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REVIEW 3 major objections 4 minor 65 references

Chern insulators in two and three dimensions: A global perspective

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper derives globally smooth formulas for the Chern number and Chern vector in terms of velocity matrix elements, and a finite-frequency conductivity that reduces to the quantum anomalous Hall effect.

desk verdict A careful reformulation of known Chern-invariant formulas with a useful optical-response derivation, but the central novelty claim is overstated and needs reframing before publication. read the letter →

arxiv 2506.04466 v2 pith:MA3EOIZ6 submitted 2025-06-04 cond-mat.mes-hall

Jason G. Kattan , J. E. Sipe This is my paper
classification cond-mat.mes-hall PACS 73.43.-f
keywords CherninsulatornumbervectorBerrycurvaturevelocitymatrixelementsquantumanomalousHalleffectcell-periodicpotentialopticalactivity
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

The paper sets out to show that the topological invariants of a Chern insulator need not be computed from locally defined Berry connections on gauge patches. Its central claim is that the two-dimensional Chern number and the three-dimensional Chern vector can be written as integrals over the full Brillouin zone of products of velocity matrix elements divided by band-energy differences, with no special handling of band-crossing points. The derivation works in a second-quantized field theory whose Hamiltonian contains a static, cell-periodic vector potential that spontaneously breaks time-reversal symmetry, modeling local magnetic moments inside each unit cell. The same formalism produces a long-wavelength conductivity tensor whose static limit is the quantum anomalous Hall effect and whose frequency dependence describes optical response, with an effective dielectric tensor satisfying generalized Kramers-Kronig relations. A sympathetic reader would care because the new formulas are globally smooth and involve only quantities straightforwardly available from Bloch states, potentially simplifying numerical evaluations of topology in real or model materials.

What carries the argument

The load-bearing machinery is the velocity operator $v(x) = p(x)/m + (\hbar/4m^2c^2)\sigma \times \nabla V_\Gamma(x)$ of the Hamiltonian with a cell-periodic vector potential, together with two identities: the flat-connection identity (28), which rewrites $\epsilon^{iab}\partial_a \xi^b_{mn}$ as a sum of products of non-Abelian Berry connections, and the velocity-matrix identity (40) linking $v^i_{nm}$ to band energies and off-diagonal Berry connections. These identities let a sum of Abelian Berry curvatures over occupied bands be re-expressed as products of velocity matrix elements with energy denominators, and the filling-factor prefactor $f_{nm}=f_n-f_m$ removes the dangerous $m=n$ terms at band degeneracies. The same velocity matrix elements then carry the response calculation, so topology and optical conductivity are handled with one set of objects.

What would settle it

Evaluate Eq. (41) and the standard gauge-patch Chern-number expression (38) for a two-band tight-binding Chern insulator with spin-orbit coupling and a cell-periodic vector potential, using the same Bloch states and a fine k mesh; the central claim is falsified if the two computations disagree on the integer Chern number, or if the local identity (28) has a nonzero residual at generic k points.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the obstruction to defining smooth Bloch frames does not obstruct the Chern invariants themselves: using the velocity operator $\mathbf{v}(x)$ and the identity $v^i_{nm}(k) = \delta_{nm}\partial E_m/\partial k^i + (i/\hbar)(E_n - E_m)\xi^i_{nm}(k)$, the authors convert the local, degenerate-singular Berry-curvature integral (38) into the global expression (41) for $C_V$ in two dimensions, and similarly produce the global Chern-vector formula (49) in three dimensions. Because the prefactor $f_{nm}=f_n-f_m$ vanishes for $m=n$, no pole contributes at band crossings, so the integrands are smooth everywhere. The paper further claims that these same velocity-matrix building blocks determine the long-wavelength conductivity tensors (53) and (55), which match the Kubo-Greenwood formula and reduce in the static limit to the quantum anomalous Hall effect.

Load-bearing premise

The load-bearing premise is that a particular relation between the wavefunction overlaps and their momentum derivatives holds for the spinor Bloch states with spin-orbit coupling and a periodic magnetic potential, away from band crossings; the global formulas are only established if this relation is valid.

Editorial extensions

If this is right

  • The Chern number $C_V$ in two dimensions and the Chern vector $\mathbf{C}_V$ in three dimensions can be evaluated from velocity matrix elements and band energies without constructing gauge patches or excising degeneracy points.
  • The global expressions vanish when any of the three discrete symmetries of the tenfold way is present, so the model's Hamiltonian is consistent with the defining symmetries of a Chern insulator.
  • The static limit of the conductivity tensors (53) and (55) reproduces the quantum anomalous Hall effect, while the finite-frequency terms yield an effective dielectric tensor that describes circular birefringence and dichroism.
  • The Kubo-Greenwood formula (57) is shown to give exactly the same conductivity tensors in both two and three dimensions when evaluated with this Hamiltonian, so the response result is not an artifact of the field-theoretic derivation.
  • Below the band gap the absorptive part of the Kubo susceptibility vanishes, so the generalized Kramers-Kronig relations reduce cleanly to the usual dispersion relations for the reactive part.

Reading between the lines

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

  • A direct numerical test of Eq. (41) on an existing two-band tight-binding model would settle whether the smooth integrand reproduces the standard integer in practice; this is the fastest way to check the paper's central claim independently.
  • Because the formulas use only velocity matrix elements and energies, they could be implemented in ab initio codes with a uniform k mesh and no gauge fixing, which may make Chern numbers and optical Hall conductivities routine outputs.
  • The finite-frequency Hall terms in the dielectric tensor imply that optical probes such as Faraday rotation could measure the Chern vector without transport contacts, provided the sample is thin enough for the long-wavelength approximation to hold.
  • The same field-theoretic machinery, with a cell-periodic vector potential, may naturally extend to time-dependent or Floquet-engineered vector potentials, yielding topological invariants for driven systems in the same velocity-matrix language.
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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

3 major / 4 minor

Summary. The paper develops a second-quantized field theory of Chern insulators in which time-reversal symmetry is broken by a cell-periodic static vector potential, and uses it to derive supposedly novel global expressions for the Chern number (2D, Eq. (41)) and the Chern vector (3D, Eq. (49)) in terms of velocity matrix elements and the full band structure. The paper also derives a long-wavelength conductivity tensor (Eqs. (53), (55)) that combines a dynamical Kubo term with a static Hall term proportional to the Chern invariant, introduces an effective dielectric tensor, and claims consistency with the Kubo-Greenwood formula (Appendix D). The central theoretical structure is coherent, and the 2D Chern-number expression indeed agrees with the standard Berry-curvature formula. However, the claimed novelty is overstated, as Eq. (41) reduces to the well-known Kubo/TKNN velocity-gauge expression for the Chern number, and the paper provides no numerical demonstration of the alleged practical advantages.

Significance. If the results are correct, the manuscript provides a self-consistent, gauge-invariant formalism for Chern invariants and optical response that incorporates spin-orbit coupling and a periodic vector potential. The 3D Chern-vector formula (49) and the finite-frequency generalization of the quantum anomalous Hall conductivity are useful extensions. The derivation of the conductivity tensor from the microscopic polarization-magnetization formalism and its equivalence to Kubo-Greenwood is a meaningful cross-check. The main limitation is novelty: Eq. (41) is essentially the standard Kubo formula for the Hall conductivity, and the paper does not clearly delineate what is new beyond what is already in the authors' prior work [36, 39]. The absence of any numerical benchmark further weakens the claim that these expressions are practically advantageous.

major comments (3)
  1. [§III.A, Eq. (41)] The novelty claim for Eq. (41) is overstated. This expression is the standard Kubo/TKNN formula for the Chern number written in velocity-matrix-element form, widely used in calculations of the anomalous Hall conductivity. The authors do not cite the original Kubo/TKNN result or any of the standard references where this velocity-gauge formula appears, nor do they explain precisely in what sense Eq. (41) is 'novel' compared to those known expressions. This is a load-bearing issue because the abstract and introduction announce 'novel expressions', and the manuscript should either demonstrate a genuine difference (e.g., the full spinor treatment or the 3D Chern vector) or explicitly frame the contribution as a rederivation in a new field-theoretic language.
  2. [§III and §V] The paper claims that the global expressions (41) and (49) circumvent numerical difficulties associated with band crossings, but no numerical example is provided to support this claim. A simple benchmark, such as the Haldane model in 2D or a lattice model with a nontrivial Chern vector in 3D, would demonstrate that the formulas yield correct integer invariants and are numerically stable in practice. Without such a demonstration, the practical-advantage claim remains an assertion rather than an established property.
  3. [§III.A and Appendix D] The derivation of Eq. (41) from Eq. (38) is only sketched as 'straightforward' and rests on identities (28) and (40), which are cited to the authors' previous work [36] rather than proved. Identity (28) is in fact the Maurer-Cartan equation for the full Bloch frame and is model-independent; the paper would be much stronger if it stated and proved this identity explicitly, rather than referencing earlier work. Similarly, the Kubo-Greenwood equivalence in Appendix D contains steps that are summarized without detailed algebra. Providing these derivations would make the central claims self-contained and easier to verify.
minor comments (4)
  1. [Introduction] There are several typographical errors: 'intrinisic' instead of 'intrinsic', and the phrase 'being are described by' is grammatically incorrect.
  2. [Appendix A] The phrase 'local of degeneracies' should be 'locus of degeneracies'.
  3. [Appendix D, Eq. (D3)] The diagonal term in Eq. (D3) appears to have incorrect dimensions: it is written as (ℏ k_i)/(2m) δ_nm, but from ∂H_k/∂k_i = ℏ v_i + (ℏ^2/m) k_i the diagonal term should be (ℏ^2/m) k_i δ_nm (up to fine details of the SOC contribution). This does not affect the subsequent result because f_nm δ_nm = 0, but the equation should be corrected for consistency.
  4. [Footnote [55]] The footnote 'See [52]' attached to Eq. (47) is opaque; it likely refers to a different reference and should be clarified or removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eqs. (41) and (49) are derived from standard kinematic identities, not from the invariants themselves.

full rationale

The central derivation proceeds from the standard chart expression (38) for the Chern number to the velocity-matrix form (41) using two identities: the flat-connection identity (28), which follows from F(k)=0 (Eq. 27) and is cited to [52] as well as [36], and the velocity-matrix identity (40), a standard Bloch-frame relation. Neither identity is defined in terms of the target Chern invariant, and no parameter is fitted to any subset of data. The Chern vector expression (49) is obtained by the same substitution on each 2-cycle. The paper's self-citations ([36], [39], [44], [46]) supply auxiliary constructions (quasi-Bloch frames, Wannier functions, conductivity tensors), but the load-bearing algebra is shown in the text and is independent of those citations. The novelty claim is arguably overstated because (41) is the well-known Kubo/velocity-gauge formula for the Chern number, and (49) is its 3D analogue; however, an overstated novelty claim is not circularity. There is no specific step in which a prediction is equivalent to an input by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted in the derivations. The model rests on physical assumptions (periodic internal magnetic field, independent-particle picture, frozen moments) and on standard mathematical results (Bloch theorem, bundle triviality, Kubo-Greenwood formula), some of which are cited to the authors' earlier papers rather than derived here. No new particles or forces are introduced.

assumptions (6)
  • domain assumption The static vector potential a_static(x) is cell-periodic and generates a lattice-periodic magnetic field b_static(x) that persists in the ground state, spontaneously breaking time-reversal symmetry.
    Defines the model Hamiltonian (7); physically motivated by local magnetic moments in Chern insulators like MnBi2Te4, but not derived from a microscopic spin model.
  • domain assumption The independent-particle approximation holds, with electron-electron interactions neglected or absorbed at mean-field level into V_Gamma(x).
    Stated in Sec. II; limits applicability to correlated systems like moire Chern insulators where interactions are essential.
  • standard math The infinite-rank Bloch bundle over the Brillouin zone is trivializable, so a flat connection exists and quasi-Bloch frames are global.
    Invoked after Eq. (17) via Kuiper's theorem (Ref. [5]); standard result for infinite-rank Hilbert bundles, but not proved in the paper.
  • standard math The velocity-matrix-element identity (40), v^i_nm = delta_nm (1/hbar) d_i E_m + (i/hbar)(E_n - E_m) xi^i_nm, is valid for the spinor Hamiltonian with spin-orbit coupling.
    Cited to the authors' prior work [36]; standard relation in Bloch theory, but the spin-orbit-modified velocity operator (39) makes the proof nontrivial.
  • standard math The Kubo-Greenwood formula (57) is the correct benchmark for the long-wavelength conductivity.
    Appendix D; standard linear-response result, used to validate the derived conductivity tensor.
  • domain assumption The magnetic field b_static is treated as a fixed classical background, ignoring back-reaction and orbital effects of the electrons on the local moments.
    Frozen-ion and frozen-moment approximation; physically reasonable for strongly ordered moments, but a simplification.

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Cite this review

Pith. "Pith review of Chern insulators in two and three dimensions: A global perspective." pith.science (2026). https://pith.science/paper/MA3EOIZ6

@misc{pith2026250604466,
  author       = {Pith},
  title        = {Pith review of: Chern insulators in two and three dimensions: A global perspective},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MA3EOIZ6}},
  note         = {Machine review of arXiv:2506.04466}
}
read the original abstract

We introduce a second-quantized field theory for Chern insulators in which the Hamiltonian features a static vector potential that has the periodicity of the crystal's lattice and spontaneously breaks time-reversal symmetry in the system's ground state. Such a vector potential generates a magnetic field at the microscopic level that may be thought of as arising from local moments associated with one or more magnetic ions in each unit cell. Considering spinor electrons, we study the Chern invariants characterizing the topology of the occupied valence bands of Chern insulators in both two and three dimensions - the Chern number and the Chern vector, respectively - and we derive novel expressions for these topological invariants that are globally defined across the Brillouin zone and involve the full band structure of the system. We also study the long-wavelength response of a Chern insulator to electromagnetic fields at finite frequency, generalizing the quantum anomalous Hall effect in the static limit to the optical regime.

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Works this paper leans on

65 extracted references · 44 canonical work pages

  1. [39]

    Ghosh, Y

    B. Ghosh, Y. Onishi, S.-Y. Xu, H. Lin, L. Fu, and A. Ban- sil, Sci. Adv.10, eado1761 (2024)

    work page 2024

  2. [36]

    quasi-Bloch states

    that one can always form a collection of globally de- fined “quasi-Bloch states” throughk-dependent unitary transformations of the Bloch states, from which a set of exponentially localized Wannier functions for a Chern in- sulator can be defined, albeit ones that include at least some of the unoccupied conduction bands as well as the occupied valence band...

    work page

  3. [1]

    Time-reversal symmetry Time-reversal transformations are represented by an antiunitary operator ˆTacting on the electronic Fock space, and we denote byTthe corresponding time- reversal operator on the single-particle Hilbert space. The time-reversal transformation of the electron field op- erator is given by [4] ˆT ˆψσ(x) ˆT −1 = (UT ) σ′ σ ˆψσ′(x),(B2) w...

    work page

  4. [2]

    The particle-hole transformation of the electron field operator is ˆC ˆψσ(x) ˆC −1 = (U ∗ C) σ′ σ ˆψ† σ′(x),(B8) whereU C is a unitary operator acting on the spin sec- tor

    Particle-hole symmetry Particle-hole transformations are represented by auni- taryoperator ˆCacting on the electronic Fock space, and we denote byCthe corresponding particle-hole operator on the single-particle Hilbert space. The particle-hole transformation of the electron field operator is ˆC ˆψσ(x) ˆC −1 = (U ∗ C) σ′ σ ˆψ† σ′(x),(B8) whereU C is a unit...

    work page

  5. [3]

    The sublattice transformation of the elec- tron field operator is ˆS ˆψσ(x) ˆS−1 = (U ∗ S) σ′ σ ˆψ† σ′(x),(B10) whereU S =U ∗ CU ∗ T is a unitary operator acting on the spin sector

    Sublattice symmetry Sublattice transformations are represented by an an- tiunitary operator ˆS= ˆT◦ ˆCacting on the electronic Fock space that is the composition of the particle-hole and time-reversal operators, and we denote bySthe corresponding sublattice operator on the single-particle Hilbert space. The sublattice transformation of the elec- tron fiel...

    work page

  6. [4]

    S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Lud- wig, New J. Phys.12, 065010 (2010)

    2010

  7. [5]

    J. A. Sobota, Y. He, and Z.-X. Shen, Rev. Mod. Phys. 93, 025006 (2021)

    2021

  8. [6]

    B. A. Bernevig, C. Felser, and H. Beidenkopf, Nature 16 603, 41 (2022)

    work page 2022

Show all 65 references
  1. [7]

    Kitaev, AIP Conference Proceedings1134, 22 (2009)

    A. Kitaev, AIP Conference Proceedings1134, 22 (2009)

    2009

  2. [8]

    Vanderbilt,Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators(Cambridge University Press, 2018)

    D. Vanderbilt,Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators(Cambridge University Press, 2018)

    2018

  3. [9]

    D. S. Freed and G. W. Moore, Ann. Henri Poincare14 (2013)

    2013

  4. [10]

    A. W. W. Ludwig, Physica Scripta2016, 014001 (2015)

    2015

  5. [11]

    R. M. Kaufmann, S. Khlebnikov, and B. Wehe- fritz–Kaufmann, J. Geom. Phys.158, 103892 (2020)

    2020

  6. [12]

    Chang, C.-X

    C.-Z. Chang, C.-X. Liu, and A. H. MacDonald, Rev. Mod. Phys.95, 011002 (2023)

    2023

  7. [13]

    Resta, Euro

    R. Resta, Euro. Phys. J. B79, 121 (2011)

    2011

  8. [14]

    Monaco, Chern and fu–kane–mele invariants as topo- logical obstructions, inAdvances in Quantum Mechan- ics: Contemporary Trends and Open Problems, edited by A

    D. Monaco, Chern and fu–kane–mele invariants as topo- logical obstructions, inAdvances in Quantum Mechan- ics: Contemporary Trends and Open Problems, edited by A. Michelangeli and G. Dell’Antonio (Springer Inter- national Publishing, Cham, 2017) pp. 201–222

    2017

  9. [15]

    G. W. Winkler, A. A. Soluyanov, and M. Troyer, Phys. Rev. B93, 035453 (2016)

    2016

  10. [16]

    R. Yu, W. Zhang, H.-J. Zhang, S.-C. Zhang, X. Dai, and Z. Fang, Science329, 61 (2010)

    2010

  11. [17]

    Varjas, F

    D. Varjas, F. de Juan, and Y.-M. Lu, Phys. Rev. B96, 035115 (2017)

    2017

  12. [18]

    Chang, W

    C.-Z. Chang, W. Zhao, D. Y. Kim, H. Zhang, B. A. Assaf, D. Heiman, S.-C. Zhang, C. Liu, M. H. W. Chan, and J. S. Moodera, Nature Mat.14, 473 (2015)

    2015

  13. [19]

    Y. Ou, C. Liu, G. Jiang, Y. Feng, D. Zhao, W. Wu, X.-X. Wang, W. Li, C. Song, L.-L. Wang, W. Wang, W. Wu, Y. Wang, K. He, X.-C. Ma, and Q.-K. Xue, Adv. Mat. 30, 1703062 (2018)

    2018

  14. [20]

    F. D. M. Haldane, Phys. Rev. Lett.61, 2015 (1988)

    1988

  15. [21]

    Y. Deng, Y. Yu, M. Z. Shi, Z. Guo, Z. Xu, J. Wang, X. H. Chen, and Y. Zhang, Science367, 895 (2020), https://www.science.org/doi/pdf/10.1126/science.aax8156

    2020 doi

  16. [22]

    Serlin, C

    M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, and A. F. Young, Science367, 900 (2020)

    2020

  17. [23]

    T. Li, S. Jiang, B. Shen, Y. Zhang, L. Li, Z. Tao, T. De- vakul, K. Watanabe, T. Taniguchi, L. Fu, J. Shan, and K. F. Mak, Nature600, 641 (2021)

    2021

  18. [24]

    Resta, J

    R. Resta, J. Phys.: Cond. Matt.22, 123201 (2010)

    2010

  19. [25]

    Sticlet, F

    D. Sticlet, F. Pi´ echon, J.-N. Fuchs, P. Kalugin, and P. Si- mon, Phys. Rev. B85, 165456 (2012)

    2012

  20. [26]

    R. D. King-Smith and D. Vanderbilt, Phys. Rev. B47, 1651 (1993)

    1993

  21. [27]

    Resta, Rev

    R. Resta, Rev. Mod. Phys.66, 899 (1994)

    1994

  22. [28]

    Y. Liu, J. Li, and Q. Liu, Nano Letters23, 8650 (2023)

    2023

  23. [29]

    J. Shi, G. Vignale, D. Xiao, and Q. Niu, Phys. Rev. Lett. 99, 197202 (2007)

    2007

  24. [30]

    A. M. Essin, J. E. Moore, and D. Vanderbilt, Phys. Rev. Lett.102, 146805 (2009)

    2009

  25. [31]

    Ceresoli, T

    D. Ceresoli, T. Thonhauser, D. Vanderbilt, and R. Resta, Phys. Rev. B74, 024408 (2006)

    2006

  26. [32]

    Liu, X.-B

    T. Liu, X.-B. Qiang, H.-Z. Lu, and X. C. Xie, Nat. Sci. Rev.12, nwae334 (2024)

    2024

  27. [33]

    S. Liu, T. Ohtsuki, and R. Shindou, Phys. Rev. Lett. 116, 066401 (2016)

    2016

  28. [34]

    Liu, C.-B

    Z.-R. Liu, C.-B. Hua, T. Peng, and B. Zhou, Phys. Rev. B105, 245301 (2022)

    2022

  29. [35]

    Tse and A

    W.-K. Tse and A. H. MacDonald, Phys. Rev. B84, 205327 (2011)

    2011

  30. [37]

    Ozawa and B

    T. Ozawa and B. Mera, Phys. Rev. B104, 045103 (2021)

    2021

  31. [38]

    Mera and T

    B. Mera and T. Ozawa, Phys. Rev. B104, 045104 (2021)

    2021

  32. [40]

    P. T. Mahon, J. G. Kattan, and J. E. Sipe, Phys. Rev. B 107, 115110 (2023)

    2023

  33. [41]

    S. M. Girvin and K. Yang,Modern Condensed Matter Physics(Cambridge University Press, 2019)

    2019

  34. [42]

    Bloch, Zeitschrift f¨ ur Physik52, 555 (1929)

    F. Bloch, Zeitschrift f¨ ur Physik52, 555 (1929)

    1929

  35. [43]

    J. G. Kattan, A. H. Duff, and J. E. Sipe, Phys. Rev. B 111, 075202 (2025)

    2025

  36. [44]

    Lucarini, J

    V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen,Kramers-Kronig relations in optical materials research, Vol. 110 (Springer, 2005)

    2005

  37. [45]

    R. Kubo, J. Phys. Soc. Japan12, 570 (1957)

    1957

  38. [46]

    D. A. Greenwood, Proc. Phys. Soc.71, 585 (1958)

    1958

  39. [47]

    P. T. Mahon and J. E. Sipe, Phys. Rev. Research2, 043110 (2020)

    2020

  40. [48]

    A. H. Duff and J. E. Sipe, Phys. Rev. B106, 085413 (2022)

    2022

  41. [49]

    Fruchart and D

    M. Fruchart and D. Carpentier, Comptes Rendus Physique14, 779 (2013)

    2013

  42. [50]

    J. G. Kattan and J. E. Sipe, Phys. Rev. A107, 032820 (2023)

    2023

  43. [51]

    Panati, Ann

    G. Panati, Ann. Henri Poincare8, 995 (2007)

    2007

  44. [52]

    Fruchart, D

    M. Fruchart, D. Carpentier, and K. Gawedzki, Europhys. Lett.106, 60002 (2014)

    2014

  45. [53]

    Hamilton,Mathematical Gauge Theory(Springer In- ternational Publishing, 2017)

    M. Hamilton,Mathematical Gauge Theory(Springer In- ternational Publishing, 2017)

    2017

  46. [54]

    Kriegl and P

    A. Kriegl and P. Michor,The Convenient Setting of Global Analysis, Mathematical Surveys (American Math- ematical Society, 1997)

    1997

  47. [55]

    Nakahara,Geometry, Topology and Physics (2nd ed.) (CRC Press, 2003)

    M. Nakahara,Geometry, Topology and Physics (2nd ed.) (CRC Press, 2003)

    2003

  48. [56]

    A. Bohm, L. J. Boya, A. Mostafazadeh, and G. Rudolph, J. Geom. Phys.12, 13 (1993)

    1993

  49. [57]

    Brouder, G

    C. Brouder, G. Panati, M. Calandra, C. Mourougane, and N. Marzari, Phys. Rev. Lett.98, 046402 (2007)

    2007

  50. [58]

    To properly define this expression, a small region con- taining each point in the locus of degeneracies must be removed, and then the contributions of these regions to the integral over the BZ d must be evaluated using the generalized Stokes theorem [39]

  51. [59]

    P. T. Mahon, R. A. Muniz, and J. E. Sipe, Phys. Rev. B 99, 235140 (2019)

    2019

  52. [60]

    J. D. Jackson,Classical electrodynamics, 3rd ed. (Wiley, New York, NY, 1999)

    1999

  53. [61]

    Lei and A

    C. Lei and A. H. MacDonald, Phys. Rev. B108, 125424 (2023)

    2023

  54. [62]

    Bakry and L

    M. Bakry and L. Klinkenbusch, Advances in Radio Sci- ence16, 23 (2018)

    2018

  55. [63]

    H. J. Monkhorst and J. D. Pack, Phys. Rev. B13, 5188 (1976)

    1976

  56. [64]

    P. T. Mahon and J. E. Sipe, Phys. Rev. Research2, 033126 (2020)

    2020

  57. [65]

    A. H. Duff, A. Lau, and J. E. Sipe, Phys. Rev. B108, 235206 (2023)

    2023

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