Quantum weight and low-loss EELS signatures of Wannier quantum geometry in black phosphorus

Sharona Horta; Vinayak M. Kulkarni

REVIEW 2 major objections 5 minor 26 references

Quantum weight and low-loss EELS signatures of Wannier quantum geometry in black phosphorus

T0 review · 2 major / 5 minor · reviewed 2026-07-10 · grok-4.5

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Pith's one-line read Black phosphorus has nearly isotropic in-plane quantum weight, so low-loss EELS can read integrated quantum geometry despite strong optical anisotropy.

desk verdict 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. read the letter →

arxiv 2607.08438 v1 pith:5FNQPCRY submitted 2026-07-09 cond-mat.mtrl-sci

Vinayak M. Kulkarni , Sharona Horta This is my paper · ORCID

classification cond-mat.mtrl-sci

keywords quantumweightmetricblackphosphoruslow-lossEELSWannierfunctionsSouza–Wilkens–Martinsumrulegeometry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What 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.

The 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.

This 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.

Referee Report

2 major / 5 minor

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.

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 (2)

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
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.

minor comments (5)

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.
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).
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.
Appendix A: the production Wannier grid (10 imes8 imes10) and the denser diagnostic grids are stated; a brief remark on why the sparser 8 imes4 imes8 grid under-determines the near-gap Hamiltonian (already mentioned in Sec. VIII C) could be cross-referenced in the appendix for reproducibility.
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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: restricted quantum weights and the Kzz/Kxx ratio are computed from the Wannier projectors and checked against the dielectric integral; neither is forced by definition or by a self-citation chain.

full rationale

The derivation chain is DFT structure → 32-band Wannier H0(k) → gauge-invariant projectors → restricted occupied-manifold weight Kii(ωc) (Eq. 20) and independent-particle Im εii, with the restricted SWM sum rule used as a numerical consistency check rather than as a definition of the reported ratio. The near-isotropy Kzz/Kxx = 0.972 ± 0.005 is a material-specific integral over the computed matrix elements and is not fixed by normalization, by the scissor (which the paper states does not rotate projectors), or by the cutoff ωc (Table III plateau). The Hubbard–Hartree drift is obtained from self-consistent orbital occupations on the same Hamiltonian, not fitted to the target geometry. Self-citations involving co-author Horta ([22], [25]) concern EELS/STEM methodology context and do not underwrite the geometric numbers. External ARPES references ([7,8]) are independent. No self-definitional loop, fitted-input-as-prediction, uniqueness import, or renaming of a known result is present. Residual concerns about scissor/cutoff systematics are correctness risks, not circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 1 invented entities

The central numerical claims rest on a standard DFT–Wannier pipeline plus three modeling choices that are not fixed by prior experiment: the scissor regularization of the PBE gap, the energy cutoff that defines the restricted weight, and the orbital Hubbard U used for the interaction scan. No new particles or forces are invented; the quantum weight itself is the established SWM object.

free parameters (3)

restricted cutoff ωc = 0.20 eV (nominal)
Chosen in the 0.15–0.25 eV window to exclude zero-loss and PBE subgap artifacts; results form a plateau but the absolute scale of Kii depends on this choice.
Hubbard U (orbital-resolved Hartree) = 1–2 eV (scan)
Model coupling scanned at 0, 1, 2 eV with no double-counting correction against DFT; used to extract the drift slope of Kzz/Kxx.
scissor shift of conduction subspace
Rigid shift applied to regularize the near-semimetallic PBE gap while preserving projectors; magnitude is set to produce a small positive direct gap.

assumptions (4)

domain assumption Independent-particle RPA dielectric response in the small-q dipole limit equals the longitudinal loss function measured by low-loss EELS after Kramers–Kronig analysis.
Used throughout Sec. VII–VIII to equate projector quantum weight with Im ε moments; local fields, excitons, and finite-q effects are neglected.
standard math The Souza–Wilkens–Martin sum rule (and its restricted version above ωc) holds for the occupied-manifold quantum weight of the Wannier Hamiltonian.
Eqs. (19)–(20); standard result applied pair-by-pair to the scissored Hamiltonian.
ad hoc to paper Static orbital-resolved Hartree self-energy on the Wannier basis captures the leading weak-coupling geometric renormalization of black phosphorus.
Motivated by the momentum-shell analysis of Sec. V B; dynamical correlations and double-counting are explicitly omitted.
domain assumption Conventional-cell folding degeneracies and intra-valence near-degeneracies render raw single-band metrics singular, so only masked or occupied-manifold quantities are meaningful.
Demonstrated in Sec. VIII B and Fig. 4; standard consequence of band theory.

invented entities (1)

restricted quantum weight Kii(ωc) independent evidence
purpose: To define a well-posed, experimentally accessible geometric integral that excludes the zero-loss region and PBE/Wannier subgap artifacts while still obeying a sum rule.
Introduced in Sec. VIII C; it is a cutoff version of the standard SWM weight rather than a new physical object, but the specific construction and its EELS identification are paper-specific.

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Pith. "Pith review of Quantum weight and low-loss EELS signatures of Wannier quantum geometry in black phosphorus." pith.science (2026). https://pith.science/paper/5FNQPCRY

@misc{pith2026260708438,
  author       = {Pith},
  title        = {Pith review of: Quantum weight and low-loss EELS signatures of Wannier quantum geometry in black phosphorus},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5FNQPCRY}},
  note         = {Machine review of arXiv:2607.08438}
}

abstract

Quantum geometry is now experimentally accessible in crystalline solids, with black phosphorus providing a key platform through polarization-resolved angle-resolved photoemission spectroscopy. We develop a first-principles framework that connects the momentum-resolved quantum metric of black phosphorus to a complementary bulk observable: the direction-resolved quantum weight measurable through low-loss electron energy-loss spectroscopy (EELS). A 32-band DFT--Wannier Hamiltonian is used to compute both single-band and occupied-manifold geometric quantities from analytic momentum derivatives. We show that the raw single-band quantum metric of the top valence band is not globally meaningful in the conventional cell because folding degeneracies and intra-valence near degeneracies produce true isolated-band singularities; masked maps and occupied-manifold projectors are therefore essential. Because semilocal PBE produces near-gap semimetallic pockets and spurious subgap interpolation features, we introduce an experimentally motivated restricted quantum weight $K_{ii}(\omega_c)$, which obeys the corresponding restricted Souza--Wilkens--Martin sum rule and is the appropriate quantity for low-loss EELS once the zero-loss region is excluded. The restricted in-plane quantum weight is nearly isotropic, $K_{zz}/K_{xx}=0.972\pm0.005$ (armchair/zigzag), despite the strong band-mass anisotropy and armchair-only absorption onset of black phosphorus. Orbital-resolved Hubbard--Hartree corrections leave the absolute quantum weights rigid at the sub-percent level while producing a small but resolved armchair-directed drift of $K_{zz}/K_{xx}$, approximately $+0.46\%$ per eV of $U$. These results identify low-loss EELS spectral moments as a practical probe of integrated quantum geometry in an anisotropic layered material.

Figures

Figures reproduced from arXiv: 2607.08438 by the authors.

**Figure 3.** FIG. 3. Pseudospin renormalization on the Bloch sphere. De [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** FIG. 4. Single-band quantum geometry of the top valence [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: FIG. 5. Directional dielectric response of bulk black phosphorus from the scissor-corrected Wannier model (independent-particle [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6. Interaction flow of the in-plane quantum-weight [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7. Band structure of the same bulk 32-band black [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

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Pith tools

Reviewed July 10, 2026 · model on record in the stance chip above.

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