs BCS coherence lengths in 2D","1/T1T curve explained by interplay of BKT, BCS, and scattering lengths","s-wave phase fluctuations give normal-state Hebel-Slichter-like 1/T1T peak"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001734,"raw_usage":{"total_tokens":7082,"prompt_tokens":1404,"completion_tokens":5678,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":1020,"completion_tokens_details":{"reasoning_tokens":5588}},"tokens_in":1020,"tokens_out":5678,"duration_ms":40288,"temperature":1.0,"reasoning_tokens":5588,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T12:03:47.350463+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a sign-problem-free quantum Monte Carlo simulation of the attractive-U Hubbard model in two dimensions and compute the NMR relaxation rate $1/T_1T$, or perform NMR on a quasi-2D $s$-wave superconductor with tunable disorder. The central claim would be falsified if the normal-state $1/T_1T$ does not begin to drop when the BKT correlation length reaches $\\xi_{\\mathrm{BCS}}$, if an $s$-wave system with small $\\Gamma_0$ shows no normal-state coherence peak just above $T_c$, if a $d$-wave system shows such a peak, or if the onset of the $s$-wave peak obeys a ratio $\\xi(T_{\\mathrm{coh}})/\\ell$ far from $2.22(3)$ while the Gaussian correlator is in force.","supporting_citations":[{"cited_title":"Kwon and A","cited_arxiv_id":null,"evidence_quote":"Supplies the phase-fluctuating self-energy and the charge-pseudogap criterion that the magnetic-pseudogap analysis directly extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Gaussian pairing correlation and the d-wave Fermi-arc framework from which the d-wave vertex-cancellation is inherited."},{"cited_title":"Korringa, Physica16, 601 (1950)","cited_arxiv_id":null,"evidence_quote":"Defines the BCS coherence length $\\xi_{\\mathrm{BCS}} = v_F/\\pi\\Delta_0$ that sets the magnetic-pseudogap scale."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Basis for the $1/T_1T \\sim N(0)^2$ relation and for regularizing the coherence-peak divergence with a background scattering rate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Recent Monte Carlo study of BKT s-wave and d-wave superconductors cited as numerical support for the normal-state coherence peak and its d-wave absence."},{"cited_title":"Song, Y.-L","cited_arxiv_id":null,"evidence_quote":"Experimental NMR data on a layered FeSe superconductor with BKT behavior and pseudogap, used as a possible observation of the predicted magnetic pseudogap."}],"review_version":1}