{"id":"6a3322db-8f86-42b1-887d-44086625df5b","arxiv_id":"2412.14381","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Pressure boosts superfluid density by 75% in 4H-NbSe2 and 20% in 2H-NbS2, while Tc and the superconducting gaps stay nearly constant.","lead":"Squeezing two layered superconductors, 4H-NbSe2 and 2H-NbS2, increases their superfluid density by up to 75 percent, while barely changing the temperature at which they become superconducting. The result challenges the idea that superconductivity in these materials is governed mainly by competition with charge-density-wave order, and it links them to cuprate and iron-based superconductors.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 75% superfluid-density enhancement in 4H-NbSe2 depends on setting the unmeasured clean-limit factor 1/(1+ξ/l) in Eq. (1) to unity; if the mean free path is short or pressure-dependent, the reported effect could be inflated.","rationale":"I agree with the reader that the clean-limit assumption is the weakest point. I found no internal inconsistency in the gap analysis: the (s+s)-wave fits reproduce the data, and the pressure-independent gap values in Fig. 8 are consistent with the claim that pairing is unaffected. The measured quantity σ_sc itself is credible given the group's established methodology. The load-bearing issue is the step from σ_sc to n_s/m*. Eq. (1) is explicit: the conversion depends on 1/(1+ξ/l), and the paper sets this to unity because ξ is small. But small ξ does not justify l ≫ ξ; it only says the correction is small if l ≫ 3.5 nm. There is no estimate of l, and pressure is known from the paper's own transport data to alter electronic scattering (Hall sign change, magnetoresistance). Therefore the 75% figure is not yet uniquely pinned to superfluid density. A similar concern applies to the cross-material comparison in Fig. 5, where differing mean free paths could align or misalign slopes. The proposed residual-resistivity measurement under pressure, combined with a re-analysis using the full factor, is feasible with the same setups and would settle the issue. Because the reader already assigned CONDITIONAL, my recommendation is UNCHANGED: the paper should add the sensitivity analysis or the l(p) data before the quantitative headline is accepted as stated.","tokens_in":18002,"tokens_out":5475,"duration_ms":48513,"concrete_test":"Measure the residual resistivity ρ0(T→0) on crystals from the same batches at the same pressures used for µSR, and estimate the mean free path from the Drude expression l = m* v_F/(n e^2 ρ0) using literature Fermi-surface parameters. Then recompute the normalized n_s/m* from σ_sc at each pressure using the full expression λ_eff^{-2} = (4π n_s e^2/(m* c^2))/(1+ξ(p)/l(p)), with ξ(p) obtained from the pressure-dependent Hc2 already reported. If the inferred 75% enhancement is reduced by more than about 20 percentage points relative to the clean-limit value, the headline claim is not robust to the clean-limit assumption; if the recomputed enhancement remains within quoted statistical errors, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claim is the 75% increase in n_s/m* under pressure, obtained from the TF-µSR relaxation rate σ_sc. The conversion uses Eq. (1), σ_sc ∝ λ_eff^{-2} = (4π n_s e^2)/(m* c^2) × 1/(1+ξ/l), and the paragraph following Eq. (1) states that ξ/l has negligible effect because ξ ≈ 3.5–3.8 nm and 'no accurate estimate for l is available'. This is the least secure link in the argument. A finite mean free path does not by itself bias relative changes, but a pressure-dependent l would. Under 2 GPa the CDW is partially suppressed, the Hall response changes sign around T*, and the Fermi surface is argued to be reconstructed; any of these can alter the quasiparticle scattering rate. If, for example, l increases from about 10 nm to 25 nm between 0 and 2 GPa, the factor 1/(1+ξ/l) changes by roughly 25%, directly mimicking a superfluid-density enhancement. Without a measurement or bound on l(p), the 75% figure cannot be attributed purely to n_s/m*. The same caveat applies to the 20% result for 2H-NbS2 and to the cross-material slope in Fig. 5. This is not a claim that the effect is spurious; it is a statement that the size of the headline effect is conditional on an untested assumption. The paper itself flags the missing l, but does not supply the sensitivity analysis needed to judge the magnitude of the resulting error.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript reports transverse-field muon spin rotation (TF-muSR) and magnetotransport experiments on 4H-NbSe2 and 2H-NbS2 under hydrostatic pressure. The authors extract the superconducting transition temperature, the vortex-lattice relaxation rate, and the effective penetration depth, converting the latter into superfluid density ns/m*. They report a 75% increase in superfluid density for 4H-NbSe2 at 2.05 GPa, a 20% increase for 2H-NbS2 at 1.8 GPa, and compare these with a 32% increase previously reported for 2H-NbSe2. Magnetotransport is used to map the CDW onset and a Hall sign-reversal temperature in 4H-NbSe2, showing only modest CDW suppression under pressure. The temperature dependence of the relaxation rate is analyzed with a two-gap (s+s)-wave model, yielding nearly pressure-independent gaps. The paper argues for an unconventional correlation between Tc and superfluid density across TMDs and draws analogies with cuprates, kagome, and iron-based superconductors.","tokens_in":18377,"tokens_out":5935,"duration_ms":53310,"significance":"If the clean-limit conversion is valid, the reported 75% enhancement of superfluid density in 4H-NbSe2 is a striking and quantitatively important result. The study is valuable because it uses a direct bulk probe (TF-muSR), covers both a CDW-bearing and a CDW-free compound, and provides a systematic pressure comparison with 2H-NbSe2. The finding that the superconducting gaps are essentially pressure-independent while the superfluid density changes substantially would impose strong constraints on models of superconductivity in TMDs. However, the headline numbers depend on an unmeasured mean free path through Eq. (1), and the cross-material slope claim in Fig. 5 is not quantified. These issues need to be resolved before the quantitative claims are fully supported.","major_comments":[{"comment":"The central quantitative claim, the 75% enhancement of ns/m* in 4H-NbSe2 reported in Fig. 4(c), is obtained from Eq. (1) by setting the clean-limit factor 1/(1+xi/l) to unity. The text correctly states that no accurate estimate of l is available and justifies this by the small coherence length xi ~ 3.5-3.8 nm. This justification is not sufficient: for a finite mean free path of order 10 nm, the factor is about 0.74, and if l grows from 10 nm to 25 nm under pressure, the conversion factor alone changes by about 18%, a substantial fraction of the reported 75% enhancement. Since the Hall and magnetoresistance data in Fig. 2 indicate Fermi-surface reconstruction and changing scattering under pressure, a pressure-dependent l is plausible. The authors should either provide a bound on l(p) from independent measurements or supply a sensitivity analysis showing how the reported relative changes of sigma_sc translate into ns/m* for a range of xi/l values. Without this, the 75% and 20% figures cannot be attributed purely to changes in superfluid density.","section":"Eq. (1) and the following paragraph"},{"comment":"The statement that \"the slopes for 2H-NbSe2, 4H-NbSe2 and 2H-NbS2 are the same\" is not supported by any quantitative fit or uncertainty estimate. Figure 5 shows only a few points without visible error bars, and the three TMD data sets cover different pressure ranges and different numbers of points. To claim a universal Tc-ns/m* correlation, the authors should report slopes with confidence intervals and goodness-of-fit measures; otherwise the comparison with cuprates and iron-based superconductors remains qualitative.","section":"Fig. 5 and the discussion of it"},{"comment":"The pressure-cell background subtraction below Tc introduces a model-dependent linear coupling between sigma_pc and DeltaB_dia, with an unspecified function C(T). The value sigma_pc(T>Tc)=0.25 micros^-1 is given, but C(T) is not defined or constrained. Because roughly 60% of muons stop in the pressure cell rather than in the sample, the authors should demonstrate that the extracted sigma_sc(p) is robust to this modeling choice, or provide the form of C(T) and its uncertainty.","section":"Methods, Eq. (6) and the following text"},{"comment":"The 75% value is quoted from zero-temperature values obtained through the (s+s)-wave fits, which include two gaps and a weight factor, rather than directly from the low-temperature plateau of sigma_sc. Since the abstract and main text emphasize relative changes, the authors should show explicitly that the pressure-induced enhancement persists when the low-T data (for example, sigma_sc at 0.3 K) are ratioed directly, separated from the model-dependent two-gap extrapolation.","section":"Fig. 4 and the paragraph citing extracted parameters"}],"minor_comments":[{"comment":"The functional form in Eq. (3) is described as a two-component fit, but the sum runs from i=0 to 2, while the moment formulas in Eqs. (4) and (5) use As,1+As,2 as the denominator. Please fix the index convention.","section":"Methods, Eq. (3)"},{"comment":"Reference [26] is a placeholder with a 'LINK' for Supplemental Material; the actual link and the content of the Supplemental Material should be resolved before publication.","section":"Reference [26]"},{"comment":"The phrase 'emergence of a secondary CDW order appears to induce Fermi-surface reconstruction' is speculative: the transport data show a Hall sign reversal but do not directly identify a secondary CDW order. Please soften this statement or provide supporting evidence.","section":"Discussion of the Hall effect"},{"comment":"In the pressure-cell analysis, the text introduces sigma_ns and sigma_pc but does not repeat the separation sigma_sc = sqrt(sigma^2 - sigma_ns^2) for the pressure runs. Clarify how the sample and cell relaxation rates are separated at pressure, especially given the temperature-dependent cell term below Tc.","section":"Methods, pressure data analysis"},{"comment":"Figure 2 uses many line colors and symbols without a panel-specific legend in the caption; a consistent legend or a table of Tc, TCDW, and T* values would improve readability.","section":"Fig. 2 caption"},{"comment":"The sentence 'the ratio Tc/TF is reduced by approximately 20 times compared to cuprates' lacks a definition of TF and a reference for the cuprate value; please specify both.","section":"Final discussion"}],"recommendation":"major_revision","confidential_remarks":"The manuscript presents a direct muSR-based measurement and a systematic pressure study, and the central qualitative finding—that superfluid density grows under pressure in these TMDs—is likely robust. The main weakness is that the headline quantitative enhancement is presented with more confidence than the unmeasured mean free path allows. A sensitivity analysis or an independent estimate of l(p) would be sufficient to put the claim on firm ground. I do not see a citation or novelty concern beyond the usual need to frame the results in light of the authors' previous work on 2H-NbSe2."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a serious referee. The paper does a straightforward thing—pressure-dependent transverse-field muSR on two TMDs, 4H-NbSe2 and 2H-NbS2—and reports a large increase in vortex-lattice relaxation (interpreted as superfluid density) with pressure: 75% in 4H-NbSe2 at 2 GPa and 20% in CDW-free 2H-NbS2 at 1.8 GPa, while Tc rises only modestly and the two gaps stay about constant. The first p-T phase diagram of 4H-NbSe2 from magnetotransport is a useful addition.\n\nThe strongest part is the internal consistency. The relative change is taken from low-temperature plateaus of σsc, not from fit parameters, so the two-gap (s+s)-wave fits are not load-bearing for the main claim. The comparison between a CDW-bearing and a CDW-free compound is the right control, and the fact that 2H-NbS2 also shows an enhancement is the cleanest evidence that the effect is not simply CDW suppression.\n\nThe main soft spot is the conversion from σsc to superfluid density. Eq. (1) has the clean-limit factor 1/(1+ξ/l) set to unity because ξ is small and no estimate of l is available. That is a reasonable guess, but it is an assumption. Under pressure the Fermi surface is argued to be reconstructed and the Hall response changes, so l(p) could well change. If l grows over the pressure range, part of the 75% would be a mean-free-path effect rather than a superfluid-density increase. The paper flags the missing l, but it does not bound the error. A short paragraph with a sensitivity analysis—say, l between 10 and 50 nm—would fix this.\n\nSecond, the headline percentages have no quoted uncertainties. The data look clean, but the reader cannot judge whether 75% vs 60% matters. Third, the cross-family comparison to cuprates, kagome, and iron-based systems in Fig. 5 is qualitative. The slopes for the TMDs cluster together, and that is defensible; calling it a shared mechanism is a stretch beyond the data.\n\nNone of these sink the central claim. The direction of the effect—pressure increases the low-temperature relaxation rate in both systems—is solid. The magnitude is conditional.\n\nThis paper is for people working on TMD superconductivity and pressure-muSR; it will be cited. Send it to peer review. The authors should add error bars, justify or measure l, and soften the cross-family language.","headline":"Solid pressure-muSR data on two TMDs, with a real caveat about the clean-limit conversion that inflates the headline 75% number; worth peer review.","tokens_in":19026,"tokens_out":2346,"would_cite":true,"duration_ms":20103,"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":"Hydrostatic pressure raises the superfluid density of the layered superconductor 4H-NbSe2 by 75%, while leaving its superconducting gaps nearly unchanged.","keywords":["superfluid density","muon spin rotation","hydrostatic pressure","charge density wave","transition metal dichalcogenides","4H-NbSe2","2H-NbS2","unconventional superconductivity"],"falsifier":"Measure the residual resistivity or use de Haas–van Alphen oscillations on the same crystals under pressure to estimate the mean free path $l$. If $l$ decreases by roughly a factor of two over 2 GPa, the $1/(1+\\xi/l)$ correction would grow enough that the apparent 75% rise in $\\lambda_{\\rm eff}^{-2}$ could be largely a mean-free-path artifact rather than a true increase in $n_s/m^*$.","tokens_in":17771,"feed_emoji":"🧲","tokens_out":9166,"duration_ms":66626,"temperature":0.7,"pith_summary":"This paper reports that compressing the layered superconductor 4H-NbSe$_2$ to about 2 GPa increases its superfluid density—the density of electrons that carry the superconducting current—by 75%, while the superconducting transition temperature rises only modestly. A smaller but robust 20% increase is seen in 2H-NbS$_2$, a compound with no charge-density-wave order. The authors argue that the large superfluid response cannot be attributed simply to pressure suppressing the competing charge-density-wave state, because the CDW is only mildly reduced and the increase is even larger in the compound without a CDW. The near-universal correlation between superfluid density and $T_c$ across these TMDs, cuprates, kagome, and iron-based superconductors suggests a common unconventional pairing mechanism. If correct, this points to pressure as a clean tuning knob for superfluid stiffness that is largely decoupled from $T_c$.","feed_headline":"Pressure boosts superfluid density 75% in 4H-NbSe2","feed_subtitle":"Muon spin rotation shows the rise is not tied to charge-density-wave collapse, hinting at a shared unconventional mechanism.","key_machinery":"The measurement engine is transverse-field muon spin rotation (TF-µSR): implanted muons precess in the vortex lattice of the superconductor, and the Gaussian relaxation rate $\\sigma_{sc}$ is converted to an effective magnetic penetration depth $\\lambda_{\\rm eff}$ through the vortex-lattice relation $\\sigma_{sc}/\\gamma_\\mu = 0.06091\\,\\Phi_0\\, \\lambda_{\\rm eff}^{-2}$ (Brandt's result). Under the clean-limit assumption (coherence length $\\xi$ much larger than mean free path $l$, so the factor $1/(1+\\xi/l)\\approx 1$), $\\lambda_{\\rm eff}^{-2}$ is proportional to the superfluid density $n_s/m^*$. The temperature dependence is analyzed with an $(s+s)$-wave two-gap model with pressure-independent weight factors, so the zero-temperature superfluid density can be tracked versus pressure. Magnetotransport (resistivity, Hall, magnetoresistance) independently locates $T_c$ and the CDW onset.","core_discovery":"The central claim is that hydrostatic pressure strongly enhances the zero-temperature superfluid density $n_s/m^*$ in niobium-based transition metal dichalcogenides, with the largest effect in 4H-NbSe$_2$: a 75% increase relative to ambient pressure at 2.05 GPa. In 2H-NbS$_2$, which has no charge-density-wave order, the enhancement is 20% at 1.8 GPa, and in 2H-NbSe$_2$ it is 32% at 2.2 GPa (from prior work). In all cases the superconducting gaps remain essentially pressure-independent, and $T_c$ changes only by about 0.2–0.9 K. Because the CDW onset temperature falls by only about 20% over the same pressure range in the selenides, the authors conclude that the superfluid-density enhancement is not a simple consequence of CDW suppression, and they highlight a pressure-independent slope in the $T_c$ versus $n_s/m^*$ scaling that resembles cuprate and iron-based superconductors.","pith_inferences":["A testable extension not pursued here is to measure the mean free path (from residual resistivity or quantum oscillations) under the same pressures; this would test whether the clean-limit assumption holds and whether the 75% is entirely a superfluid-density effect.","The authors' explanation invokes pressure-driven changes in electron-phonon coupling, $p$-$d$ hybridization, and a saddle point near the Fermi level; first-principles band-structure calculations under pressure could verify which of these dominates.","The scaling plot places these TMDs on a Uemura-type correlation, which in cuprates has been read as BEC-like pairing; confirming that would require an independent measure of the effective mass $m^*$ to separate changes in $n_s$ from changes in $m^*$.","If the effect is a stiffness enhancement decoupled from $T_c$, uniaxial strain or chemical pressure in related TMDs might achieve similar or larger superfluid-density gains without raising $T_c$."],"forward_implications":["A 75% increase in superfluid density at only 2 GPa means pressure can nearly double the supercurrent-carrying capacity of 4H-NbSe$_2$ without a corresponding rise in $T_c$, a response that standard BCS theory does not predict.","The same slope in the $T_c$ vs $n_s/m^*$ scaling across 2H-NbSe$_2$, 4H-NbSe$_2$, and 2H-NbS$_2$ indicates a common microscopic mechanism for the pressure effect in the NbX$_2$ family.","Because 2H-NbS$_2$ shows a 20% enhancement with no CDW, a quantitative theory of these compounds must explain superfluid-density growth through a channel other than CDW suppression—such as pressure-driven changes in electron-phonon coupling, $p$-$d$ hybridization, or Fermi-surface topology.","The pressure independence of the two superconducting gaps constrains the pairing: pressure changes the condensate fraction without changing the gap magnitudes, which is consistent with phase-fluctuation or stiffness-driven pictures.","The addition of these TMDs to the Uemura-type scaling plot (with cuprates, iron-based, and kagome superconductors) suggests that a common mechanism may link superfluid density and $T_c$ across very different materials."],"supporting_citations":[{"why":"Prior µSR study of 2H-NbSe2; supplies the 32% baseline and the comparative pressure data, plus the sample synthesis route.","marker":"[1]"},{"why":"Provides the 1T′-MoTe2 data point in the Tc–superfluid-density plot and the high-pressure µSR methodology.","marker":"[2]"},{"why":"Characterizes 4H-NbSe2 (structure, CDW, superconductivity, upper critical field used for the coherence-length estimate).","marker":"[9]"},{"why":"Reports the upper critical field and FFLO evidence in NbS2 used to estimate the coherence length.","marker":"[10]"},{"why":"Derives the vortex-lattice relation connecting the muon relaxation rate to the penetration depth, the quantitative backbone of the superfluid-density extraction.","marker":"[27]"},{"why":"Pressure-dependent µSR on a Fe-based superconductor, providing the comparison data in the scaling plot.","marker":"[34]"},{"why":"Pressure-dependent µSR on cuprate La2-xBaxCuO4, providing the cuprate comparison in the scaling plot.","marker":"[35]"},{"why":"Original Uemura correlation between Tc and superfluid density in cuprates; the benchmark for the claimed unconventional scaling.","marker":"[46]"}],"fun_headline_variants":["Pressure surge lifts superfluid density 75% in NbSe2","Superfluid density jumps 75% under pressure in dichalcogenide","Pressure boosts superfluid density with and without CDW","Pressure reveals unconventional superfluid scaling in TMDs","75% superfluid density gain under pressure in NbSe2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The reported superfluid-density increase assumes the clean limit, where the mean free path is long enough that the $\\xi/l$ correction in the penetration-depth relation is negligible; the paper does not measure the mean free path.","fun_headline_variants_meta":{"raw":{"variants":["Pressure surge lifts superfluid density 75% in NbSe2","Superfluid density jumps 75% under pressure in dichalcogenide","Pressure boosts superfluid density with and without CDW","Pressure reveals unconventional superfluid scaling in TMDs","75% superfluid density gain under pressure in NbSe2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000323,"raw_usage":{"total_tokens":1934,"prompt_tokens":1182,"completion_tokens":752,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":798,"completion_tokens_details":{"reasoning_tokens":667}},"tokens_in":798,"tokens_out":752,"duration_ms":6491,"temperature":1.0,"reasoning_tokens":667,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:16:58.990022+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the residual resistivity or use de Haas–van Alphen oscillations on the same crystals under pressure to estimate the mean free path $l$. If $l$ decreases by roughly a factor of two over 2 GPa, the $1/(1+\\xi/l)$ correction would grow enough that the apparent 75% rise in $\\lambda_{\\rm eff}^{-2}$ could be largely a mean-free-path artifact rather than a true increase in $n_s/m^*$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Prior µSR study of 2H-NbSe2; supplies the 32% baseline and the comparative pressure data, plus the sample synthesis route."},{"cited_title":"pan- cake","cited_arxiv_id":null,"evidence_quote":"Provides the 1T′-MoTe2 data point in the Tc–superfluid-density plot and the high-pressure µSR methodology."},{"cited_title":"Zhang, A","cited_arxiv_id":null,"evidence_quote":"Characterizes 4H-NbSe2 (structure, CDW, superconductivity, upper critical field used for the coherence-length estimate)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports the upper critical field and FFLO evidence in NbS2 used to estimate the coherence length."},{"cited_title":"[34, 56–62]","cited_arxiv_id":null,"evidence_quote":"Derives the vortex-lattice relation connecting the muon relaxation rate to the penetration depth, the quantitative backbone of the superfluid-density extraction."},{"cited_title":"El Youbi, S","cited_arxiv_id":null,"evidence_quote":"Pressure-dependent µSR on a Fe-based superconductor, providing the comparison data in the scaling plot."},{"cited_title":"Guguchia, A","cited_arxiv_id":null,"evidence_quote":"Pressure-dependent µSR on cuprate La2-xBaxCuO4, providing the cuprate comparison in the scaling plot."},{"cited_title":"Majumdar, D","cited_arxiv_id":null,"evidence_quote":"Original Uemura correlation between Tc and superfluid density in cuprates; the benchmark for the claimed unconventional scaling."}],"review_version":1}