Pith. sign in

REVIEW 4 major objections 6 minor 53 references

Higher-Order Topological States with Cleavage-Dependent Dirac Mass

T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Cleaving an obstructed atomic insulator along the right direction yields $e/2$ corner charges on exactly two of four corners, with dangling-bond tilt as the deciding factor.

desk verdict A plausible new half-corner HOTI mechanism with a cleavage-direction rule that is, for now, an inference from eight patterns rather than a proven criterion. read the letter →

arxiv 2608.10520 v1 pith:VC365HKD submitted 2026-08-11 cond-mat.mes-hall

Hongyu Chen , Min Long , Chuang Chen , Zi Yang Meng This is my paper · ORCID
classification cond-mat.mes-hall
keywords higher-ordertopologicalinsulatorcornerzeromodesfractionalchargeobstructedatomicDiracmasscrystallinecleavagebond-centerorbitalsentanglemententropy
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 whether cleaving an obstructed atomic insulator produces higher-order topological corner states is decided not by the symmetry of the cut alone but by the orientation of the dangling bonds left exposed at the corners. In the tight-binding model studied here, the exposed nearest-neighbor bonds act as the mass of a Dirac fermion at the edge, and corner zero modes appear only when those bonds tilt toward the corner region. When the criterion is met, exactly two diagonal corners carry $e/2$ fractional charge while the other two are vacancies of interstice charge, a pattern enforced by $C_2T$ or $C_2$ symmetry. The claim matters because it turns cleavage direction into a predictive control knob for higher-order topology, beyond what local charge profiles alone can say.

What carries the argument

The load-bearing object is the two-band Dirac Hamiltonian $H(k)=\hat{d}_i(k)\tau_i$ obtained by keeping only the valence bonds at the high-symmetry interstice site (labeled $4c$) that act as the Dirac mass. The sign of the mass on each edge is fixed by the rotation symmetry of the superlattice, giving $(+,+,-,-)$ for $C_2T$/$C_2$ and $(+,-,+,-)$ for $C_4$, and corners connecting edges of opposite sign accumulate $e/2$ zero modes. The paper extracts this model from the full tight-binding Hamiltonian and verifies the cleavage rule by tracing the winding of $\hat{d}(k)$ along the four Brillouin-zone edges and by computing bond-center localized orbitals that expose the interstice vacancies at the empty corners.

What would settle it

Within the same tight-binding model, construct a ninth cleavage pattern whose corner regions expose nearest-neighbor dangling bonds aligned straight along the edges rather than tilted toward the corner; a zero-energy corner doublet in that spectrum would falsify the directionality criterion, as would a tilted-bond pattern that fails to produce corner states.

Watch

Extended reading notes

Core claim

Starting from an obstructed atomic insulator whose low-energy description is a gapped Dirac fermion, the paper shows that the sign pattern of the Dirac mass on the four edges is set by the superlattice symmetry: with $C_2T$ or $C_2$ symmetry the signs run $(+,+,-,-)$, so two opposite corners connect edges of opposite mass and host double-degenerate $e/2$ fractional charges, while the other two corners are empty interstice vacancies. The central discovery is that this zero-mode structure requires the nearest-neighbor dangling bonds at the corner to be not only exposed but also sloped toward the corner; comparing eight cleavage patterns, the four that have corner states are exactly those whose dangling bonds show this angular tilt, and the four that do not have it, even with higher $C_4$ symmetry, are trivial. The directionality is corroborated by the winding of the $\hat{d}$-vector along the Brillouin-zone edges in a simplified two-band model, where only two edges wind fully and the other two turn back. A further result is that the topological corners have lower site-resolved entanglement entropy but higher energy than the bulk, an ordering opposite to the energy distribution.

Load-bearing premise

The rule that corner zero modes appear exactly when the exposed nearest-neighbor dangling bonds tilt toward the corner is extracted from a simplified two-band model that keeps only the valence bonds acting as the Dirac mass, so the argument assumes the edge mass sign pattern is the only quantity controlling the zero modes.

Editorial extensions

If this is right

  • Cleavage direction becomes a switch: the same obstructed atomic insulator can be cut into a higher-order topological insulator with $e/2$ corner charges or into a trivial edge-polarized insulator by choosing different edge orientations.
  • Higher-order corner states can be protected by $C_2T$ or $C_2$ symmetry alone, with only two charged corners, rather than requiring $C_4$ or higher rotation symmetry.
  • The interstice charge vacancies at the empty corners are not defects but the topological partners of the fractional charges, so local charge profiles can diagnose the phase.
  • Corner zero modes coexist with a real-space pattern in which the topological corners have lower entanglement entropy but higher energy than the bulk, implying a boundary compensation between energy and entanglement.

Reading between the lines

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

  • The directionality rule should be testable in classical-wave analogues: a microwave or acoustic lattice built from the same hopping and flux pattern should show corner modes for tilted cuts but not for straight-bond cuts.
  • If the cleavage rule generalizes to three dimensions, a three-dimensional obstructed atomic insulator might be switched between hinge-conducting and hinge-insulating phases by the orientation of exposed hinge bonds.
  • The energy-entropy compensation at the corners suggests a boundary thermodynamic relation that could be probed site-by-site in quantum simulators or ultracold atom lattices.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. This manuscript studies a tight-binding model of an obstructed atomic insulator, Eq. (1), with nearest-neighbor hopping t~=1, next-nearest-neighbor hopping t=sqrt(2)/4 with flux phase phi=±3π/4, and third-neighbor hopping t'=-1/4. The lattice is cleaved into eight superlattice patterns by shifting atomic positions, rotating crystalline orientation, and flipping magnetic flux. Exact diagonalization shows zero-energy, double-degenerate e/2 fractional corner charges in patterns 1, 4, 5, and 8, while the two opposite corners host vacancies of interstice charge at WP 4c. The authors attribute the existence and location of the corner modes to the sign pattern of a Dirac mass on the edges, set by C2T or C2 symmetry, and to the exposure and angular tilt of nearest-neighbor dangling bonds at the corners. This is supported by a simplified two-band model whose d-vector winds along two edges for the C2T/C2 patterns. The paper also reports lower site-resolved 'entanglement' entropy and higher energy at the topological corners compared with the bulk.

Significance. The numerical observation that four of eight systematically enumerated cleavage patterns host zero-energy corner modes, together with the explicit winding analysis of the simplified d-vector model, is a concrete and reproducible contribution to the phenomenology of higher-order topological states in obstructed atomic insulators. The proposed cleavage-directionality rule, if rigorously established, would be practically valuable for predicting corner modes from real-space bond geometry. The manuscript is also transparent about its model parameters and enumerates the eight patterns in detail. However, the central rule is not yet established as a robust criterion: it rests on a drastic reduction of the Hamiltonian, a real-space bond-tilt interpretation that is asserted rather than derived, and a single parameter set. The vacancy picture is partly definitional, and the entropy diagnostic is mislabeled. These issues are substantial but fixable within the scope of the manuscript, so major revision is appropriate.

major comments (4)
  1. [Interstice Charge Vacancies, Eq. (2)] The vacancy claim is constructed by definition rather than derived from an independent calculation. In Eq. (2), P_Bond is defined as the average of P_Atom over adjacent atoms, so P_Bond=0 automatically whenever all adjacent atoms are removed. Thus the statement that the empty corners host 'vacancies of interstice charge' is a property of the chosen trial Wannier functions, not a separate physical finding. The paper should support the vacancy picture with a trial-function-independent quantity, such as the integrated charge density in a corner region, or explicitly state that the vacancies are a convention of the bond-center Wannier construction.
  2. [Dirac Mass and Dangling Bonds, Fig. 4] The central directionality criterion is not robustly established. The simplified two-band model is obtained 'by neglecting the irrelevant hoppings and only keeping the valence bonds as the Dirac mass' (Fig. 4 caption), but the omitted next-nearest-neighbor hopping t=sqrt(2)/4≈0.35 and third-neighbor hopping t'=-1/4 are not small. Nothing in the manuscript rules out that these terms change the sign of the edge Dirac mass or create or annihilate corner modes for some cleavage geometry. Because all eight patterns are computed at a single parameter set, the 'if and only if' character of the cleavage rule is not demonstrated. The authors should scan the (t, t', phi) parameter space for at least one topologically nontrivial and one trivial pattern, or provide a symmetry-based argument that the edge-mass sign pattern is independent of these hoppings.
  3. [Dirac Mass and Dangling Bonds, paragraph after Fig. 4] The connection between the reciprocal-space winding anisotropy and the real-space statement that dangling bonds must 'expose in, and subtly slope toward' the corner regions is asserted via 'formally captured by the connection between GSuperlattice and GWP=4c', but no derivation is given. This is a load-bearing logical step, since the bond-tilt criterion is the paper's main predictive claim. The authors should provide an explicit mapping from the edge-mass sign pattern to the bond orientation at each corner, or prove that the winding condition is equivalent to the geometric tilt condition, rather than inferring this from eight patterns.
  4. [Energy and Entropy Distribution, S(r) definition] The quantity called 'binary entanglement entropy' is not an entanglement entropy. The expression S(r) = -sum_n [ |a_{alpha,r}^n|^2 ln(|a_{alpha,r}^n|^2) + (1-|a_{alpha,r}^n|^2) ln(1-|a_{alpha,r}^n|^2) ] is a Shannon entropy of single-particle probability amplitudes, not the von Neumann entropy of a reduced density matrix. Therefore the claim that topological corners acquire lower entanglement entropy than the bulk is unsupported. At minimum the terminology should be changed to 'site-resolved probability entropy' and the physical interpretation should be revised accordingly.
minor comments (6)
  1. [Author affiliations] The affiliation contains a typo: 'Ch ina' should be 'China'.
  2. [References] Reference [47] (Wen and Zee, classification of abelian quantum Hall states) does not appear to be the correct citation for a C4-symmetric HOTI with four corner states; it is cited alongside Refs. [4] and [22] in the Dirac-mass discussion.
  3. [Table I] The parenthetical symmetry labels for patterns 5-8 in Table I are difficult to parse; a separate column or clearer notation would improve readability.
  4. [Supplemental Material] Several conclusions rely on Supplemental Material figures S1-S19 that are not included in the submitted manuscript, so the reader cannot verify the supporting data.
  5. [Fig. 4] Labels such as '2/g155' in Fig. 4 are unexplained and should be clarified or removed.
  6. [Fig. 3 caption] The phrase '0 leads to the dimensionless translation phase e^{ik·0}=1' in the Fig. 3 caption is confusing and should be rewritten or removed.

Circularity Check

1 steps flagged · score 6.0 of 10

The 'complementary vacancies of interstice charge' at empty corners reduce by construction to the definition of bond-center Wannier functions, while the e/2 corner states and the Dirac-mass directionality rule retain independent content.

  1. self definitional [Section 'Interstice Charge Vacancies', after Eq. (2), page 2-3]
    "In our nearest-neighbor approximation, for a high-symmetry WP, P_Bond_n = 0 when the adjacent atoms are all cut off, termed 'vacancies' of interstice charge."

    The 'vacancy' is defined as the condition P_Bond_n = 0. Equation (2) defines P_Bond_n as an average of atom-projected amplitudes over adjacent bonded atoms, so whenever all adjacent atoms are cut off, P_Bond_n vanishes by the construction of the trial function. The paper then presents 'anomalous vacancies at WP 4c' at the empty corners as a discovered feature and builds the 'balanced geometry/neutrality' claim on it. This part of the central narrative is a direct consequence of the Wannier-function definition, not an independent physical result. The e/2 fractional corner charges and their locations are computed separately and are not compromised, so the circularity is partial.

full rationale

The e/2 fractional corner charges and the zero-mode spectra are obtained by diagonalizing the tight-binding Hamiltonian (1) and are independent of the bond-center Wannier trial functions; no parameter is fitted and then renamed as a prediction. The Dirac-mass directionality rule is derived from a simplified two-band model of the same Hamiltonian, and while its robustness to neglected hoppings (t and t') is not established and the real-space bond-tilt connection is asserted rather than proven, this is a correctness/robustness concern rather than a circular reduction. The only self-citation, Ref. [17], supports the polarized-edge interpretation but is not load-bearing. The one clear circular step is the vacancy claim: Eq. (2) defines bond-center weight as an average over adjacent atom amplitudes, and the paper explicitly states that P_Bond_n = 0 when adjacent atoms are cut off, so 'vacancies at empty corners' follow by definition. Thus the vacancy half of the abstract's central claim reduces to the trial-function construction, warranting a score of 6 for partial circularity; the topological corner-state discovery itself remains independent.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central result rests on a hand-tuned tight-binding model in a specific OAI phase, on the assumption that the low-energy Dirac-mass description is complete, and on an ad hoc Wannier construction for interstitial charge. No new entities are introduced, but the model parameters act as tuning knobs that are not justified by external data or a broader parameter scan.

free parameters (3)
  • NN hopping amplitude t~ = 1 (set as energy unit)
    Chosen by hand to place the system in the obstructed atomic insulator phase with an intermediate Dirac-metal transition; no parameter robustness study is presented.
  • NNN hopping amplitude t and flux phase phi = t = sqrt(2)/4, phi = +/- 3pi/4
    Hand-tuned to realize the OAI phase and the desired symmetry pattern; the sign of phi sets the magnetic flux in the two sublattice squares.
  • Third-neighbor hopping t' = -1/4
    Hand-chosen as part of the model that realizes the OAI; the paper does not explore the range of t' over which the half-corner states persist.
assumptions (4)
  • standard math Exact diagonalization of the tight-binding Hamiltonian and Wannier function construction are reliable tools for this model.
    The paper assumes standard numerical methods for band structures, eigenstates, and Wannier functions; these are accepted in the field.
  • domain assumption The low-energy physics of the OAI is captured by a single Dirac fermion whose mass is set by the nearest-neighbor bond at WP 4c.
    Invoked in the 'Dirac Mass and Dangling Bonds' section to connect the WP 4c charge to the Dirac mass; not derived from a controlled expansion.
  • domain assumption The superlattice symmetry group G_Superlattice determines the sign pattern of the Dirac mass on the edges.
    Used to predict which edges have opposite mass signs and thus which corners host zero modes; this is a symmetry-based assumption about the edge effective theory.
  • ad hoc to paper Bond-center Wannier trial functions with Gaussian volumes assigned by Eq. (2) correctly represent interstitial charge.
    The construction P_Bond = (1/n) sum P_Atom is specific to this paper; the resulting vacancies at cut corners are defined relative to this construction, not measured independently.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Higher-Order Topological States with Cleavage-Dependent Dirac Mass." pith.science (2026). https://pith.science/paper/VC365HKD

@misc{pith2026260810520,
  author       = {Pith},
  title        = {Pith review of: Higher-Order Topological States with Cleavage-Dependent Dirac Mass},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VC365HKD}},
  note         = {Machine review of arXiv:2608.10520}
}
read the original abstract

Topological quantum chemistry based on local charge profiles lacks predictive power for the crystalline cleavage of higher-order topological insulators (HOTIs). By cleaving an obstructed atomic insulator, we discover a topological phase characterized by e/2 fractional charges localized at precisely half of the corners, while the remaining empty corners host complementary vacancies of interstice charge. These zero-energy charge-vacancies and topological corners form a spatially balanced geometry, confined separately by C2 rotation symmetry. Crucially, we demonstrate that the emergence of corner zero modes dictates that specific dangling bonds-acting as the mass of a Dirac fermion-must explicitly expose in, and subtly slope toward, the corner regions. This strict directionality is verified by the anisotropic evolution of the mass term within a (2+1)-dimensional parameter space. Moreover, we find that the topological corners acquire lower entanglement entropy compared to the bulk, a behavior opposite to that of the real-space energy distribution.

Figures

Figures reproduced from arXiv: 2608.10520 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

53 extracted references · 43 canonical work pages

  1. [1]

    and the C4-symmetric Hamiltonian [ 22] [see SM [ 36] Sec. E]. The spectra of thus obtained Hamiltonian are in (2 + 1)-dimensional space of {kx, ky, E(kx, ky)}, and can reflect the system’s topological properties [ 49]. By tracing the winding of ˆd- vectors along the four edges of the Brillouin zone, we find that the winding along four edges of the Brillouin...

    work page

  2. [2]

    W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Quantized electric multipole insulators, Phys. Rev. B 96, 245115 (2017)

    work page 2017

  3. [3]

    3 (a) and (b)], 4 [SM [ 36] Figs

    For the topologically nontrivial patterns 1 [Figs. 3 (a) and (b)], 4 [SM [ 36] Figs. S16(b) and S16(c)], 5 [SM [ 36] Figs. S13(b) and S13(c)], 8 [SM [ 36] Figs. S16(d) and S17(e)], the corners without topological charge are large crys- talline interstices and become the prominent vacancies after the charge is redistributed via bonding. Conversely, for the...

    work page

  4. [4]

    Xiao, M.-C

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

    2010

  5. [5]

    Z. Song, Z. Fang, and C. Fang, (d-2)-dimensional edge states of rotation symmetry protected topological states, Phys. Rev. Lett. 119, 246402 (2017)

    work page 2017

  6. [6]

    W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Electric multipole moments, topological multipole mo- ment pumping, and chiral hinge states in crystalline in- sulators, Science 357, 61 (2017)

    work page 2017

  7. [7]

    Ezawa, Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lat- tices, Phys

    M. Ezawa, Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lat- tices, Phys. Rev. Lett. 120, 026801 (2018)

    2018

  8. [8]

    M. J. Park, Y. Kim, G. Y. Cho, and S. B. Lee, Higher- order topological insulator in twisted bilayer graphene, Phys. Rev. Lett. 123, 216803 (2019)

    work page 2019

Show all 53 references
  1. [9]

    Schindler, A

    F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. Parkin, B. A. Bernevig, and T. Neupert, Higher-order topological insulators, Sci. Adv. 4, eaat0346 (2018)

    2018

  2. [10]

    N. Mao, H. Wang, Y. Dai, B. Huang, and C. Niu, Higher- order topological insulators in two-dimensional dirac ma- terials, npj Comput. Mater. 8, 154 (2022)

    2022

  3. [11]

    W. A. Benalcazar, T. Li, and T. L. Hughes, Quantization of fractional corner charge in cn- symmetric higher-order topological crystalline insulato rs, Phys. Rev. B 99, 245151 (2019)

    2019

  4. [12]

    Takahashi, T

    R. Takahashi, T. Zhang, and S. Murakami, General- ized bulk-edge correspondence for higher-order topologi- cal phases, Phys. Rev. B 103, 205123 (2021)

    2021

  5. [13]

    R. Chen, T. Liu, C. M. Wang, H. Z. Lu, and X. C. Xie, Higher-order topological insulators in quasicrystal s, Phys. Rev. Lett. 127, 066801 (2021)

    2021

  6. [14]

    Queiroz and A

    R. Queiroz and A. Stern, Symmetry- protected bulk-boundary correspondence, Phys. Rev. Lett. 123, 036802 (2019)

    2019

  7. [15]

    R. X. Zhang, F. Wu, and S. D. Sarma, Mo- bius insulator and higher-order topology in mnbi2te4, 6 Phys. Rev. Lett. 124, 136407 (2020)

    2020

  8. [16]

    Lin and C

    L. Lin and C. Lee, Higher-order topology in a band-insulated crystalline superconductor, Commun. Phys. 7, 393 (2024)

    2024

  9. [17]

    Langbehn, Y

    J. Langbehn, Y. Peng, L. Trifunovic, F. von Oppen, and P. W. Brouwer, Reflection-symmetric second-order topological insulators and superconductors , Phys. Rev. Lett. 119, 246401 (2017)

    2017

  10. [18]

    Y. Han, C. Cui, X. P. Li, T. T. Zhang, Z. Zhang, Z. M. Yu, and Y. Yao, Higher-order topological phases in mag- netic semimetals, Phys. Rev. Lett. 133, 176602 (2024)

    2024

  11. [19]

    Zhang, M

    L. Zhang, M. Long, Y. Zhang, Z. Y. Meng, and X.-Y. Song, Discrete Shift and Polarization from Re- sponse to Symmetry Defects in Interacting Topo- logical Phases, arXiv e-prints , arXiv:2510.19483 (2025) , arXiv:2510.19483 [cond-mat.str-el]

    2025 arXiv

  12. [20]

    O. Pozo, C. Repellin, and A. G. Grushin, Quantization of the charge pump in higher-order topological insulators, Phys. Rev. Lett. 123, 247401 (2019)

    2019

  13. [21]

    Y. Xu, Z. Song, Z. Wang, H. Weng, and X. Dai, Higher-order topology of the axion insulator euin2as2, Phys. Rev. Lett. 122, 256402 (2019)

    2019

  14. [22]

    C. H. Hsu, P. Stano, J. Klinovaja, and D. Loss, Majo- rana kramers pairs in higher-order topological insulators , Phys. Rev. Lett. 121, 196801 (2018)

    2018

  15. [23]

    Y. Wu, H. Jiang, J. Liu, H. Liu, and X. C. Xie, Higher-order topology in a magnetized plasmonic crys- tal, Phys. Rev. Lett. 125, 036801 (2020)

    2020

  16. [24]

    May-Mann and T

    J. May-Mann and T. L. Hughes, Fractional corner charges in a three-dimensional second-order topological insulator, Phys. Rev. B 106, L241113 (2022)

    2022

  17. [25]

    M. S. Hossain, F. Schindler, R. Islam, Z. Muham- mad, Y.-X. Jiang, Z.-J. Cheng, Q. Zhang, T. Hou, H. Chen, M. Litskevich, B. Casas, J.-X. Yin, T. A. Cochran, M. Yahyavi, X. P. Yang, L. Balicas, G. Chang, W. Zhao, T. Neupert, and M. Z. Hasan, Observation of higher-order topolog...

    2024

  18. [26]

    R. J. Slager, A. Mesaros, V. Juriˇ ci´ c, and J. Zaanen, Th e space group classification of topological band-insulators , Nat. Phys. 9, 98 (2013)

    2013

  19. [27]

    F. Tang, H. C. Po, A. Vishwanath, and X. Wan, Compre- hensive search for topological materials using symmetry indicators, Sci. Adv. 5, eaau8725 (2019)

    2019

  20. [28]

    H. C. Po, A. Vishwanath, and H. Watanabe, Symmetry- based indicators of band topology in the 230 space groups, Nat. Commun. 8, 50 (2017)

    2017

  21. [29]

    Bradlyn, L

    B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature 547, 298 (2017)

    2017

  22. [30]

    Z. D. Song, L. Elcoro, and B. A. Bernevig, Twisted bulk-boundary correspondence of fragile topology, Science 367, 794 (2020)

    2020

  23. [31]

    Nelson, T

    A. Nelson, T. Neupert, T. Bzduˇ sek, and A. Alexan- dradinata, Multicellularity of topological states, Phys. Rev. Lett. 126, 216404 (2021)

    2021

  24. [32]

    Hwang, V

    Y. Hwang, V. Gupta, F. Schindler, L. Elcoro, Z. Song, B. A. Bernevig, and B. Bradlyn, Topo- logical quantum chemistry of non-hermitian systems, arXiv:2505.09697 (2025)

    2025 arXiv

  25. [33]

    H. C. Po, H. Watanabe, and A. Vishwanath, Fragile topology and wannier obstructions, Phys. Rev. Lett. 121, 126402 (2018)

    2018

  26. [34]

    Elcoro, B

    L. Elcoro, B. J. Wieder, Z. Song, Y. Xu, B. Bradlyn, and B. A. Bernevig, Magnetic topological quantum chemistry, Nat. Commun. 12, 5965 (2021)

    2021

  27. [35]

    Wang, Z.-Y

    F.-J. Wang, Z.-Y. Xiao, R. Queiroz, B. A. Bernevig, A. Stern, and Z.-D. Song, Topological origin of the quantized bulk quadrupole moment, Nat. Commun. 15, 3069 (2024)

    2024

  28. [36]

    L. Wang, Y. Jiang, J. Liu, S. Zhang, J. Li, P. Liu, Y. Sun, H. Weng, and X. Chen, Higher-order topolog- ical phases in a three-dimensional topological insulator, Phys. Rev. B 106, 155144 (2022)

    2022

  29. [37]

    M. I. Aroyo, J. M. Perez-Mato, D. Orobengoa, E. Tasci, G. de la Flor, and A. Kirov, Crys- tallography online: Bilbao crystallographic server, Bulg. Chem. Commun. 43, 183 (2011)

    2011

  30. [38]

    It also includes Refs

    Supplemental Material [URL will be inserted by pub- lisher] provides details of model, wavefunction distribu- tion of in-gap corner and edge states, Wannier function, Dirac fermionic Lagrangian, coarse-graining for mass- dependent Hamiltonian, formula of density-weighted en- e...

  31. [39]

    Marzari and D

    N. Marzari and D. Vanderbilt, Maximally localized gen- eralized wannier functions for composite energy bands, Phys. Rev. B 56, 12847 (1997)

    1997

  32. [40]

    A. A. Soluyanov and D. Vanderbilt, Wan- nier representation of z2 topological insulators, Phys. Rev. B 83, 035108 (2011)

    2011

  33. [41]

    Marzari, A

    N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Maximally local- ized wannier functions: Theory and applications, Rev. Mod. Phys. 84, 1419 (2012)

    2012

  34. [42]

    Khalaf, W

    E. Khalaf, W. A. Benalcazar, T. L. Hughes, and R. Queiroz, Boundary-obstructed topological phases, Phys. Rev. Res. 3, 013239 (2021)

    2021

  35. [43]

    X.-J. Luo, M. Wang, and F. Wu, Higher-order topolog- ical phases in a two-dimensional topological insulator, Phys. Rev. B 107, 235127 (2023)

    2023

  36. [44]

    C. Xu, J. Li, Y. Xu, Z. Bi, and Y. Zhang, Fractional corner charges in higher-order topological insulators with interaction, Proc. Natl. Acad. Sci. 121, e2316749121 (2024)

    2024

  37. [45]

    T. M. Gunawardana, A. M. Turner, and R. Barnett, Higher-order topology and corner states in a topological crystalline insulator, Phys. Rev. Res. 6, 023046 (2024)

    2024

  38. [46]

    H. C. Po, L. Zou, A. Vishwanath, and T. Senthil, Origin of mott insulating behavior and su- perconductivity in twisted bilayer graphene, Phys. Rev. X 8, 031089 (2018)

    2018

  39. [47]

    Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, All dirac and weyl semimetals and all topological insulators in the 230 space groups, Phys. Rev. Lett. 123, 036401 (2019)

    2019

  40. [48]

    Koepernik, O

    K. Koepernik, O. Janson, Y. Sun, and J. V. D. Brink, Higher-order topology in a magnetic topological insula- tor, Phys. Rev. B 107, 235135 (2023)

    2023

  41. [49]

    Wen and A

    X.-G. Wen and A. Zee, Classification of abelian quantum hall states and matrix formulation of topological fluids, Phys. Rev. Lett. 69, 953 (1992)

    1992

  42. [50]

    Y. Ren, Z. Qiao, and Q. Niu, Topological phase transitions in two-dimensional systems, Phys. Rev. Lett. 124, 166804 (2020)

    2020

  43. [51]

    Qi, Y.-S

    X.-L. Qi, Y.-S. Wu, and S.-C. Zhang, Topological quan- tization of the spin hall effect in two-dimensional insula- tors, Phys. Rev. B 74, 085308 (2006)

    2006

  44. [52]

    HPC2021, High performance computing cluster , Infor- 7 mation Technology Services, The University of Hong Kong (2021)

    2021

  45. [53]

    Beijing PARATERA Tech Co., Ltd., Paratera high performance computing platform , Beijing PARATERA Tech Co., Ltd. (2021)

    2021

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.