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Orbital Embedding and the Physical Definition of Quantum Geometry

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Pith's one-line read This paper argues that the standard k-derivative formula for the quantum geometric tensor is convention-dependent and unphysical unless intra-unit-cell orbital positions are included, and that the correct convention-independent object is de

desk verdict The convention-dependence claim is right and the SSH demonstration is clean, but the k·p failure section rests on an unproven projection of the current, so that part needs a real downfolding or a clear subleading bound before it is published. read the letter →

arxiv 2607.21882 v1 pith:PBHOEUAU submitted 2026-07-24 cond-mat.str-el

Chang-geun Oh , Shuichi Murakami This is my paper · ORCID
classification cond-mat.str-el
keywords quantumgeometrictensormetricBerrycurvatureorbitalembeddingPeierlssubstitutionk·pperturbationtheorySu-Schrieffer-Heegermodelengineering
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

This paper argues that the standard way of computing the quantum geometric tensor (QGT) — by differentiating the occupied-band projector with respect to crystal momentum — is not physically well defined unless the Bloch basis explicitly includes the positions of orbitals within the unit cell (Convention B). The authors show that in the widely used Convention A, which omits these intra-cell positions, the computed quantum metric and even Berry-curvature-related response become dependent on an arbitrary choice of unit-cell origin; in the SSH model the Wannier spread and optical conductivity change when the same physical chain is partitioned differently, which is unphysical. The resolution is a convention-independent QGT built from the covariant derivative DμP = i[P, r̂μ], which they show is the version uniquely mandated by the Peierls-substitution derivation of the physical current. A direct corollary is a warning for k·p effective theories: when the gap comes from bond-order (inter-sublattice) terms, the embedding-operator correction enters at the same order as the standard current vertex, so H_eff alone does not fix the optical response. The consequence is 'geometric engineering': spectral and geometric properties can in principle be tuned independently.

What carries the argument

The central object is the covariant derivative D_μP ≡ i[P, r̂_μ], where P is the projector onto the occupied subspace and r̂ = R̂ + X̂ is the full position operator (lattice position plus intra-cell orbital position). Inserted into Q^phys_μν = Tr[D_μP(1−P)D_νP], it yields a QGT that is invariant under the choice of Bloch convention, because in Convention A it reduces to ∂_μP − i[X̂_μ, P] and in Convention B it reduces to ∂_μP. The derivation relies on minimal coupling: the physical current in the Peierls-substituted tight-binding Hamiltonian is −e/ℏ ∂_k H in Convention B, while in Convention A it acquires the extra (ie/ℏ)[X̂_μ, H] term, which is exactly the term that the partial-derivative Q

What would settle it

For a tight-binding SSH chain with known t1, t2 and sublattice separation a/2, compute the interband optical conductivity exactly (Convention B) and from the minimal two-band k·p Hamiltonian H_eff(q) alone; if the exact and k·p results agree at leading order in the gap, the claim of a leading-order failure is falsified.

Watch

Extended reading notes

Core claim

The central claim is that the physical QGT is not the partial-derivative object Q^∂_μν = Tr[∂_μP(1−P)∂_νP] but the covariant object Q^phys_μν = Tr[D_μP(1−P)D_νP] with D_μP = i[P, r̂_μ] and r̂ = R̂ + X̂, the full position operator. In a basis that embeds orbital positions in the Bloch phase (Convention B), this reduces to the ordinary derivative; in the common cell-periodic basis (Convention A) it subtracts the missing intra-cell commutator −i[X̂, P]. Using the SSH model, the authors show that omitting this term violates the elementary requirement that bulk observables be invariant under shifting the unit-cell boundary: the BZ-integrated metric and the optical conductivity become asymmetric u

Load-bearing premise

The load-bearing premise is that the physical current of the low-energy projected subspace is exactly the projection of the full-lattice current (Eq. 11); if a proper downfolding from high-energy bands adds comparable non-local corrections, the claimed leading-order failure of standard k·p with bond-ordered gaps could be modified.

Editorial extensions

If this is right

  • Any calculation that feeds a Convention-A partial-derivative quantum metric into optical-conductivity, Wannier-spread, or similar formulas can be wrong at leading order; results should be quoted together with the orbital-embedding convention.
  • For low-energy k·p models with bond-ordered gaps, H_eff is insufficient data; the effective embedding operator X_eff is needed to compute current and geometric response, and the correction survives to zeroth order in momentum.
  • The optical conductivity of a nearly flat band is strongly sensitive to embedding, since the discrepancy ratio scales as (1 + cm/v)^2 and diverges as v → 0.
  • Geometric engineering becomes a design principle: by varying intra-cell atomic positions while keeping hoppings constant, one can change geometric observables without changing the band dispersion.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the same covariant-derivative argument should apply to time-dependent or nonlinear response calculations, where the missing X̂ commutators can alter not just the QGT but higher-order optical tensors; a systematic rewiring of existing formulas in convention A is implied.
  • Editorial inference: the 'commuting vs noncommuting' classification suggests a sharp practical criterion for when a k·p theory is safe—if the gap matrix commutes with X_eff (on-site mass), the standard description is adequate at leading order; otherwise it is not.
  • Editorial inference: the embedding dependence also offers a route to experimentally probe 'where' electrons really sit in a band: measured quantum geometry could be inverted to infer X_eff for a given H_eff, effectively measuring the orbital embedding that photoemission resolves only indirectly.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper argues that the conventional QGT defined via k-derivatives of the Bloch projector, Eq. (1), is convention-dependent because the Bloch phase may or may not include intra-cell orbital positions (Conventions A and B). Using the SSH model, the authors show that Convention A gives a BZ-integrated metric and an interband optical conductivity that are not invariant under a shift of the unit-cell partition (t1↔t2), which is unphysical. They introduce a convention-independent QGT via the covariant derivative D_μP = i[P, r̂_μ], Eqs. (9)-(10), which in Convention B reduces to the standard derivative and in Convention A adds -i[X̂_μ,P]. They show this object is consistent with the Peierls-substitution derivation of the current. They further claim that low-energy k·p effective Hamiltonians for bond-ordered gapped systems require an embedding operator X_eff in addition to H_eff, and they illustrate the effect in a massive Dirac model. Finally they propose "geometric engineering" of classical metamaterials.

Significance. The central identification of Qphys with the covariant-derivative expression is internally consistent and well demonstrated. The SSH calculation is a clean counterexample to the routine use of Eq. (1) in Convention A. If the effective-current part can be placed on firmer footing, the paper would be an important cautionary contribution: it would explain known embedding ambiguities and provide a practical rule (use Convention B or the covariant derivative) for computing geometric observables. The "geometric engineering" proposal is stimulating though speculative. The paper does not ship code, but the analytical and numerical derivations are simple enough to be checked. The definition of Qphys is not circular: it is defined via the position operator and checked against the Peierls current.

major comments (2)
  1. [Implications for Low-Energy Effective Theories, Eq. (13)] The effective current J_eff,μ = -e/ℏ ∂_{qμ} H_eff + (ie/ℏ)[X_eff,μ, H_eff] is stated as "applying Eq. (8) to the projected theory" without a downfolding derivation. Starting from the exact projected current P J P = -e/ℏ P ∂_k H P + (ie/ℏ) P[X,H]P, the commutator term contains cross terms P X Q H P - P H Q X P with Q=1-P, which are not expressible through X_eff and H_eff alone. A proper Schrieffer-Wolff or downfolding treatment of both H and J would in general generate additional corrections. The paper provides no estimate or bound showing these are subleading in the regime where the bond-ordered gap is the relevant low-energy scale. Since Eq. (13), the flat-band amplification, and the "leading-order failure of k·p" claim all depend on Eq. (11), this conclusion is not established. Please derive the effective current from a concrete microscopic lattice model (e.g., a few-band model with hi
  2. [Implications for Low-Energy Effective Theories, Eqs. (12)-(13)] Even within the proposed two-band Dirac model, X_eff = -(c/2)σ_z is introduced as an "equivalently one may choose" input, not derived from a lattice embedding. The ratio (1 + cm/v)^2 is therefore a property of an assumed current operator, not a general consequence of the Peierls substitution. To support the general claim that standard k·p theories fail at leading order for bond-ordered gaps, the authors should show that this X_eff is the correct low-energy projection of a concrete lattice Hamiltonian and that the exact lattice optical conductivity matches the effective-theory calculation in the relevant frequency window.
minor comments (5)
  1. [Abstract and Conclusion] The word "uniquely" is stronger than the presented argument: the Peierls derivation shows consistency and naturalness of D_μP = i[P, r̂_μ], but it does not prove that no other convention-independent QGT can describe the same physical observables. I suggest softening the claim unless a no-go statement is supplied.
  2. [Eq. (9)] The commutator with r̂ in an infinite periodic system should be defined more carefully (e.g., through Bloch matrix elements or the k-space representation). The citation to Resta is appropriate, but the domain of the operator and the meaning of i[P, r̂_μ] on periodic Bloch states deserve a brief comment.
  3. [Conclusion] The text says "we proved" for the k·p failure. In light of the missing downfolding derivation for Eq. (11), phrasing such as "we illustrate" or "we argue" would be more precise.
  4. [Supplemental Material] The main text refers to the Supplemental Materials for SSH algebra and numerical details, but no supplement is included with the arXiv submission. The SSH calculation should be sketched in the main text or the supplement should be made available.
  5. [Fig. 3] The statement that the mechanical SSH chains have identical vibrational bands for different intra-cell spacings assumes ideal springs whose force constants are independent of the spacing. This assumption should be stated explicitly.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the physical QGT is defined independently of the observables and the central claim does not reduce to a fit or a self-citation.

full rationale

The paper's central construction is self-contained rather than circular. Qphys is defined by Eq. (9)-(10) through the covariant derivative D_mu P = i[P, rhat_mu], and the argument that this is the physically relevant quantity rests on the Peierls-substitution derivation of the current (Eq. (7)-(8)), which is a standard external input. The SSH demonstration is a direct calculation showing that Convention A violates the bulk-invariance requirement f(t1,t2)=f(t2,t1) while Convention B restores it; this is a check, not a fitting of the conclusion. The paper does cite its own Ref. [10] for Eq. (3), but Eq. (3) is independently established and also cited to Ref. [12], so the self-citation is not load-bearing. The word "uniquely mandated" overstates the strength of the argument, and the effective-theory current in Eq. (11) is asserted via "applying Eq. (8) to the projected theory" without displaying the projection algebra; however, at q=0 the cross terms between low- and high-energy subspaces vanish when P is a spectral projector, so the leading-order conclusion is not shown to be circular. No fitted parameter is relabeled as a prediction, and no central result is equivalent to an input by construction. The identified concerns are correctness/overclaim issues rather than circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central derivation has no fitted constants; the only hand-chosen input is the illustrative embedding parameter c in the k·p example. The main structural assumptions are the Peierls current, the position-operator covariant derivative, the projection of the current to the effective theory, and the validity of the two-band optical formula. Qphys and X_eff are definitions/projections of existing operators, not newly invented entities.

free parameters (1)
  • embedding parameter c (effective orbital separation) = c = 0, 1/3, 1/2 in Fig. 2; also cm/v in Eq. (13)
    Chosen by hand in the 1D massive Dirac example; the predicted optical conductivity ratio (1 + cm/v)^2 depends on c, which is the point that H_eff alone does not fix the response. It is an input of the demonstration, not a fitted constant.
assumptions (5)
  • domain assumption Peierls substitution with full orbital coordinates gives the physical current operator
    Used in Eqs. (7)-(8) to derive the current in Conventions A and B; standard in tight-binding, but is the load-bearing anchor for the 'physical' label.
  • domain assumption The position operator r̂ = R̂ + X̂ is a valid operator on the periodic Hilbert space and D_μP = i[P, r̂_μ] is the correct covariant derivative
    Invoked in Eq. (9) to define Qphys; the paper cites Resta [51,52] but does not rigorously define r̂ on a torus. In the B basis it reduces to ∂P, so it is internally consistent, but its formal standing is assumed.
  • ad hoc to paper The effective current is obtained by projecting the full current onto the low-energy subspace, giving Eq. (11)
    This is not derived; high-energy virtual processes can alter the projected current. Load-bearing for the k·p failure claim.
  • domain assumption The optical conductivity formula Eq. (3) with the quantum metric is valid for two-band systems
    Cited from Refs. [10,12]; used to identify the convention-B metric as physical.
  • domain assumption Bulk physical observables (Wannier spread, optical conductivity) must be invariant under arbitrary repositioning of the unit-cell boundary
    Used in the SSH paradox test; accepted physical requirement, not derived.

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Cite this review

Pith. "Pith review of Orbital Embedding and the Physical Definition of Quantum Geometry." pith.science (2026). https://pith.science/paper/PBHOEUAU

@misc{pith2026260721882,
  author       = {Pith},
  title        = {Pith review of: Orbital Embedding and the Physical Definition of Quantum Geometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PBHOEUAU}},
  note         = {Machine review of arXiv:2607.21882}
}
abstract

The Quantum Geometric Tensor, encompassing the quantum metric and Berry curvature, is a central concept in modern condensed matter physics. However, its standard calculation via $k$-derivatives of the Bloch projector conceals a fundamental ambiguity regarding the choice of unit-cell convention, specifically in the treatment of intra-cell orbital positions (i.e., with or without the orbital position $e^{ikx_\alpha}$). We resolve this inconsistency by introducing a convention-independent physical QGT defined via a covariant derivative that explicitly incorporates the full position operator. We demonstrate that this formulation is uniquely mandated by the microscopic derivation of the physical current via the Peierls substitution. Notably, we uncover a leading-order failure in standard $k \cdot p$ effective theories for systems with bond-ordered gaps, identifying a need for caution in their application. Finally, we propose geometric engineering as a new design paradigm, enabling the independent tuning of geometric responses without altering the energy dispersion.

Figures

Figures reproduced from arXiv: 2607.21882 by the authors.

Figure 1
Figure 1. FIG. 1. The unit-cell paradox in the SSH model. (a) Lattice struc [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Impact of effective orbital embedding in the one-dimensional [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Spectral invariance under geometric engineering. (a) Me [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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Reference graph

Works this paper leans on

53 extracted references · 4 linked inside Pith · cited by 1 Pith paper

  1. [42]

    S. Lee, S. H. Lee, and B.-J. Yang, Embedding independent length scale of flat bands, arXiv preprint arXiv:2511.02240 (2025)

    arXiv 2025

  2. [1]

    Provost and G

    J. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Commun. Math. Phys.76, 289 (1980)

    1980

  3. [2]

    Y .-Q. Ma, S. Chen, H. Fan, and W.-M. Liu, Abelian and non- abelian quantum geometric tensor, Phys. Rev. B81, 245129 (2010)

    2010

  4. [3]

    M. V . Berry, The quantum phase, five years after, inGeometric Phases in Physics, edited by F. Wilczek and E. Shapere (1989) pp. 3–28

    1989

  5. [4]

    A. Gao, N. Nagaosa, N. Ni, and S.-Y . Xu, Quantum geom- etry phenomena in condensed matter systems, arXiv preprint arXiv:2508.00469 (2025)

    arXiv 2025

  6. [5]

    T ¨orm¨a, S

    P. T ¨orm¨a, S. Peotta, and B. A. Bernevig, Superconductivity, superfluidity and quantum geometry in twisted multilayer sys- tems, Nat. Rev. Phys.4, 528 (2022)

    2022

  7. [6]

    J. Yu, B. A. Bernevig, R. Queiroz, E. Rossi, P. T¨orm¨a, and B.-J. Yang, Quantum geometry in quantum materials, npj Quantum Materials10, 101 (2025)

    2025

  8. [7]

    T ¨orm¨a, Essay: Where can quantum geometry lead us?, Phys

    P. T ¨orm¨a, Essay: Where can quantum geometry lead us?, Phys. 6 Rev. Lett.131, 240001 (2023)

    2023

Show all 53 references
  1. [8]

    Nagaosa, J

    N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Anomalous hall effect, Reviews of modern physics82, 1539 (2010)

    2010

  2. [9]

    Xiao, M.-C

    D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on elec- tronic properties, Rev. Mod. Phys.82, 1959 (2010)

    1959

  3. [10]

    Oh, S.-W

    C.-g. Oh, S.-W. Kim, K. W. Kim, B. Monserrat, and J.-W. Rhim, Mass-invariant universal optical conductivity from quantum ge- ometry, Science Advances12, eady2033 (2026)

    2026

  4. [11]

    Oh and S.-W

    C.-g. Oh and S.-W. Kim, Color and transparency from quantum geometry, arXiv preprint arXiv:2507.20904 (2025)

    2025 arXiv

  5. [12]

    Ezawa, Analytic approach to quantum metric and optical conductivity in dirac models with parabolic mass in arbitrary dimensions, Physical Review B110, 195437 (2024)

    M. Ezawa, Analytic approach to quantum metric and optical conductivity in dirac models with parabolic mass in arbitrary dimensions, Physical Review B110, 195437 (2024)

    2024

  6. [13]

    K. Das, S. Lahiri, R. B. Atencia, D. Culcer, and A. Agarwal, In- trinsic nonlinear conductivities induced by the quantum metric, Physical Review B108, L201405 (2023)

    2023

  7. [14]

    Gao, Y .-F

    A. Gao, Y .-F. Liu, J.-X. Qiu, B. Ghosh, T. V . Trevisan, Y . On- ishi, C. Hu, T. Qian, H.-J. Tien, S.-W. Chen,et al., Quantum metric nonlinear Hall effect in a topological antiferromagnetic heterostructure, Science381, 181 (2023)

    2023

  8. [15]

    A.-K. Wu, S. Sarkar, X. Wan, K. Sun, and S.-Z. Lin, Quantum- metric-induced quantum hall conductance inversion and reen- trant transition in fractional Chern insulators, Physical Review Research6, L032063 (2024)

    2024

  9. [16]

    Mitscherling and T

    J. Mitscherling and T. Holder, Bound on resistivity in flat-band materials due to the quantum metric, Physical Review B105, 085154 (2022)

    2022

  10. [17]

    Huhtinen and P

    K.-E. Huhtinen and P. T ¨orm¨a, Conductivity in flat bands from the kubo-greenwood formula, Physical Review B108, 155108 (2023)

    2023

  11. [18]

    Ghosh, Y

    B. Ghosh, Y . Onishi, S.-Y . Xu, H. Lin, L. Fu, and A. Bansil, Probing quantum geometry through optical conductivity and magnetic circular dichroism, Science Advances10, eado1761 (2024)

    2024

  12. [19]

    Neupert, C

    T. Neupert, C. Chamon, and C. Mudry, Measuring the quan- tum geometry of bloch bands with current noise, Physical Re- view B—Condensed Matter and Materials Physics87, 245103 (2013)

    2013

  13. [20]

    C.-g. Oh, K. W. Kim, and J.-W. Rhim, Thermoelectric trans- port driven by the hilbert–schmidt distance, Advanced Science , 2411313 (2024)

    2024

  14. [21]

    N. Wang, D. Kaplan, Z. Zhang, T. Holder, N. Cao, A. Wang, X. Zhou, F. Zhou, Z. Jiang, C. Zhang,et al., Quantum-metric- induced nonlinear transport in a topological antiferromagnet, Nature621, 487 (2023)

    2023

  15. [22]

    J. Han, T. Uchimura, Y . Araki, J.-Y . Yoon, Y . Takeuchi, Y . Ya- mane, S. Kanai, J. Ieda, H. Ohno, and S. Fukami, Room- temperature flexible manipulation of the quantum-metric struc- ture in a topological chiral antiferromagnet, Nature Physics20, 1110 (2024)

    2024

  16. [23]

    Han, J.-W

    S.-H. Han, J.-W. Rhim, and C.-g. Oh, Klein tunneling in quan- tum geometric semimetals, arXiv preprint arXiv:2601.12797 (2026)

    2026

  17. [24]

    Peotta and P

    S. Peotta and P. T¨orm¨a, Superfluidity in topologically nontrivial flat bands, Nat. Commun.6, 8944 (2015)

    2015

  18. [25]

    Liang, T

    L. Liang, T. I. Vanhala, S. Peotta, T. Siro, A. Harju, and P. T ¨orm¨a, Band geometry, Berry curvature, and superfluid weight, Physical Review B95, 024515 (2017)

    2017

  19. [26]

    C.-g. Oh, H. Watanabe, and N. Tsuji, Role of quantum geom- etry in the competition between higgs mode and quasiparticles in third-harmonic generation of superconductors, arXiv preprint arXiv:2512.01200 (2025)

    2025

  20. [27]

    J.-W. Rhim, K. Kim, and B.-J. Yang, Quantum distance and anomalous landau levels of flat bands, Nature584, 59 (2020)

    2020

  21. [28]

    J. Jung, H. Lim, and B.-J. Yang, Quantum geometry and lan- dau levels of quadratic band crossings, Physical Review B109, 035134 (2024)

    2024

  22. [29]

    Oh, J.-W

    C.-g. Oh, J.-W. Rhim, and B.-J. Yang, Revisiting the magnetic responses of bilayer graphene from the perspective of quantum distance, Physical Review B110, 155412 (2024)

    2024

  23. [30]

    Hwang, J.-W

    Y . Hwang, J.-W. Rhim, and B.-J. Yang, Geometric characteriza- tion of anomalous Landau levels of isolated flat bands, Nature communications12, 6433 (2021)

    2021

  24. [31]

    C.-g. Oh, T. Kitamura, A. Daido, J.-W. Rhim, and Y . Yanase, Magnetic phase transitions driven by quantum geometry, Phys- ical Review Research8, 023096 (2026)

    2026

  25. [32]

    Kitamura, A

    T. Kitamura, A. Daido, and Y . Yanase, Spin-triplet supercon- ductivity from quantum-geometry-induced ferromagnetic fluc- tuation, Physical Review Letters132, 036001 (2024)

    2024

  26. [33]

    Y . Gao, S. A. Yang, and Q. Niu, Geometrical effects in orbital magnetic susceptibility, Physical Review B91, 214405 (2015)

    2015

  27. [34]

    Shimizu, C.-g

    M. Shimizu, C.-g. Oh, and Y . Yanase, Magnetic fluctuations driven by quantum geometry, arXiv preprint arXiv:2602.14511 (2026)

    2026

  28. [35]

    C.-g. Oh, D. Cho, S. Y . Park, and J.-W. Rhim, Bulk-interface correspondence from quantum distance in flat band systems, Communications Physics5, 320 (2022)

    2022

  29. [36]

    Kim, C.-g

    H. Kim, C.-g. Oh, and J.-W. Rhim, General construction scheme for geometrically nontrivial flat band models, Commu- nications Physics6, 305 (2023)

    2023

  30. [37]

    J. Yu, C. J. Ciccarino, R. Bianco, I. Errea, P. Narang, and B. A. Bernevig, Non-trivial quantum geometry and the strength of electron–phonon coupling, Nature Physics20, 1262 (2024)

    2024

  31. [38]

    Komissarov, T

    I. Komissarov, T. Holder, and R. Queiroz, The quantum geomet- ric origin of capacitance in insulators, Nature Communications 15, 4621 (2024)

    2024

  32. [39]

    S. H. Simon and M. S. Rudner, Contrasting lattice geometry de- pendent versus independent quantities: Ramifications for Berry curvature, energy gaps, and dynamics, Physical Review B102, 165148 (2020)

    2020

  33. [40]

    Sastges, I

    M. Sastges, I. Baek, H. Lee, H.-W. Lee, Y . Mokrousov, and D. Go, Modern approach to orbital Hall effect based on wannier picture of solids, arXiv preprint arXiv:2604.08280 (2026)

    2026 arXiv

  34. [41]

    H. Lee, I. Baek, M. Sastges, Y . Mokrousov, H.-W. Lee, and D. Go, Anatomy of the modern theory of orbital magnetism from first-principles: term-by-term analysis in the gauge- covariant formalism, arXiv preprint arXiv:2603.19875 (2026)

    2026 arXiv

  35. [43]

    Marzari and D

    N. Marzari and D. Vanderbilt, Maximally localized generalized wannier functions for composite energy bands, Physical Review B56, 12847 (1997)

    1997

  36. [44]

    W. J. Jankowski, J. J. Thompson, B. Monserrat, and R.-J. Slager, Excitonic topology and quantum geometry in organic semiconductors, Nature Communications16, 4661 (2025)

    2025

  37. [45]

    S. Chen, S. Chaudhary, G. Refael, and C. Lewandowski, En- hancing shift current response via virtual multiband transitions, Communications Physics7, 250 (2024)

    2024

  38. [46]

    Zeng, W.-X

    X.-L. Zeng, W.-X. Lai, Y .-W. Wei, and Y .-Q. Ma, Quantum geometric tensor and the topological characterization of the extended su–schrieffer–heeger model, Chinese Physics B33, 030310 (2024)

    2024

  39. [47]

    T. Xu, T. Morimoto, and J. E. Moore, Nonlinear optical effects in inversion-symmetry-breaking superconductors, Physical Re- view B100, 220501 (2019)

    2019

  40. [48]

    Terada, S

    I. Terada, S. Kitamura, H. Watanabe, and H. Ikeda, Multi- 7 tunneling effect of nonreciprocal landau-zener tunneling: In- sights from dc field responses, Physical Review B111, 064315 (2025)

    2025

  41. [49]

    W.-P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in poly- acetylene, Physical Review Letters42, 1698 (1979)

    1979

  42. [50]

    Ventura, D

    G. Ventura, D. Passos, J. Lopes dos Santos, J. Viana Par- ente Lopes, and N. M. Peres, Gauge covariances and nonlinear optical responses, Physical Review B96, 035431 (2017)

    2017

  43. [51]

    Resta, Quantum-mechanical position operator in extended systems, Physical Review Letters80, 1800 (1998)

    R. Resta, Quantum-mechanical position operator in extended systems, Physical Review Letters80, 1800 (1998)

    1998

  44. [52]

    Resta, Macroscopic polarization in crystalline dielectrics: the geometric phase approach, Reviews of modern physics66, 899 (1994)

    R. Resta, Macroscopic polarization in crystalline dielectrics: the geometric phase approach, Reviews of modern physics66, 899 (1994)

    1994

  45. [53]

    C.-Y . Hou, C. Chamon, and C. Mudry, Electron fractionaliza- tion in two-dimensional graphenelike structures, Physical Re- view Letters98, 186809 (2007)

    2007

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.