{"id":"030f3864-11a8-4abc-9e9f-23062e94eff2","arxiv_id":"2411.18518","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":8,"one_line_summary":"Muon spin relaxation and DFT+U indicate that the kagome MOF Cu3(HOTP)2 is a clean, gapless 2D quantum spin liquid whose excitations are best matched by a Z2-linear Dirac model with additional singlet excitations.","lead":"Muon spin relaxation shows that the kagome metal-organic framework Cu3(HOTP)2 does not magnetically order down to 50 mK, and its spin fluctuations follow 2D diffusion, consistent with a quantum spin liquid. The study also extracts entanglement-related parameters from the muon data and argues that a Z2-linear Dirac model best describes the low-energy excitations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 2D-diffusion assignment of the muon data is not tested against alternative spectral-density models; all derived D2D, A^2, entanglement, and Dirac-model conclusions inherit this unvalidated identification.","rationale":"The reader's identification of the temperature-independent hyperfine coupling as weak is correct but somewhat narrow. The deeper issue is that Eq. (2) is the only bridge between raw muon spectra and every derived physical quantity; the field-dependence data are shown for only two temperatures and the text gives no model-selection statistics. I therefore checked whether this is the most load-bearing step. If the 2D-diffusion form is not uniquely favoured, then the '2D lattice' statement in the abstract is not proven by this experiment, and the subsequent entanglement and Z2-Dirac model selection rests on parameters whose meaning is not fixed. I am not claiming the model is wrong: the no-ordering result is robust, prior thermodynamic data support a QSL-like ground state, and the raw data are deposited, which makes the proposed refit feasible. The concrete test comparing AIC/BIC across spectral-density families would directly settle whether the D2D values and their T-dependence are unique. Until that test is done, the appropriate verdict remains conditional.","tokens_in":17706,"tokens_out":12240,"duration_ms":120803,"concrete_test":"Refit the published raw LF-muon data (DOI:10.5286/ISIS.E.RB2010775) at every temperature with at least four spectral-density models: Eq. (1) 2D diffusion; Lorentzian J(omega)=C/(1+(omega/omega0)^2); Fourier transform of a stretched exponential; and Eq. (1) plus the low-field Lorentzian term of Sec. III C 5. Compare maximum-likelihood fits via AIC/BIC and residual plots. Then recompute D2D(T) and A^2(T) for every model not decisively rejected; if D2D shifts by more than the reported error bar under an equally good fit, the 2D-diffusion identification and its downstream QSL conclusions are underdetermined.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Everything quantitative in Secs. III C 2-C 6 and Table I hangs on the pair (D2D, A^2) obtained by fitting Eq. (2) with the single functional form J2D(omega) from Eq. (1). The paper states (Fig. 7) that 2D diffusion gives 'a good representation' and that a 1D power law is 'much poorer', but no residuals, chi-squared values, or information criteria are reported, and no other spectral shapes are tested. A Lorentzian, a stretched-exponential spectrum, or a two-component 2D+localized model could in principle reproduce the same field dependence within noise; if so, D2D(T) in Fig. 8(a) and the A^2 drop in Fig. 8(b) are not uniquely determined, and the localized fraction (Sec. III C 5), the quantum Fisher information FQ (Eq. 6), the quantum-critical exponents (Table I), and the dimensional-reduction estimate would all lose their quantitative foundation. The assumption that A is temperature-independent (Sec. III C 5, 'We do not expect strong T dependence in A') is part of this misspecification risk: a T-dependent muon-site population or hyperfine tensor would produce the observed A^2 drop without any localized spin excitations. The no-ordering conclusion from ZF muons is robust, but the '2D spin diffusion' evidence is not.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents muon spin relaxation (muSR), DFT+U, and modelling results for the kagome metal-organic framework Cu3(HOTP)2. The authors report the absence of magnetic ordering down to 50 mK, interpret the longitudinal-field relaxation in terms of a 2D spin-diffusion spectral density, and extract a temperature-dependent diffusion rate D2D and an effective hyperfine coupling A^2. From these they infer a classical-quantum crossover, a localized fraction of excitations at low temperature, a dimensional reduction of interlayer coupling, and an entanglement length via the quantum Fisher information. Combining these with literature specific-heat and susceptibility exponents, they argue that a Z2-linear Dirac QSL model with additional singlet excitations provides the best match to experiment.","tokens_in":18040,"tokens_out":4257,"duration_ms":40517,"significance":"The no-ordering result is robust and the compound is a clean candidate kagome QSL with a conveniently small exchange scale J ~ 2 K; the muon data are deposited and the material is of clear interest to the frustrated-magnetism community. The paper also proposes a relatively new route to estimating entanglement from muSR. However, the quantitative claims about localized excitations, entanglement growth, and QSL model selection all depend on the unvalidated assumption that the relaxation is described by the specific 2D-diffusion spectral density of Eq. (1) with a temperature-independent hyperfine coupling. If that identification is wrong, most of the quantitative results lose their foundation. The paper is therefore a useful experimental contribution but its central interpretive claims require substantial additional support.","major_comments":[{"comment":"The identification of the spectral density as 2D spin diffusion is load-bearing for the rest of the paper, but it is supported only by visual comparison. No residuals, chi-squared values, or information criteria are reported, and the competing 1D power law is dismissed as 'much poorer' without quantitative support. A Lorentzian, stretched-exponential, or a two-component 2D+localized spectrum could in principle reproduce the same field dependence within noise; if so, D2D(T) and A^2(T) in Fig. 8 are not uniquely determined. The authors should report goodness-of-fit statistics and explicitly test alternative spectral shapes before using D2D and A^2 in the subsequent analysis.","section":"III C 2, Eq. (2) and Fig. 7"},{"comment":"The drop of the effective A^2 to about 2/3 of its high-temperature value is assigned to a reduced diffusive fraction, based on the statement 'We do not expect strong T dependence in A'. This is an assumption, not a demonstrated result. A temperature-dependent muon site population or hyperfine tensor would produce the same A^2 drop without any localized spin excitations. The later interpretation of a low-field Lorentzian component as localized excitations therefore also rests on this assumption. The authors should either provide independent evidence for a temperature-independent A or quantify the systematic uncertainty this introduces.","section":"III C 5, Fig. 8(b)"},{"comment":"The quantum Fisher information FQ is computed from the same spectral density J2D(ω) that was fitted to the muon relaxation rates in Eq. (2). The observation that FQ follows the universal T^-3/4 law is therefore not an independent test of the theoretical prediction; it is a consequence of the fitted spectral shape. The paper should explicitly state that the T^-3/4 agreement is a model-based consistency check, not an independent validation, and should present the comparison with that caveat.","section":"III C 6, Eq. (6)"},{"comment":"The DFT+U calculation of J is tuned by varying U until the calculated J matches the experimental value of 2 K. The agreement is therefore a fit, not a first-principles prediction. The paper should show the full J(U) dependence with the experimental uncertainty and justify that the chosen U values are within the expected accuracy of DFT+U for this system. As written, the statement that the experimental J is 'matched for reasonable values of U' overstates the predictive content.","section":"III A 3, Fig. 3(b)"},{"comment":"The model selection in Table I combines the muon-derived exponent nD with the literature exponents nχ and nC via the scaling relations of Eqs. (7)-(9). Since nD is extracted from the model-dependent D2D(T), and nχ and nC come from a different experiment, the conclusion favouring a Z2-linear Dirac QSL is only as strong as the weakest of these inputs. If the 2D-diffusion identification is not secured, the exclusion of spinon-Fermi-surface and quartic-dispersion models in Table I is premature. The discussion should be softened to reflect this dependence.","section":"IV A 1, Table I"}],"minor_comments":[{"comment":"There is a typo: 'orgin' should be 'origin'.","section":"IV B"},{"comment":"There is a typo: 'fittted' should be 'fitted'.","section":"III C 4"},{"comment":"The 'power law fit' in Fig. 7 is not defined; the exponent and functional form should be stated in the caption or text.","section":"III C 2, Fig. 7"},{"comment":"The estimate of the hyperfine coupling A0 uses a single model molecule and one muon site. The uncertainty in A0 should be discussed, since A0 is used as the reference for fdiff in Fig. 8(b).","section":"II C"},{"comment":"The term 'Z2-linear Dirac state' is used without a definition of the spinon dispersion or a discussion of how it differs from the U(1) Dirac state in the present context; a brief explanation would improve accessibility.","section":"IV A 2"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a valuable experimental dataset and a solid no-ordering result for a clean kagome MOF. The main risk is that the central interpretive claims are built on an unvalidated spectral-density model, and the DFT+U part is a parameter match rather than a prediction. In revision, the authors should either provide quantitative model-selection evidence for 2D diffusion or substantially restrict the claims to those that are robust to the spectral-shape uncertainty. The paper currently overreaches in its QSL-model conclusions, but the issues are fixable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The paper has a solid core: the muon data convincingly show no magnetic ordering down to 50 mK, and the field-dependent relaxation is fit reasonably by a 2D spin diffusion spectral density. That makes Cu3(HOTP)2 a useful new kagome QSL platform, especially since it lacks the defect spins that complicate herbertsmithite. The electrostatic stacking-frustration model is a nice, concrete contribution, and the estimate of interlayer coupling (0.02(1) K at 25 K, with a large low-T anisotropy) is physically sensible and broadly consistent with the broken-symmetry DFT.\n\nThe paper is less convincing where it builds on the fitted spectral density. The reader and the stress test are right: J2D(ω) is the only functional form tried; a 1D power law is dismissed without reporting residuals or an information criterion. Everything quantitative—D2D(T), the A^2 drop, the localized fraction, FQ, the exponent nD, and the Table I model selection—inherits that choice. If a Lorentzian or a two-component spectrum fits the same field scans within noise, the entanglement and Dirac-model conclusions lose their foundation. The assumption that A is temperature-independent is also load-bearing; a T-dependent muon site population or hyperfine tensor could produce the observed A^2 reduction without any localized spin excitations. The authors flag this assumption, which is honest, but they don't test it.\n\nThe Table I discrimination is weaker than it looks. nD = 0.03(3) is consistent with zero, and nχ and nC are taken from previous work. The conclusion that a linear Dirac spinon plus quadratic singlets matches experiment is a plausible consistency argument, not a unique model selection. The DFT+U J matching by tuning U is a known limitation, though the range shown is reasonable.\n\nSo the central no-ordering claim holds, and the 2D diffusion picture is plausible; the specific quantitative conclusions should be treated as model-dependent interpretations. I'd send this to peer review, asking the authors to add model-comparison statistics, address the T-dependence of A, and soften the entanglement claims. A serious referee can get good value from this paper.","headline":"The no-ordering result and the clean kagome platform are solid; the entanglement, Dirac-model, and dimensional-reduction conclusions are model-dependent and need less confidence.","tokens_in":18614,"tokens_out":3224,"would_cite":true,"duration_ms":30060,"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":"Muon spin relaxation shows Cu3(HOTP)2 is a gapless kagome quantum spin liquid down to 50 mK.","keywords":["quantum spin liquid","kagome lattice","muon spin relaxation","metal-organic framework","spin diffusion","quantum entanglement","Z2 Dirac spin liquid","Cu3(HOTP)2"],"falsifier":"Measure the muon hyperfine coupling directly versus temperature, for example via avoided level-crossing resonance on the same powder; a temperature-dependent A would invalidate the localized-fraction and entanglement-length interpretation. Separately, high-resolution inelastic neutron scattering that resolves a spin gap above about 0.02 J would rule out the claimed gapless Z2-linear Dirac spectrum.","tokens_in":17514,"feed_emoji":"🧲","tokens_out":8568,"duration_ms":72894,"temperature":0.7,"pith_summary":"Cu3(HOTP)2, a semiconducting metal-organic framework with S=1/2 copper spins on a kagome lattice, is argued here to be a clean realization of a gapless quantum spin liquid. Muon spin relaxation shows no magnetic order down to 50 mK, and the field dependence of the relaxation rate matches the spectral density of spins diffusing in two dimensions rather than a one-dimensional power law. Combining the muon-derived exponent nD = 0.03 with reported susceptibility and specific heat exponents rules out spinon Fermi surface, quadratic, and quartic dispersions, and singles out a Z2-linear Dirac spectrum with linearly dispersing spinons and quadratic singlet excitations. The paper also presents a DFT+U calculation that reproduces the intra-layer exchange J ≈ 2 K, an electrostatic model in which frustrated layer stacking explains the material's semiconducting behavior, and muon evidence that the spin layers decouple further and the entanglement length grows roughly threefold as the system enters the quantum regime.","feed_headline":"Kagome MOF stays a spin liquid down to 50 millikelvin","feed_subtitle":"Muon relaxation fits 2D spin diffusion and singles out a Z2-linear Dirac spectrum for the excitations.","key_machinery":"The central object is the 2D spin-diffusion spectral density J2D(ω), the Fourier transform of S2D(t) = [exp(−2D2D t) I0(2D2D t)]^2, which enters the muon relaxation rate as λ(B) = ($A^{2}$/4) J2D(γ_e B_LF). Fitting this formula to the field dependence of the relaxation rate yields the in-layer spin diffusion rate D2D and the effective hyperfine scaling $A^{2}$. From D2D the paper derives the mean free path through D2D = v/(2l), identifies it with an entanglement length, and independently estimates entanglement from the quantum Fisher information FQ built from the same spectral density. DFT+U broken-symmetry calculations provide the intra-layer exchange and interlayer coupling, while a model of electrostatically frustrated slip stacking explains the disordered stacking that turns the ordered metallic band structure into the observed 2D semiconductor.","core_discovery":"The central claim is that the kagome layers of Cu3(HOTP)2 host a gapless Z2-linear Dirac quantum spin liquid and that muon spin relaxation can see both its diffusive spin dynamics and its entanglement. The evidence is a chain: zero-field and longitudinal-field muon data show no static order to 50 mK; the relaxation rate versus field follows the 2D spin-diffusion spectral density J2D(ω), not a 1D power law; the fitted diffusion rate D2D is nearly constant below about J/kB = 2.6 K and increases about threefold in the paramagnetic regime; and the effective squared hyperfine coupling falls to about 2/3 of its high-temperature value, which the authors interpret as localization of one-third of the spin excitations. When the muon exponent nD = 0.03 is combined with the reported susceptibility exponent nχ = −0.31 and specific heat exponent nC = 0.52, spinon Fermi surface, quadratic, and quartic spinon models are excluded, while linearly dispersing spinons with correlation-length exponent ν = 0.673(7) plus quadratic singlet (vison-like) excitations reproduce all three. The paper further estimates that the effective interlayer coupling drops by roughly two orders of magnitude at low temperature, attributing this dimensional reduction to quantum entanglement reducing the number of unentangled spins available for interlayer exchange.","pith_inferences":["If the Z2-linear Dirac assignment is correct, the vison (singlet) sector should show up in thermal transport or a thermal Hall signal at temperatures below roughly J/kB; this is a testable prediction the paper does not make.","The same muon-based inversion from J2D(ω) to D2D and FQ could be applied to other layered kagome MOFs with tunable J, turning the entanglement metric into a systematic probe of how frustration, layer spacing, and stacking disorder shape spin-liquid behavior.","A single-crystal or aligned-film muon measurement could test whether part of the low-field Lorentzian component assigned to localized excitations is actually a powder-averaging artifact of the 2D spectral density.","Controlling the stacking slip angle, for example by intercalation or applied pressure, could tune the interlayer decoupling and thereby provide a direct test of the entanglement-based explanation of dimensional reduction."],"forward_implications":["Cu3(HOTP)2 becomes a benchmark kagome quantum spin liquid: the interlayer magnetic coupling is tiny, the exchange scale J ≈ 2 K puts both the quantum and classical regimes within reach of a single experiment, and there are no interlayer metal sites to create the defect spins that complicate herbertsmithite.","The three measured exponents nD = 0.03, nχ = −0.31, and nC = 0.52 form a combined constraint that any proposed kagome QSL model must satisfy, and they already exclude Fermi-surface, quadratic, and quartic spinon dispersions.","If the low-temperature interlayer decoupling is real, it implies that quantum entanglement itself can suppress interlayer magnetic exchange, giving a dimensional-reduction mechanism in a layered QSL analogous to that found near quantum critical points in dimer systems.","The electrostatic frustration model for layer stacking predicts intrinsically disordered, non-periodic stacking in this MOF family, linking the structural disorder to the measured semiconducting rather than metallic transport.","The upper bound on the spin gap of roughly 0.02 J places this material among the gapless or nearly gapless kagome candidates, sharpening the experimental distinction between gapped and gapless QSL scenarios."],"supporting_citations":[{"why":"Supplies the prior experimental characterization—susceptibility J/kB ≈ 2 K, specific heat exponent nC = 0.52, susceptibility exponent nχ = −0.31, no ordering to 38 mK—that the muon results are combined with to discriminate QSL models.","marker":"[29]"},{"why":"Provides the 2D spin-diffusion autocorrelation function S2D(t) whose Fourier transform J2D(ω) is the central spectral density used to fit the muon relaxation field dependence.","marker":"[56]"},{"why":"Defines the quantum Fisher information metric FQ and its universal T^{-3/4} high-temperature behavior, used to quantify entanglement from the measured spectral density.","marker":"[58]"},{"why":"Reports the Z2-linear Dirac variational state whose fluctuation spectrum matches the U(1) Dirac state; the paper's best-match model for the measured exponents.","marker":"[16]"},{"why":"Gives the Z2 spin liquid dynamics formulas, including the momentum relaxation time power law 2/ν − 3 used to derive nD from the correlation length exponent ν.","marker":"[61]"},{"why":"Establishes the muon-based method for extracting nD and nC in a layered QSL (1T-TaS2) and the critical-exponent relations used to compare models.","marker":"[54]"},{"why":"Supplies the precise O(N) critical exponents, in particular ν = 0.672 for O(2), against which the measured ν = 0.673(7) is matched.","marker":"[62]"},{"why":"Reports the synthesis of M3(HHTP)2 MOFs including the Cu compound whose sample preparation recipe is followed here.","marker":"[25]"}],"fun_headline_variants":["Kagome MOF confirms 2D Dirac spin liquid at 50 mK","Muon spin relaxation in kagome MOF unveils Dirac quantum spin liquid","Kagome MOF spin liquid: no order at 50 mK, Dirac spectrum","Z2-linear Dirac spin liquid in kagome MOF at 50 mK"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The analysis assumes the muon's effective hyperfine coupling A is temperature-independent, so the measured drop in $A^{2}$ at low temperature can be assigned entirely to a reduced fraction of diffusive spin excitations; if A itself changed with temperature, or the muon site shifted, the localized fraction and entanglement-length estimates would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Kagome MOF confirms 2D Dirac spin liquid at 50 mK","Muon spin relaxation in kagome MOF unveils Dirac quantum spin liquid","Kagome MOF spin liquid: no order at 50 mK, Dirac spectrum","Z2-linear Dirac spin liquid in kagome MOF at 50 mK"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000969,"raw_usage":{"total_tokens":4195,"prompt_tokens":1095,"completion_tokens":3100,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":711,"completion_tokens_details":{"reasoning_tokens":3011}},"tokens_in":711,"tokens_out":3100,"duration_ms":23271,"temperature":1.0,"reasoning_tokens":3011,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:07:20.829468+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the muon hyperfine coupling directly versus temperature, for example via avoided level-crossing resonance on the same powder; a temperature-dependent A would invalidate the localized-fraction and entanglement-length interpretation. Separately, high-resolution inelastic neutron scattering that resolves a spin gap above about 0.02 J would rule out the claimed gapless Z2-linear Dirac spectrum.","supporting_citations":[{"cited_title":"Nishimoto, N","cited_arxiv_id":null,"evidence_quote":"Supplies the prior experimental characterization—susceptibility J/kB ≈ 2 K, specific heat exponent nC = 0.52, susceptibility exponent nχ = −0.31, no ordering to 38 mK—that the muon results are combined with to discriminate QSL models."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the 2D spin-diffusion autocorrelation function S2D(t) whose Fourier transform J2D(ω) is the central spectral density used to fit the muon relaxation field dependence."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the quantum Fisher information metric FQ and its universal T^{-3/4} high-temperature behavior, used to quantify entanglement from the measured spectral density."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports the Z2-linear Dirac variational state whose fluctuation spectrum matches the U(1) Dirac state; the paper's best-match model for the measured exponents."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Z2 spin liquid dynamics formulas, including the momentum relaxation time power law 2/ν − 3 used to derive nD from the correlation length exponent ν."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the muon-based method for extracting nD and nC in a layered QSL (1T-TaS2) and the critical-exponent relations used to compare models."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the precise O(N) critical exponents, in particular ν = 0.672 for O(2), against which the measured ν = 0.673(7) is matched."},{"cited_title":"Nakano and T","cited_arxiv_id":null,"evidence_quote":"Reports the synthesis of M3(HHTP)2 MOFs including the Cu compound whose sample preparation recipe is followed here."}],"review_version":1}