{"id":"5790ef04-fbce-41c1-bae8-667de82ce846","arxiv_id":"2501.09063","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"The Fermi-level band of monolayer NbSe2 is shown by STM to be an obstructed atomic insulator, with its Wannier center at the empty 1c site, via a new method that extracts orbital correlation functions from scanning tunneling images.","lead":"STM measurements of monolayer NbSe2, combined with a new deconvolution scheme using ab initio Wannier functions, show that the Fermi-level band has its Wannier center at an empty lattice site, making it an obstructed atomic insulator. This is the first quantitative experimental identification of such a phase and introduces a general STM-based tool for mapping orbital correlations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Table S2 contradicts the headline: one of three tips gives O1≈0.12–0.13 while the main text reports O1≈0.03; without a stated exclusion criterion or error bars, the data do not establish an unambiguous OA identification.","rationale":"In good faith, the paper is substantial: it derives the Tersoff-Hamann framework carefully, constructs Wannier functions, enforces symmetry constraints, performs bias-dependent conductance checks, and presents multiple datasets. The method is plausible, and the independent check at V≈−2 V does support the origin assignment. However, the core empirical claim is quantitative, and the paper's own supplementary Table S2 contains a dataset that disagrees with the headline result. This is not primarily a DFT-origin or Wannier-basis circularity objection; it is an internal-data consistency failure. The experiment may well be correct, but as reported it is not unambiguous. I therefore keep the reader's CONDITIONAL verdict but sharpen its basis: the missing error bars are not merely an omission, because the tabulated values across tips show a discrepancy large enough to affect the conclusion. I only partially agree with the reader's weakest assumption, since I locate the strongest issue in the tip-to-tip scatter and absence of an error model rather than only in the DFT-origin uncertainty.","tokens_in":54948,"tokens_out":6441,"duration_ms":73989,"concrete_test":"Pool all rows of Table S2 and report O1 with per-tip random effects: compute a 95% confidence interval for O1 under both fixed- and random-effects models and plot the three tip groups separately. If the confidence interval excludes 0, or if between-tip heterogeneity is substantial, the claim O1≈0 is not supported. As a second check, re-fit the same data with O0 free and with O3–O7 included, to test whether Tip 2's excess O1 is an artifact of the constrained three-correlator model; report which rows were excluded and why.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing quantity is the correlator O1: it is the UA order parameter, expected to be 1/3 in the fictitious UA limit and 0 in the OA limit, and the paper's central claim is O1≈0 with O2≈1/3. The supplementary Table S2, however, contains three rows from Tip 2 (sets 1–3) with fitted values O1=0.124, 0.129, 0.132 and O2=0.309, 0.307, 0.306. These are not small: O1 is about 40% of its UA value, and in the parameterization O1=(1/3)cosθ, O2=(1/3)sinθ the implied angle is θ≈67°, far from the θ=90° OA limit. The main-text value O1=0.030 comes from Tip 3, and Tip 1 gives O1≈0.02; the statement that all measurements consistently yield O1≈0 and O2≈1/3 is therefore not supported by the authors' own table. No error bars or tip-selection criterion are given, so one cannot decide whether Tip 2 is an outlier, a systematic tip-dependent effect, or evidence that the truncated three-correlator model is misspecified. The fitted tip height is nearly the same for Tip 2 (z=3.25–3.30 Å) and Tip 3, so the discrepancy is not absorbed by the fitted height. Because the central claim is explicitly one of unambiguous quantitative identification, this internal inconsistency is a correctness risk independent of any DFT-basis or origin-assignment concern.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript claims to provide the first unambiguous quantitative experimental identification of an obstructed atomic (OA) phase, using monolayer 1H-NbSe2 as the test case. The authors measure constant-height STM tunneling current maps at two biases bracketing the Fermi-level quasi-flat band, combine them to obtain the band's charge density distribution (CDD), and Fourier-transform the CDD to obtain A(Q,z). They then fit A(Q,z) against a linear model A(Q,z) = sum_{ΔR,i,j} B_ij(Q,z,ΔR) ρ_ij(ΔR), where the spatial factor B_ij is computed from ab initio Wannier functions, to extract inter-orbital correlators ρ_ij(ΔR). With a three-correlator truncation (O0, O1, O2), the main-text fit gives O1≈0.030 and O2≈0.332, which the authors interpret as the OA limit (O1=0, O2=1/3) with the Wannier center at the empty 1c position. The paper also presents bias-dependent STS contrast as an independent check of the unit-cell site assignment, and argues that the method is general and transferable to other materials.","tokens_in":55324,"tokens_out":6392,"duration_ms":65413,"significance":"If the central claim holds, this is a significant conceptual and experimental advance: it provides a quantitative real-space probe of a band-topology classification that goes beyond Chern numbers, namely the obstructed versus unobstructed atomic distinction, and it demonstrates a general technique for extracting orbital-off-diagonal correlators from the spatially diagonal STM density. The theoretical derivation of the relation between the CDD and the correlator matrix is carried out carefully in the Supplementary Information, with explicit normalization constraints and symmetry analysis. The paper also deserves credit for reporting multiple experimental data sets with different tips and for including an independent bias-dependent STS validation of the site assignment. The main scientific risk is not the derivation but the robustness of the quantitative fitting pipeline and the consistency of the reported data with the headline claim of an 'unambiguous' OA identification.","major_comments":[{"comment":"The statement that 'all measurements consistently yield O1≈0 and O2≈1/3' is not supported by the paper's own Table S2. Three Tip-2 data sets give fitted O1 = 0.124, 0.129, 0.132 and O2 = 0.309, 0.307, 0.306, while the main-text values O1≈0.030, O2≈0.332 come from Tip 3. O1 = 0.13 is about 40% of its unobstructed-atomic value 1/3, and in the parameterization O1=(1/3)cosθ, O2=(1/3)sinθ these Tip-2 values imply θ≈67° rather than the θ=90° OA limit. No error bars, goodness-of-fit values, or tip-selection/exclusion criteria are given, so a reader cannot determine whether Tip 2 is an outlier, a tip-dependent systematic effect, or evidence that the three-correlator model is misspecified. Since the central claim depends quantitatively on O1 being near zero, this internal inconsistency is load-bearing and must be resolved, either by including all data in a proper statistical treatment or by justifying the exclusion of Tip 2 with objective, pre-specified criteria.","section":"Supplementary Table S2; main text Fig. 3(i)-(j)"},{"comment":"The tip-sample distance z is a fitted parameter, selected as the minimum of the symmetrized error metric ϵsym, but no confidence intervals are reported for z, O1, or O2. Figure 3(i) shows that the extracted correlators vary substantially with the assumed z (the main text reports z=3.25 Å as the optimum, while Table S2 contains fitted z values of 3.25, 3.30, and 3.70 Å across data sets). For example, Tip 3 set 8, with fitted z=3.70 Å, gives O1=0.080 and O2=0.324, which is less cleanly in the OA limit. Because the error metric is symmetrized and the unsymmetrized error is reported as flat in z, the claim that the minimum at z=3.25 Å determines the correlators quantitatively needs an uncertainty propagation analysis. The authors should provide error bars and show that O1≈0, O2≈1/3 is stable within the experimental uncertainties in z and in the measured A(Q).","section":"Main text Fig. 3(i)-(j); SI IV.2, Eq. (S4.27)"},{"comment":"The deconvolution is not parameter-free and is partly circular: the spatial factor B_ij(Q,z,ΔR) is computed from the same ab initio DFT framework that predicts the OA phase, and the unit-cell origin in the experimental data is assigned using the DFT-computed brightness hierarchy (1b brightest, 1a darkest). If the Wannier functions are inaccurate, or if the s-like three-orbital basis does not capture the true orbital content of the band, the fitted correlators will be biased toward the DFT prediction. The bias-dependent STS comparison in Fig. 4 validates the site assignment but does not validate the quantitative O1 and O2 values. I recommend adding robustness tests, for example refitting with Wannier functions from different gauge choices or disentanglement windows, using an 11-band basis including Se p orbitals, and trying all three possible C3z origin assignments, to demonstrate that the O1≈0, O2≈1/3 conclusion is stable rather than imposed by the input basis.","section":"Main text Eq. (2); SI IV.1d"},{"comment":"The fit assumes that only O0, O1, and O2 are nonzero, an approximation justified by the DFT values in Table S1. Table S1 itself shows that O3-O7 are not strictly zero (e.g., O3=-0.075, O4=0.038), and if the true correlator matrix of the real material has non-negligible additional elements, the fitted O1 and O2 will absorb them. This model-misspecification possibility is consistent with the anomalous Tip-2 results. The paper should report per-data-set values of the error metrics ϵ and ϵsym, and perform a sensitivity analysis that includes O3-O7 in the fit, to show that the central OA identification is not an artifact of the three-correlator truncation.","section":"SI IV.2, Eqs. (S4.24)-(S4.27); Table S1"}],"minor_comments":[{"comment":"The four Fourier-transform panels showing amplitude and phase are not labeled with the corresponding reciprocal-lattice vectors Q; adding such labels would make the figure much easier to interpret.","section":"Main text, Figs. 3(e)-(h)"},{"comment":"The normalization conventions for A(Q,z) differ between the main text (1/Ω) and the SI (1/(NΩ0)); these should be stated consistently so that Eq. (2) can be reproduced directly from the SI.","section":"Main text, Eq. (2) and SI Eq. (S2.33)"},{"comment":"The column header 'O_i (for 1≤i≤3)' is confusing because the table lists O0, O1, and O2; please relabel as O0, O1, O2.","section":"Supplementary Table S2"},{"comment":"The abstract says 'first direct experimental evidence' while the introduction says 'first direct quantitative experimental evidence'; the wording should be unified, especially because 'quantitative' is central to the paper's claim.","section":"Introduction and abstract"},{"comment":"The claimed 'more than 94% overlap' between the compact model and the ab initio wave function is attributed to '[33], to be published'. Since that reference is not yet available, please include the numerical details needed to verify this number in the supplement.","section":"Main text and Ref. [33]"}],"recommendation":"major_revision","confidential_remarks":"The Table S2 discrepancy is the key issue: the authors' own tip-dependent data contradict the 'unambiguous' OA identification. I recommend major revision rather than rejection because the problem is potentially fixable within the manuscript's scope by a full statistical treatment of all data, explicit tip-selection criteria, and robustness tests against the Wannier-basis and origin-assignment choices. I would also ask the editor to ensure that the novelty claim is carefully positioned relative to the related work in Ref. [37] (arXiv:2412.02813), which addresses a similar problem in TMDs with a different method."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: read this for the method, not the headline. The paper does something new. It extracts orbital-off-diagonal correlation functions from the spatially diagonal STM charge density, by deconvolving with ab initio Wannier functions, and uses this to claim the first quantitative identification of an obstructed atomic (OA) phase. The approach is well-motivated, the supplement is careful, and the bias-dependent conductance maps are a genuinely independent check on the unit-cell origin. Credit where due: the derivation in App. II is thorough, and the authors are transparent about leaning on unpublished theory (Ref. 33) and about the concurrent WSe2 work (Ref. 37).\n\nThe soft spot is the data that carries the load. The central parameter is O1, the UA order parameter: OA limit O1=0, UA limit O1=1/3. The main text reports O1=0.030, O2=0.332 from Tip 3. But Table S2 lists everything: Tip 1 gives O1=0.014-0.022, Tip 3 gives 0.030-0.043, and Tip 2 gives O1=0.124, 0.129, 0.132 with O2 around 0.307. That is not 'all measurements consistently yield O1=0.' Tip 2's O1 is about 40% of the UA value, and the implied angle in the O1=(1/3)cos(theta) parameterization is about 67 degrees, not 90. The fitted z is nearly the same for Tip 2, so the discrepancy is not absorbed by tip height. No error bars, no stated exclusion criterion. As presented, the data cannot support 'unambiguous'; it supports a conditional identification pending a decision on Tip 2.\n\nThe other soft spot is structural. The fit assumes the three-correlator truncation and the Wannier basis, both sourced from the same ab initio framework that predicted the OA phase. That makes the analysis pipeline theory-loaded and limits its ability to falsify the hypothesis. The bias-dependent check helps pin the unit-cell origin, which is the largest purely geometric ambiguity, but it does not quantify the systematic uncertainty entering through the Wannier functions.\n\nWho this is for: groups working on topological quantum chemistry, TMDs, and quantitative STM. The method has legs beyond NbSe2. The paper deserves a serious referee, but the authors should report Tip 2 separately, provide error estimates, and either justify a tip-selection rule or show that the three-correlator model is misspecified. Without that, the claim should be softened from 'unambiguous' to 'consistent with.'","headline":"A genuinely new STM deconvolution method and a plausible OA identification, but Table S2 contains a three-tip spread that undercuts the word 'unambiguous.'","tokens_in":55856,"tokens_out":2096,"would_cite":true,"duration_ms":21390,"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":"Monolayer NbSe2's Fermi-level band is an obstructed atomic phase whose Wannier center sits on an empty site, as established by a new STM deconvolution of orbital correlators.","keywords":["obstructed atomic insulator","Wannier center","inter-orbital correlation functions","scanning tunneling microscopy","monolayer 1H-NbSe2","charge density distribution","topological quantum chemistry","flat band"],"falsifier":"Re-fit the same STM-derived $A(Q,z)$ with Wannier functions obtained from a different exchange-correlation functional or with spin-orbit coupling included, and simultaneously check that the fitted tip height $z=3.25$ Å agrees with the physical setpoint for the stabilization parameters used in the measurement; if $O_1$ moves toward $1/3$, $O_2$ toward $0$, or the optimal $z$ becomes unphysical under a plausible alternative basis, the obstructed-atomic conclusion fails.","tokens_in":54759,"feed_emoji":"🔬","tokens_out":11366,"duration_ms":106210,"temperature":0.7,"pith_summary":"Topologically trivial insulators split into unobstructed atomic phases, whose Wannier centers sit on occupied atoms, and obstructed atomic phases, whose centers sit at empty sites in the unit cell. This paper claims that the quasi-flat band crossing the Fermi level of monolayer 1H-NbSe2 is an obstructed atomic phase, and that this is the first unambiguous quantitative identification of such a phase. The evidence comes from converting constant-height STM current maps at two bias voltages into the band's charge density distribution, then deconvolving that density with Wannier orbital wave functions from ab initio calculations to extract the inter-orbital correlation matrix. The fitted correlators land at O1 ≈ 0.03 and O2 ≈ 0.33, which is the obstructed signature (Wannier center at the empty 1c site), not the unobstructed one (O1 ≈ 1/3, O2 ≈ 0). The identification matters because obstructed bands are predicted to be common and are tied to flat-band geometry, superconductivity, and electron-phonon effects.","feed_headline":"STM places NbSe2's flat band on an empty lattice site","feed_subtitle":"Deconvolving STM maps extracts orbital correlators, giving the first quantitative sighting of an obstructed atomic phase.","key_machinery":"The load-bearing object is the orbital correlator matrix $\\rho_{ij}(\\Delta R)=\\langle\\Phi|\\,c^{\\dagger}_{R,i}c_{R+\\Delta R,j}\\,|\\Phi\\rangle$ for the fully filled quasi-flat band. The identity that converts the STM map into physics is $A(Q,z)=\\sum_{\\Delta R,i,j}B_{ij}(Q,z,\\Delta R)\\,\\rho_{ij}(\\Delta R)$: since the STM density is a spatially diagonal observable, the off-diagonal orbital correlations are recovered only through the finite spatial extent of the Wannier orbitals encoded in $B$. The obstructed character is read off two correlators: $O_1=\\rho_{12}(0)$ and $O_2=\\rho_{23}(a_1)$, which take values $(0,1/3)$ in the obstructed limit and $(1/3,0)$ in the fictitious unobstructed limit. A minimal three-orbital tight-binding model with real hoppings $t=-0.784$ eV and $E_0=1.733$ eV generates compact Wannier orbitals with the same correlator pattern, which is what makes the fitted numbers interpretable as an optimally compact obstructed atomic phase.","core_discovery":"On the paper's own terms, the discovery is that the Fermi-level quasi-flat band of monolayer 1H-NbSe2 realizes an optimally compact obstructed atomic phase. Its Wannier center lies at the 1c position, an empty site in the unit cell, rather than at the Nb (1a) or Se (1b) positions. The central experimental relation writes the measured band charge density in Fourier space as $A(Q,z)=\\sum_{\\Delta R,i,j} B_{ij}(Q,z,\\Delta R)\\,\\rho_{ij}(\\Delta R)$, where $B$ is computed from ab initio Wannier wave functions and $\\rho_{ij}(\\Delta R)$ is the orbital-correlation matrix of the filled band. Fitting only the three dominant correlators gives $O_0=1/3$, $O_1=0.030$, and $O_2=0.332$ at the best-fit tip height $z=3.25$ Å. This is the obstructed-atomic limit ($O_1\\to 0$, $O_2\\to 1/3$), opposite to the unobstructed limit ($O_1\\to 1/3$, $O_2\\to 0$), and the band's wave function has more than 94% overlap with an optimally compact obstructed-atomic model on the triangular lattice.","pith_inferences":["Beyond the paper: repeating the same extraction below the 33 K charge-density-wave transition could show whether the 3×3 reconstruction moves or splits the Wannier center, turning the method into a direct probe of CDW order parameters.","Beyond the paper: the fitted correlation matrix is exactly the kind of input needed to evaluate geometric bounds on superfluid weight and electron-phonon coupling, so a single STM measurement could be converted into quantitative constraints on NbSe2's superconductivity.","Beyond the paper: re-fitting the same experimental density with Wannier functions from different density functionals, different cutoffs, or with spin-orbit coupling included would reveal how much of the obstructed-atomic conclusion is inherited from the ab initio input rather than from the measured maps."],"forward_implications":["The Fermi-level band of monolayer 1H-NbSe2 has its Wannier center at the empty 1c site, confirming the obstructed-atomic assignment for this material.","The same two-bias STM measurement, deconvolved with the ab initio spatial factor, gives a general route to identifying obstructed atomic bands in other monolayer transition-metal dichalcogenides.","The extracted correlators and the better-than-94% overlap with the optimally compact triangular-lattice model validate a minimal three-orbital description of the band, a controlled starting point for models of superconductivity and charge-density-wave order.","Because obstructed bands have an enlarged quantum geometric tensor, the measurement connects STM density maps to geometric lower bounds on superfluid weight and to electron-phonon coupling in flat bands."],"supporting_citations":[{"why":"Supply the band-representation classification used to identify the quasi-flat band as A1@1c, an obstructed atomic band.","marker":"[1, 2]"},{"why":"Introduce the real-space invariants that separate unobstructed from obstructed atomic insulators, the category the paper is testing.","marker":"[8–11]"},{"why":"Provides the tight-binding model of NbSe2 with t = -0.784 eV and E0 = 1.733 eV and the 94% overlap with the optimally compact OA wavefunction.","marker":"[33]"},{"why":"Defines the optimally compact obstructed-atomic model on the triangular lattice used as the reference for the overlap.","marker":"[14]"},{"why":"Tersoff-Hamann approximation, the basis for reading the two-bias tunneling current maps as the band's charge density distribution.","marker":"[35, 36]"},{"why":"Wannier90 maximally-localized Wannier functions, from which the spatial factor B_ij(Q,z,ΔR) is computed.","marker":"[57–60]"},{"why":"Concurrent real-space imaging work that also fixes the Se atom position in STM images, corroborating the unit-cell origin.","marker":"[37]"}],"fun_headline_variants":["First direct proof: NbSe2 hosts obstructed atomic phase","STM deconvolution finds flat band centered on empty site","Obstructed atomic phase in NbSe2 caught by orbital correlators","Wannier center on empty lattice site: first quantitative sighting","NbSe2's flat band sits on a vacancy, not an atom"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The result stands or falls on the assumption that the atomic-like orbital shapes used in the deconvolution, taken from density-functional calculations, are the true orbital content of the Fermi-level band, and that the darkest and brightest spots in the STM map correctly mark the Nb and Se sites; if either assumption fails, the extracted correlators are biased toward the theoretical prediction.","fun_headline_variants_meta":{"raw":{"variants":["First direct proof: NbSe2 hosts obstructed atomic phase","STM deconvolution finds flat band centered on empty site","Obstructed atomic phase in NbSe2 caught by orbital correlators","Wannier center on empty lattice site: first quantitative sighting","NbSe2's flat band sits on a vacancy, not an atom"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000794,"raw_usage":{"total_tokens":3555,"prompt_tokens":1063,"completion_tokens":2492,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":679,"completion_tokens_details":{"reasoning_tokens":2402}},"tokens_in":679,"tokens_out":2492,"duration_ms":17964,"temperature":1.0,"reasoning_tokens":2402,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:10:57.097235+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-fit the same STM-derived $A(Q,z)$ with Wannier functions obtained from a different exchange-correlation functional or with spin-orbit coupling included, and simultaneously check that the fitted tip height $z=3.25$ Å agrees with the physical setpoint for the stabilization parameters used in the measurement; if $O_1$ moves toward $1/3$, $O_2$ toward $0$, or the optimal $z$ becomes unphysical under a plausible alternative basis, the obstructed-atomic conclusion fails.","supporting_citations":[{"cited_title":"This is due to contributions from the mirror-odd sector, whose Wannier centers are located at the1c site, as indicated in Fig","cited_arxiv_id":null,"evidence_quote":"Defines the optimally compact obstructed-atomic model on the triangular lattice used as the reference for the overlap."}],"review_version":1}