{"id":"487645af-3866-4fa3-87c9-ed8aaf5171a7","arxiv_id":"2607.08438","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Restricted in-plane quantum weight of black phosphorus is nearly isotropic (Kzz/Kxx ≈ 0.972) and rigid under orbital Hartree corrections, making low-loss EELS spectral moments a practical bulk probe of Wannier quantum geometry.","lead":"First-principles calculations show black phosphorus has nearly isotropic in-plane quantum weight despite strong band-mass and optical anisotropy, and that low-loss EELS moments can measure it. This gives experimentalists a bulk, direction-resolved probe of integrated quantum geometry complementary to ARPES.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.5","headline":"The scissor-plus-cutoff regularization may still leave residual near-gap projector contamination that artificially enforces in-plane isotropy of Kzz/Kxx.","rationale":"The reader correctly isolates the scissor-plus-cutoff step (Sec. VIII C, Eq. (20)) as the weakest assumption supporting the headline isotropy and EELS-probe claims. The paper is transparent about PBE near-semimetallicity and supplies convergence tables, yet the physical content of the restricted integral still hinges on the claim that residual near-gap contamination is isotropic or negligible once ωc is applied. Because absolute weights and their ratio are both obtained from the same regularized projectors, an independent gap-opening method is the cleanest falsification test. No stronger internal inconsistency appears; the occupied-manifold formulation and sum-rule checks are sound. Verdict therefore remains CONDITIONAL, with the same high-confidence computational content and the same experimental-EELS caveat.","tokens_in":14065,"tokens_out":596,"duration_ms":6395,"concrete_test":"Recompute Table II Kxx and Kzz on the identical 32-band model after replacing the rigid scissor by a hybrid or GW-corrected conduction manifold (or by a larger frozen-window Wannierization that already opens a robust gap without scissor). If Kzz/Kxx moves outside 0.96–0.98 while the restricted SWM check remains satisfied, the isotropy claim is regularization-dependent and the strongest claim weakens.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim that restricted in-plane quantum weight is nearly isotropic (Kzz/Kxx=0.972±0.005) despite strong band-mass and onset anisotropy rests on Eq. (20) with a rigid scissor of the conduction subspace plus ωc≈0.15–0.25 eV. Sec. VIII C states that unrestricted PBE weights are ill-posed because of near-semimetallic pockets and Wannier-interpolation dips that dominate the quadratic energy denominators. The scissor is asserted not to rotate projectors, only to separate fragile subgap weight; Table III shows a double plateau. However, the same section notes that dense sampling exposes pockets missed on sparse grids, and Fig. 7 still shows a residual conduction dip on Γ–Y even after scissor. If residual near-gap pairs (or scissor-induced reordering of orbital character at the indirect extrema) continue to contribute anisotropically inside the restricted window, the near-isotropy could be an artifact of the regularization rather than a physical cancellation of armchair-onset weight by higher-energy zigzag weight. The Hubbard–Hartree drift (+0.46 %/eV) is smaller than this possible systematic and is computed on the same scissored Hamiltonian, so it does not independently validate the absolute ratio.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript develops a DFT–Wannier framework that links the quantum geometry of bulk black phosphorus to a bulk, direction-resolved observable: the restricted quantum weight K_ii(ω_c) accessible via low-loss EELS spectral moments. Using a 32-band Wannier Hamiltonian with analytic momentum derivatives, the authors show that raw single-band metrics of the top valence band are singular on conventional-cell folding planes and intra-valence near-degeneracy rings, so masked maps and occupied-manifold projectors are required. Because PBE produces near-gap semimetallic pockets and interpolation artifacts that dominate unrestricted integrals, they introduce an energy-restricted quantum weight obeying a restricted Souza–Wilkens–Martin sum rule. The central numerical claims are that the restricted in-plane weight is nearly isotropic (K_zz/K_xx = 0.972 ± 0.005) despite strong band-mass and onset anisotropy, and that orbital-resolved Hubbard–Hartree corrections leave absolute weights rigid at the sub-percent level while producing a small, resolved armchair-directed drift of the ratio (~+0.46 % per eV of U). Three falsifiable low-loss EELS signatures are predicted.","tokens_in":14377,"tokens_out":1432,"duration_ms":12608,"significance":"If the restricted-weight construction is robust, the work supplies a practical bulk complement to ARPES-based quantum-metric reconstructions in a real anisotropic layered semiconductor, with concrete, ratio-based EELS predictions that are less sensitive to absolute normalization and zero-loss subtraction. Strengths include gauge-invariant projector formulas, analytic derivatives (no finite-difference stencils), multi-grid and multi-cutoff convergence of the restricted weights (Tables II–III), an independent check of the restricted SWM sum rule against the absorptive dielectric integral, and explicit experimental signatures (onset dichroism vs near-isotropic integrated moment, plus a small interaction drift of the ratio). The careful separation of single-band vs occupied-manifold geometry and the honest labeling of the model Hartree channel are also valuable methodological contributions for the growing quantum-geometry community.","major_comments":[{"comment":"Sec. VIII C and Eq. (20): the central isotropy claim K_zz/K_xx = 0.972 ± 0.005 rests on a rigid scissor of the conduction subspace plus a lower cutoff ω_c ≈ 0.15–0.25 eV. The manuscript correctly notes that unrestricted PBE weights are ill-posed because near-gap pockets and Wannier-interpolation dips dominate the quadratic denominators, and Table III shows a double plateau. However, Fig. 7 still exhibits a residual conduction dip on Γ–Y after scissor, and dense sampling is said to expose pockets missed on sparse grids. Residual near-gap pairs (or scissor-induced reordering of orbital character at the indirect extrema) that remain inside the restricted window could contribute anisotropically and partially enforce the reported near-isotropy. Because the Hubbard–Hartree drift is computed on the same scissored Hamiltonian and is smaller than this possible systematic, it does not independentl","section":null},{"comment":"Sec. X and Table II: the stacking-direction weight K_yy is reported as the largest component (K_xx/K_yy = 0.255) but is flagged as provisional pending the Wannier position-operator correction to the velocity. Because the paper’s experimental emphasis is on the in-plane ratio, this does not overturn the main claim, yet the table presents K_yy on equal footing with K_xx and K_zz. Either complete the position-operator correction for the production numbers or move K_yy out of the primary results table and state clearly that only the in-plane ratio is claimed to be experimentally robust at present.","section":null}],"minor_comments":[{"comment":"Fig. 5 caption and Sec. VIII D: absolute plasmon peak positions and heights in Im[-1/ε] are correctly labeled qualitative, but the main text still refers to them as a “loss proxy.” A single sentence clarifying that only the restricted Im ε moments (not the loss-function peaks) are used for the SWM comparison would avoid over-reading the figure.","section":null},{"comment":"Sec. V B–C: the weak-coupling shell analysis motivates an orbital-resolved Hartree channel, but the manuscript should state more explicitly that no double-counting correction against the DFT starting point is applied and that dynamical correlations and vertex corrections are outside the model (the text already says this in places; a single consolidated caveat near Eq. (16) would help).","section":null},{"comment":"Table I and axis conventions: the zigzag/armchair assignment is clear, but several black-phosphorus papers swap a and c. Adding a one-line note that literature conventions differ would reduce confusion for experimental readers.","section":null},{"comment":"Appendix A: the production Wannier grid (10\times8\times10) and the denser diagnostic grids are stated; a brief remark on why the sparser 8\times4\times8 grid under-determines the near-gap Hamiltonian (already mentioned in Sec. VIII C) could be cross-referenced in the appendix for reproducibility.","section":null},{"comment":"Eq. (19) vs Eq. (20): the unrestricted SWM form is written with a continuum integral of Re σ/ω, while the restricted form is written as a projector sum with Emk-Enk>ω_c. A short clause equating the two under the same cutoff would make the restricted sum-rule statement fully self-contained.","section":null}],"recommendation":"major_revision","confidential_remarks":"The technical core is careful and the EELS predictions are falsifiable; the main risk is that the near-isotropy is partly an artifact of the scissor-plus-cutoff regularization of a PBE near-semimetal. If the authors can show the ratio is stable under a better-gapped starting point or a clean energy-window decomposition, the paper would be a solid contribution. Scope is appropriate for a materials/condensed-matter journal that values first-principles quantum geometry and spectroscopy connections."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The useful takeaway is simple: they give a first-principles, direction-resolved quantum weight for bulk black phosphorus that is nearly isotropic in plane (Kzz/Kxx = 0.972 ± 0.005) even though the band masses and optical onset are strongly anisotropic, and they show that a weak orbital Hartree shift leaves the absolute weights almost rigid while producing a small, resolved armchair drift. That is a concrete bulk observable you can actually compare to monochromated STEM-EELS moments, complementary to the ARPES metric work already done on this material.\n\nWhat is new is not the SWM sum rule itself, but the material-specific execution: a 32-band DFT–Wannier model with analytic derivatives, honest masking of single-band singularities from folding and near-degeneracies, and the restricted weight Kii(ωc) that excludes the zero-loss and PBE-pocket region while still obeying a restricted sum rule. They check the projector integral against the dielectric moment, show a double plateau across grids and cutoffs, and keep the interaction analysis at the level of a transparent static channel rather than overclaiming many-body geometry. The citation trail is clean and the math is standard projector geometry done carefully.\n\nThe soft spot is real but not load-bearing. PBE is near-semimetallic, so they scissor the conduction manifold and cut at ~0.2 eV. The paper is explicit that the scissor does not rotate projectors and that unrestricted weights are ill-posed; Table III and the multi-grid consistency support the plateau. Residual interpolation dips remain visible in the band plot, so a skeptic can still worry that some anisotropic near-gap weight leaks into the restricted window. That is a systematic to watch, not a reason to discard the isotropy claim. Stacking-direction Kyy is flagged as provisional pending the position-operator fix; that is minor for the in-plane story they emphasize. No code is shipped, so independent re-runs will take work.\n\nThis is for people who care about quantum geometry in real layered semiconductors and for STEM-EELS groups looking for a falsifiable ratio rather than absolute intensities. It deserves a serious referee. I would engage with it, cite the restricted-weight construction and the BP numbers when discussing bulk probes of quantum geometry, and treat the EELS predictions as worth testing.","headline":"Solid materials calculation that turns SWM quantum weight into a concrete low-loss EELS prediction for black phosphorus, with a real near-isotropy result that survives their own checks.","tokens_in":14988,"tokens_out":584,"would_cite":true,"duration_ms":6364,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Black phosphorus has nearly isotropic in-plane quantum weight, so low-loss EELS can read integrated quantum geometry despite strong optical anisotropy.","keywords":["quantum weight","quantum metric","black phosphorus","low-loss EELS","Wannier functions","Souza–Wilkens–Martin sum rule","quantum geometry"],"falsifier":"Monochromated, momentum-resolved low-loss STEM-EELS on bulk black phosphorus with q along armchair versus zigzag: after Kramers–Kronig analysis, the ratio of window-integrated Im ε moments should approach approximately 0.97 rather than the large anisotropy of the absorption onset.","tokens_in":14934,"feed_emoji":"🔬","tokens_out":660,"duration_ms":6639,"temperature":0.7,"pith_summary":"Black phosphorus is famous for direction-dependent band masses and an absorption edge that lights up only along the armchair axis. This paper shows that a bulk, direction-resolved geometric quantity—the restricted quantum weight—is nevertheless almost the same along armchair and zigzag. Using a 32-band Wannier model built from density-functional theory, the authors compute the occupied-manifold quantum metric from analytic momentum derivatives and convert it into a restricted integral that matches the frequency moments of the dielectric function once the zero-loss region is cut out. That restricted weight is nearly isotropic (ratio 0.972) and stays rigid under orbital Hartree corrections, with only a small, measurable drift of the anisotropy ratio. Because low-loss electron energy-loss spectroscopy measures exactly those directional moments, the work turns quantum geometry into a practical bulk observable for an anisotropic layered solid and supplies concrete signatures an experimenter can look for.","feed_headline":"Black phosphorus quantum weight is nearly isotropic in plane","feed_subtitle":"Low-loss EELS can therefore read integrated quantum geometry despite strong optical anisotropy","key_machinery":"The restricted quantum weight Kii(ωc): an occupied-manifold projector integral that excludes transitions below an energy cutoff, obeys the corresponding restricted Souza–Wilkens–Martin sum rule, and equals the low-loss EELS moment of Im ε once the zero-loss region is removed.","core_discovery":"In bulk black phosphorus the restricted in-plane quantum weight is nearly isotropic, Kzz/Kxx = 0.972 ± 0.005 (armchair/zigzag), even though the band masses and near-gap absorption are strongly anisotropic; absolute weights remain rigid at the sub-percent level under orbital-resolved Hubbard–Hartree corrections while the ratio drifts toward armchair by about +0.46 % per eV of U. Consequently, low-loss EELS spectral moments become a practical probe of integrated quantum geometry.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Black phosphorus quantum weight nearly isotropic despite band anisotropy","Restricted in-plane quantum weight of BP stays nearly isotropic","Low-loss EELS reads isotropic quantum weight in anisotropic BP","BP quantum weight Kzz/Kxx near 0.97 despite optical anisotropy","In-plane quantum weight rigid and nearly isotropic in black phosphorus"],"cache_read_input_tokens":128,"weakest_assumption_plain":"A rigid scissor shift of the conduction bands plus a fixed energy cutoff cleanly separates the physical interband weight that EELS measures from the near-gap artifacts of the starting calculation.","fun_headline_variants_meta":{"raw":{"variants":["Black phosphorus quantum weight nearly isotropic despite band anisotropy","Restricted in-plane quantum weight of BP stays nearly isotropic","Low-loss EELS reads isotropic quantum weight in anisotropic BP","BP quantum weight Kzz/Kxx near 0.97 despite optical anisotropy","In-plane quantum weight rigid and nearly isotropic in black phosphorus"]},"model":"grok-4.5","effort":"low","cost_usd":0.005906,"raw_usage":{"total_tokens":1653,"prompt_tokens":907,"num_sources_used":0,"completion_tokens":88,"cost_in_usd_ticks":59060000,"prompt_tokens_details":{"text_tokens":907,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":658,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":907,"tokens_out":88,"duration_ms":5773,"temperature":1.0,"reasoning_tokens":658,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T07:28:00.323460+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Monochromated, momentum-resolved low-loss STEM-EELS on bulk black phosphorus with q along armchair versus zigzag: after Kramers–Kronig analysis, the ratio of window-integrated Im ε moments should approach approximately 0.97 rather than the large anisotropy of the absorption onset.","supporting_citations":[],"review_version":1}