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One-Particle Density Matrix Framework for Mode-Shell Correspondence: Characterizing Topology in Amorphous Higher-Order Topological Insulators

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper claims that higher-order topology—corner modes and the bulk topology surrounding them—can be read off from the one-particle density matrix alone, through an exact equality between a mode index and a shell index, and that this diag

desk verdict Useful 1PDM reformulation of mode-shell correspondence, but the amorphous claim rests on an unproven bulk-cancellation assumption and Appendix B has a real derivation error. read the letter →

arxiv 2509.03632 v3 pith:6AJHTN76 submitted 2025-09-03 cond-mat.mes-hall cond-mat.dis-nn

Miguel F. Martínez , Lucien Jezequel , Jens H. Bardarson , Thomas Klein Kvorning , Julia D. Hannukainen This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.dis-nn
keywords one-particledensitymatrixmode-shellcorrespondencehigher-ordertopologicalinsulatoramorphouschiralsymmetrylocalmarkerbulk-boundaryGaussianstates
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 tries to establish that the topology of a quantum state, including higher-order topology with protected corner modes, can be determined directly from the state's one-particle density matrix, with no Hamiltonian and no translation invariance required. Its central claim is that two indices computed from the restricted density matrix, the mode index and the shell index, are exactly equal: the mode index counts chiral boundary modes, while the shell index measures the surrounding bulk topology, forming a bulk-boundary diagnostic. The authors derive this mode-shell correspondence for Gaussian states under a chiral constraint, show that in one dimension the shell index reduces to the local chiral marker, and demonstrate the framework on a C4-symmetric higher-order topological insulator, including a disordered amorphous version. If correct, the framework offers a practical way to characterize higher-order topological phases in disordered and interacting settings purely from the quantum state.

What carries the argument

The load-bearing object is the restricted one-particle density matrix ρ_A on a region A enclosing the topological modes, together with a position-space filter θ that is one on a subregion A_shell and decays to zero across a 'shell'. The mode index I_mode = 4 Tr((ρ_A − ρ_A^2) θ C) isolates chiral modes pinned at eigenvalue 1/2, while the shell index I_shell = −2 Tr(C ρ_A [ρ_A, θ]) is its algebraic rewriting in a form supported only where θ varies. The identity between the two, enforced by the chiral constraint, is what converts a boundary-mode count into a bulk diagnostic; the gradient expansion of the commutator is what connects the shell index to local real-space chiral markers.

What would settle it

Average the local shell-index density over bulk sites, away from the shell and boundaries, for many independent C4-symmetric amorphous realizations at the same amorphicity; if the mean does not vanish on length scales large compared to the correlation length, the asserted bulk cancellation fails and the near-integer total shell index would not establish a bulk-boundary diagnostic. A second check: compute the shell index for a trivial amorphous C4-symmetric insulator, which should return zero under the same prescription.

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Extended reading notes

Core claim

Working in real space, the paper defines, for a region A enclosing a chiral boundary mode, the restricted one-particle density matrix ρ_A and assumes the chiral constraint {ρ_A, C} = C. It then constructs the mode index I_mode = 4 Tr((ρ_A − ρ_A^2) θ C), where θ is a smooth filter supported on a shell within A, and the shell index I_shell = −2 Tr(C ρ_A [ρ_A, θ]). Algebraic manipulation using the cyclic property of the trace and the chiral constraint shows the two expressions are identical. Because [ρ_A, θ] vanishes away from the shell, I_shell is localized on a shell deep inside the bulk, and a gradient expansion turns it into the position-space chiral marker; in one dimension this is exactly

Load-bearing premise

The amorphous diagnostic presumes that coarse-graining the one-particle density matrix restores translation invariance in the bulk, so the positive and negative bulk fluctuations in the shell index cancel over length scales larger than the correlation length; the paper shows one realization, not an ensemble average.

Editorial extensions

If this is right

  • For C4-symmetric higher-order insulators, an odd mode index, equivalently a fractional shell index at each edge-shell intersection, certifies that the higher-order phase is intrinsic; an even mode index alone cannot distinguish intrinsic from extrinsic phases.
  • In one-dimensional translation-invariant chains, the shell index reduces to the local chiral marker and recovers the chiral winding number in that limit.
  • Because the indices are defined in real space from the one-particle density matrix, they remain well defined without translation invariance, giving a topological diagnostic for amorphous structures with enforced crystalline symmetries.
  • For any state whose one-particle density matrix has a gapped occupation spectrum, adiabatic flattening turns it into a projector, so the same mode-shell correspondence defines the topology of interacting states.
  • The framework is stated to generalize to higher dimensions and other crystalline symmetries that map parts of the shell onto one another, where equal contributions lead again to fractional quantization and intrinsic topology.

Reading between the lines

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

  • Since the indices are built only from the one-particle density matrix, the same computation should work for density matrices reconstructed from quantum simulators or state tomography, where no parent Hamiltonian is known; the paper does not pursue this measurement-oriented step.
  • The odd-mode-index criterion for intrinsic topology suggests a sharper experimental signature than a bulk invariant: observing a fractional shell index at an edge-shell intersection would certify intrinsic higher-order topology. This is an extension, not a protocol stated in the paper.
  • The amorphous claim rests on a bulk cancellation that is shown in a single realization; a natural stress test is to average the local shell-index density over many disorder realizations and check that the mean vanishes while the total index stays quantized at larger amorphicity.
  • The generalization to interacting states is argued via adiabatic flattening, but no interacting numerical example is given; testing the mode-shell correspondence on a small disordered interacting chain would confirm whether the gapped one-particle density matrix criterion is practically sufficient.
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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

4 major / 5 minor

Summary. The paper reformulates the mode-shell correspondence in terms of the one-particle density matrix (1PDM) rather than a single-particle Hamiltonian. It defines a mode index I_mode (Eq. 4) and a shell index I_shell (Eq. 7), proves their algebraic equivalence, and applies the framework to a C4-symmetric BBH higher-order topological insulator, including an amorphous realization. The paper further claims that in one dimension the shell index reduces to the local chiral marker, that a fractional/half-integer shell-boundary contribution identifies intrinsic higher-order topology, and that the framework extends to gapped interacting states. The numerical examples give I_mode = I_shell ≈ 1 in both crystalline and amorphous settings.

Significance. If validated, the 1PDM formulation would be a useful addition to the topological-marker toolbox: it is state-based, real-space, and potentially applicable to systems without translation invariance and to gapped interacting states. The trace identity leading from Eq. (5) to Eq. (7) is clean and machine-checkable by direct algebra, and the numerical results in Figs. 2 and 3 are consistent with the claimed correspondence. The paper also connects the shell index to the established local chiral marker, which provides a valuable benchmark. However, several load-bearing steps are not yet rigorously supported: the 1D equivalence proof contains an invalid manipulation, the amorphous bulk-cancellation claim is only asserted, and the intrinsic-phase criterion is stated rather than proven. These gaps currently prevent the paper from fully establishing its central claims.

major comments (4)
  1. [Appendix B, Eq. (B6)-(B7)] The derivation of the equivalence between the shell index and the local chiral marker contains an invalid step. The text states: '... using that X|x> must be zero due to translational invariance.' This is false: X|x> = x|x>, not zero. The manipulation from Eq. (B6) to Eq. (B7) is therefore not justified. Since the one-dimensional equivalence is advertised as a central result ('the shell index reduces to the local chiral marker'), this proof needs to be corrected or replaced by a valid derivation (or a precise citation to one).
  2. [Section V, Fig. 3(d)] The amorphous diagnostic rests on an unproven bulk-cancellation assumption. The text asserts that nonzero local bulk contributions to I_shell vanish after averaging over regions much larger than the correlation length because coarse graining restores translation invariance, but no proof or disorder-ensemble demonstration is supplied. Figure 3(d) shows order-one positive and negative bulk contributions in one realization, and the reported I_shell = 0.996 is a single sample value. Since I_mode itself is only approximately quantized (0.996), the exact identity I_mode = I_shell does not by itself show that the bulk contributions cancel systematically. The amorphous claim would be considerably strengthened by an ensemble average over disorder realizations, finite-size scaling, or a rigorous argument for the cancellation.
  3. [Section IV, intrinsic-phase criterion] The assertion that an odd mode index / half-integer edge-shell contribution identifies an intrinsic higher-order topological phase is stated rather than proven. The text argues that adding a one-dimensional chain changes the shell index only by integer units, but this is an assumed property, not derived or referenced. Given that the abstract explicitly claims that 'a fractional shell index implies that the higher-order phase is intrinsic,' this criterion needs a proof or a precise citation to a theorem in the existing mode-shell literature.
  4. [Section IV, bulk suppression argument] The explanation that all bulk contributions to the shell index vanish in the crystalline BBH model relies on a slice-by-slice weak-invariant argument (nonzero bulk shell index would imply a weak topological insulator with end zero-modes). This is plausible but is only sketched. Since the crystalline case is the reference point for the amorphous generalization, a more formal statement of this argument, or a reference, would be appropriate.
minor comments (5)
  1. [Appendix B] There is a duplicated sentence: 'In one dimension the chiral marker, with respect to the restricted density matrix over a region A is equal to the shell index in the same region.' appears twice in succession.
  2. [Appendix A, Eq. (A6)] The expression in Eq. (A6) writes the gradient term as '2 Tr(C rho_A [rho_A, X_i]) partial_i theta', with the derivative outside the trace and the implicit sum not shown. As written this is notationally ambiguous; it should be written as 2 Tr(C rho_A [rho_A, X_i] partial_i theta) with the sum explicit.
  3. [Eq. (12) and Section V] The filter function is defined with a parameter omega in Eq. (12), but the text immediately below says 'Here w=L_A/2 determines the position of the shell', and Section V uses omega=L_A/2. The symbol should be made consistent.
  4. [Section III] The text says theta acts as a filter 'projecting into a subregion A_shell', but theta is a smooth function, not a projector. This wording is imprecise and could confuse readers.
  5. [Section II and III] The notation A_shell ∈ A should be A_shell ⊂ A; using set membership for a subset is nonstandard and should be corrected.

Circularity Check

1 steps flagged · score 6.0 of 10

Central I_mode=I_shell equality is algebraic/tautological; amorphous bulk-cancellation is an unproven assumption.

  1. self definitional [Section III, Eqs. (5)-(7)]
    "By using the cyclic property of the trace operation and the constraint {ρ A, C}=C, the mode index in Eq. (5) can be rewritten as: 4Tr(C(ρ A −ρ 2 A)θ)= Tr(Cθ)−2Tr(Cρ A[ρA, θ]). ... In our case, the first term always vanishes ... The remaining term in Eq. (6) is only nonzero on the shell—the commutator [ρ A, θ] vanishes away from the shell ... defining the shell index: I shell =−2Tr(Cρ A[ρA, θ]). (7)"

    The shell index is not an independently defined bulk quantity that is then shown to match the mode index. Equation (7) defines I_shell as the algebraic remainder obtained by rewriting I_mode in Eq. (5) using the cyclic trace property and the chiral constraint, after dropping Tr(Cθ). Thus I_shell ≡ I_mode by construction. The numerical agreement reported in Figs. 2 and 3 (I_mode = I_shell = 0.998 and 0.996) is therefore a consequence of the definition, not an independent verification of a bulk-boundary correspondence. The physical content lies in the interpretation/localization of the operators, not in the equality itself.

full rationale

The paper is largely self-contained: the mode-shell formulas are rederived from the one-particle density matrix, and the identification with the local chiral marker in 1D and the BBH benchmarks provide external grounding. The self-citations [33,89,90] are not load-bearing as proofs, since the needed identities are derived in this paper. The central circularity is that I_shell is defined by algebraically rewriting I_mode, making the mode-shell correspondence a tautology. The amorphous extension in Section V relies on an unproven assumption that coarse graining restores translation invariance and makes bulk contributions to I_shell cancel; that is a correctness gap rather than a circularity, but it weakens the amorphous claim. Overall, because the headline equality reduces by construction while the framework still has independent benchmark content, the circularity score is moderate.

Assumptions & free parameters 7 free parameters · 8 assumptions · 0 invented entities

The framework introduces no new physical particles, forces, or dimensions. Its load-bearing assumptions are the chiral constraint on the restricted density matrix, exponential localization, band-flattenability of gapped density matrices, the coarse-grained translation invariance of amorphous states, and the unproved integer-shift property of edge modifications. The hand-chosen numerical parameters are all model or filter settings, not fitted outputs.

free parameters (7)
  • Chiral filter sharpness c (crystalline) = L_A/2 = 5
    Chosen by hand in Eq. (12) for L=20, L_A=10; no convergence study with respect to c is shown.
  • Chiral filter radius omega (w) = L_A/2 = 5
    Sets the shell location; chosen by hand to enclose the corner modes. The text uses both w and omega for this quantity.
  • Amorphous filter sharpness c = 2L_A/3 = 6.67
    Chosen to account for increased localization length, but larger c is a sharper profile than in the crystalline case, opposite to the stated rationale.
  • Amorphicity w = 0.1
    Controls Gaussian disorder in the amorphous point set; a single value is used with no w-dependence analysis.
  • Hopping cutoff r0 = 1.3
    Sets the connectivity range in Eq. (16); chosen by hand, with no sensitivity check.
  • BBH parameters gamma and eta = gamma=0.5, eta=1
    Model parameters chosen deep in the higher-order topological phase, not fit to data.
  • System size L and region size L_A = L=20, L_A=10
    Finite-size settings for all numerics; no scaling or convergence check is provided.
assumptions (8)
  • domain assumption The restricted density matrix satisfies the chiral constraint {rho_A, C} = C.
    Invoked in Section III before Eq. (6); inherited from the BdG structure of Gaussian states with chiral symmetry.
  • domain assumption The one-particle density matrix is exponentially localized with correlation length xi in the bulk.
    Used in Section II and Appendix A to justify the gradient expansion and the separation of A_shell from the boundary of A.
  • domain assumption Gapped one-particle density matrices can be adiabatically band-flattened to a projector without changing topology.
    Section II and the Discussion; this is the basis for extending the framework to interacting states.
  • ad hoc to paper Adding a one-dimensional chain to an edge changes the shell index only by integer units.
    Section IV, final paragraph; this is the key premise of the intrinsic-phase criterion but is asserted without proof.
  • domain assumption Coarse-grained translation invariance is recovered in the amorphous bulk, so bulk shell-index contributions vanish on average.
    Section V; used to explain the nonzero bulk contributions in Fig. 3(d) as averaging to zero.
  • standard math The local chiral marker equals the chiral winding number in translation-invariant limits.
    Cited from Refs. [33,91] and used in Section III and Appendix B to connect the shell index to the winding number.
  • standard math The first term Tr(C theta) vanishes for BdG one-particle density matrices.
    Section III after Eq. (6); relies on C being traceless in the BdG single-particle space when theta is position-diagonal.
  • domain assumption The four-copy construction preserves the C4 symmetry needed to pair edge-shell intersections.
    Section V; the argument that the two edge-shell intersections contribute equal half-integer values uses this symmetry.

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Cite this review

Pith. "Pith review of One-Particle Density Matrix Framework for Mode-Shell Correspondence: Characterizing Topology in Amorphous Higher-Order Topological Insulators." pith.science (2026). https://pith.science/paper/6AJHTN76

@misc{pith2026250903632,
  author       = {Pith},
  title        = {Pith review of: One-Particle Density Matrix Framework for Mode-Shell Correspondence: Characterizing Topology in Amorphous Higher-Order Topological Insulators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6AJHTN76}},
  note         = {Machine review of arXiv:2509.03632}
}
abstract

We present a framework for characterizing higher-order topological phases directly from the one-particle density matrix, without any reference to an underlying Hamiltonian. Our approach extends the mode-shell correspondence, originally formulated for single-particle Hamiltonians, to Gaussian states subject to chiral constraints. In this correspondence, the mode index counts topological boundary modes, while the shell index quantifies the bulk topology in a region surrounding the modes, providing a bulk-boundary diagnostic. In one-dimensional topological insulators, the shell index reduces to the local chiral marker, recovering the winding number in the translation-invariant limit. We apply the mode-shell correspondence to a $C_4$-symmetric higher-order topological insulator with a chiral constraint and show that a fractional shell index implies that the higher-order phase is intrinsic. The one-particle density matrix is formulated in real space, so the mode-shell correspondence also applies to models without translation invariance. By introducing structural disorder into the $C_4$-symmetric higher-order insulator, we show that the mode-shell correspondence remains a meaningful diagnostic in amorphous structures. The mode-shell correspondence generalizes to interacting states with a gapped bulk spectrum in the one-particle density matrix, providing a practical and diverse route to characterize higher-order topology from the quantum state itself.

Figures

Figures reproduced from arXiv: 2509.03632 by the authors.

Figure 1
Figure 1. Example of the choice of regions A, its complement A c , and Ashell ⊂ A in a one-dimensional chain with the topo￾logical zero modes, depicted in blue, at the two ends of the chain. The solid line represents the boundary between A and A c , and the dotted line represents the shell, the boundary of Ashell. that the topology of the band flattened one-particle den￾sity matrix, ϱ = [PITH_FULL_IMAGE:figures/full_fig_p002… view at source ↗
Figure 2
Figure 2. (a) Spectrum of the restricted one-particle density [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. (a) Spectrum of the restricted one-particle density [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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Reference graph

Works this paper leans on

97 extracted references · 70 canonical work pages · cited by 1 Pith paper

  1. [1]

    A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Lud- wig, Phys. Rev. B78, 195125 (2008)

    2008

  2. [2]

    Kitaev, AIP Conf

    A. Kitaev, AIP Conf. Proc.1134, 22 (2009)

    2009

  3. [3]

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

    2010

  4. [4]

    A. W. W. Ludwig, Phys. Scr.T168, 014001 (2015)

    2015

  5. [5]

    Cartan, Bull

    E. Cartan, Bull. Soc. Math. France54, 214 (1926)

    1926

  6. [6]

    M. R. Zirnbauer, J. Math. Phys.37, 4986 (1996)

    1996

  7. [7]

    Altland and M

    A. Altland and M. R. Zirnbauer, Phys. Rev. B55, 1142 (1997)

    1997

  8. [8]

    Ando and L

    Y. Ando and L. Fu, Annu. Rev. Condens. Matt. Phys.6, 361–381 (2015)

    2015

Show all 97 references
  1. [9]

    Neupert and F

    T. Neupert and F. Schindler, Topological crystalline in- sulators, in Topological Matter (Springer International Publishing, 2018) p. 31–61

    2018

  2. [11]

    Langbehn, Y

    J. Langbehn, Y. Peng, L. Trifunovic, F. von Oppen, and P. W. Brouwer, Phys. Rev. Lett.119, 246401 (2017)

    2017

  3. [12]

    Geier, L

    M. Geier, L. Trifunovic, M. Hoskam, and P. W. Brouwer, Phys. Rev. B97, 205135 (2018)

    2018

  4. [13]

    Schindler, A

    F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert, Sci. Adv.4(2018)

    2018

  5. [14]

    Trifunovic and P

    L. Trifunovic and P. W. Brouwer, Phys. Rev. X9, 011012 (2019)

    2019

  6. [15]

    Trifunovic and P

    L. Trifunovic and P. W. Brouwer, Phys. Status Solidi B 258(2021)

    2021

  7. [16]

    A. Y. Chaou, P. W. Brouwer, and N. Sedlmayr, Phys. Rev. B107, 035430 (2023)

    2023

  8. [17]

    Xie, H.-X

    B. Xie, H.-X. Wang, X. Zhang, P. Zhan, J.-H. Jiang, M. Lu, and Y. Chen, Nat. Rev. Phys.3, 520 (2021)

    2021

  9. [18]

    Varjas, A

    D. Varjas, A. Lau, K. P¨ oyh¨ onen, A. R. Akhmerov, D. I. Pikulin, and I. C. Fulga, Phys. Rev. Lett.123, 196401 (2019)

    2019

  10. [19]

    Agarwala, V

    A. Agarwala, V. Juriˇ ci´ c, and B. Roy, Phys. Rev. Res.2, 012067 (2020)

    2020

  11. [20]

    Tao, J.-H

    Y.-L. Tao, J.-H. Wang, and Y. Xu, SciPost Phys.15, 193 (2023)

    2023

  12. [21]

    Peng, C.-B

    T. Peng, C.-B. Hua, R. Chen, Z.-R. Liu, H.-M. Huang, and B. Zhou, Phys. Rev. B106, 125310 (2022)

    2022

  13. [22]

    A. Y. Chaou, M. Moreno-Gonzalez, A. Altland, and P. W. Brouwer, Phys. Rev. B112, 035167 (2025)

    2025

  14. [23]

    Prodan, New J

    E. Prodan, New J. Phys.12, 065003 (2010)

    2010

  15. [24]

    T. A. Loring and M. B. Hastings, EPL92, 67004 (2011)

    2011

  16. [25]

    Prodan, J

    E. Prodan, J. Phys. A44, 113001 (2011)

    2011

  17. [26]

    Bianco and R

    R. Bianco and R. Resta, Phys. Rev. B84, 241106 (2011)

    2011

  18. [27]

    T. A. Loring, Ann. Phys.356, 383 (2015)

    2015

  19. [28]

    Huang and F

    H. Huang and F. Liu, Phys. Rev. B98, 125130 (2018)

    2018

  20. [29]

    Irsigler, J.-H

    B. Irsigler, J.-H. Zheng, and W. Hofstetter, Phys. Rev. A100, 023610 (2019)

    2019

  21. [30]

    T. A. Loring, arXiv:1907.11791

    1907 arXiv

  22. [31]

    Mondragon-Shem and T

    I. Mondragon-Shem and T. L. Hughes, Phys. Rev. B110, 035146 (2024)

    2024

  23. [32]

    Jezequel, C

    L. Jezequel, C. Tauber, and P. Delplace, J. Math. Phys. 63, 121901 (2022). 9

    2022

  24. [33]

    J. D. Hannukainen, M. F. Mart ´ ınez, J. H. Bardarson, and T. K. Kvorning, Phys. Rev. Lett.129, 277601 (2022)

    2022

  25. [34]

    J. D. Hannukainen, M. F. Mart ´ ınez, J. H. Bardarson, and T. K. Kvorning, Phys. Rev. Res.6, L032045 (2024)

    2024

  26. [35]

    Marrazzo and R

    A. Marrazzo and R. Resta, Phys. Rev. B95, 121114 (2017)

    2017

  27. [36]

    Ba` u and A

    N. Ba` u and A. Marrazzo, Phys. Rev. B109, 014206 (2024)

    2024

  28. [37]

    M. D. Caio, G. M¨ oller, N. R. Cooper, and M. J. Bhaseen, Nat. Phys.15, 257–261 (2019)

    2019

  29. [38]

    d’Ornellas, R

    P. d’Ornellas, R. Barnett, and D. K. K. Lee, Phys. Rev. B106, 155124 (2022)

    2022

  30. [39]

    Kitaev, Ann

    A. Kitaev, Ann. Phys.321, 2–111 (2006)

    2006

  31. [40]

    T. A. Loring and H. Schulz-Baldes, Canad. Math. Bull. 62, 373 (2019)

    2019

  32. [41]

    T. A. Loring and H. Schulz-Baldes, J. Noncommuttal. Geom.14, 1 (2020)

    2020

  33. [42]

    Schulz-Baldes and T

    H. Schulz-Baldes and T. Stoiber, Proc. Am. Math. Soc 149, 121 (2021)

    2021

  34. [43]

    Schulz-Baldes and T

    H. Schulz-Baldes and T. Stoiber, Europhys. Lett.136, 27001 (2021)

    2021

  35. [44]

    Schulz-Baldes and T

    H. Schulz-Baldes and T. Stoiber, J. Math. Phys.64, 8 (2023)

    2023

  36. [45]

    Franca and A

    S. Franca and A. G. Grushin, Phys. Rev. B109, 195107 (2024)

    2024

  37. [46]

    Stoiber, Commun

    T. Stoiber, Commun. Math. Phys.406, 184 (2025)

    2025

  38. [47]

    Doll and H

    N. Doll and H. Schulz-Baldes, Adv. Math.392, 108038 (2021)

    2021

  39. [48]

    Cerjan, L

    A. Cerjan, L. Koekenbier, and H. Schulz-Baldes, J. Math. Phys.64(2023)

    2023

  40. [49]

    N. Doll, T. Loring, and H. Schulz-Baldes, J. Math. Phys. Anal. Geom.28, 13 (2025)

    2025

  41. [50]

    Schulz-Baldes, arXiv:2403.18948

    H. Schulz-Baldes, arXiv:2403.18948

    arXiv

  42. [51]

    Jezequel, J

    L. Jezequel, J. H. Bardarson, and A. G. Grushin, arXiv:2508.00214

    arXiv

  43. [52]

    Penrose and L

    O. Penrose and L. Onsager, Phys. Rev.104, 576 (1956)

    1956

  44. [53]

    Koch and S

    E. Koch and S. Goedecker, Solid State Commun.119, 105–109 (2001)

    2001

  45. [54]

    S. Bera, H. Schomerus, F. Heidrich-Meisner, and J. H. Bardarson, Phys. Rev. Lett.115, 046603 (2015)

    2015

  46. [55]

    S. Bera, T. Martynec, H. Schomerus, F. Heidrich- Meisner, and J. H. Bardarson, Ann. Phys. (Berlin)529, 1600356 (2017)

    2017

  47. [56]

    T. L. M. Lezama, S. Bera, H. Schomerus, F. Heidrich- Meisner, and J. H. Bardarson, Phys. Rev. B96, 060202(R) (2017)

    2017

  48. [57]

    Kells, N

    G. Kells, N. Moran, and D. Meidan, Phys. Rev. B97, 085425 (2018)

    2018

  49. [58]

    Agarwala and V

    A. Agarwala and V. B. Shenoy, Phys. Rev. Lett.118, 236402 (2017)

    2017

  50. [59]

    Mansha and Y

    S. Mansha and Y. D. Chong, Phys. Rev. B96, 121405 (2017)

    2017

  51. [60]

    Xiao and S

    M. Xiao and S. Fan, Phys. Rev. B96, 100202 (2017)

    2017

  52. [61]

    N. P. Mitchell, L. M. Nash, D. Hexner, A. M. Turner, and W. T. M. Irvine, Nat. Phys.14, 380 (2018)

    2018

  53. [62]

    Bourne and E

    C. Bourne and E. Prodan, J. Phys. A: Math. Theor.51, 235202 (2018)

    2018

  54. [63]

    P¨ oyh¨ onen, I

    K. P¨ oyh¨ onen, I. Sahlberg, A. Weststr¨ om, and T. Ojanen, Nat. Commun.9, 2103 (2018)

    2018

  55. [64]

    E. L. Minarelli, K. P¨ oyh¨ onen, G. A. R. van Dalum, T. Ojanen, and L. Fritz, Phys. Rev. B99, 165413 (2019)

    2019

  56. [65]

    Chern, Europhys

    G.-W. Chern, Europhys. Lett.126, 37002 (2019)

    2019

  57. [66]

    Mano and T

    T. Mano and T. Ohtsuki, JPSJ88, 123704 (2019)

    2019

  58. [67]

    Costa, G

    M. Costa, G. R. Schleder, M. Buongiorno Nardelli, C. Lewenkopf, and A. Fazzio, Nano Lett.19, 8941 (2019)

    2019

  59. [68]

    Marsal, D

    Q. Marsal, D. Varjas, and A. G. Grushin, PNAS117, 30260 (2020)

    2020

  60. [69]

    Sahlberg, A

    I. Sahlberg, A. Weststr¨ om, K. P¨ oyh¨ onen, and T. Ojanen, Phys. Rev. Res.2, 013053 (2020)

    2020

  61. [70]

    M. N. Ivaki, I. Sahlberg, and T. Ojanen, Phys. Rev. Res. 2, 043301 (2020)

    2020

  62. [71]

    Wang, Y.-B

    J.-H. Wang, Y.-B. Yang, N. Dai, and Y. Xu, Phys. Rev. Lett.126, 206404 (2021)

    2021

  63. [72]

    Focassio, G

    B. Focassio, G. R. Schleder, M. Costa, A. Fazzio, and C. Lewenkopf, 2d Mater.8, 025032 (2021)

    2021

  64. [73]

    N. P. Mitchell, A. M. Turner, and W. T. M. Irvine, Phys. Rev. E104, 025007 (2021)

    2021

  65. [74]

    Spring, A

    H. Spring, A. Akhmerov, and D. Varjas, SciPost Phys. 11, 022 (2021)

    2021

  66. [75]

    C. Wang, T. Cheng, Z. Liu, F. Liu, and H. Huang, Phys. Rev. Lett.128, 056401 (2022)

    2022

  67. [76]

    A. J. Ur ´ ıa-´Alvarez, D. Molpeceres-Mingo, and J. J. Pala- cios, Phys. Rev. B105, 155128 (2022)

    2022

  68. [77]

    Mu˜ noz Segovia, P

    D. Mu˜ noz Segovia, P. Corbae, D. Varjas, F. Hellman, S. M. Griffin, and A. G. Grushin, Phys. Rev. Res.5, L042011 (2023)

    2023

  69. [78]

    Cassella, P

    G. Cassella, P. d’Ornellas, T. Hodson, W. M. H. Natori, and J. Knolle, Nat. Commun.14, 6663 (2023)

    2023

  70. [79]

    A. G. Grushin and C. Repellin, Phys. Rev. Lett.130, 186702 (2023)

    2023

  71. [80]

    Corbae, J

    P. Corbae, J. D. Hannukainen, Q. Marsal, D. Mu˜ noz- Segovia, and A. G. Grushin, EPL142, 16001 (2023)

    2023

  72. [81]

    Manna, S

    S. Manna, S. K. Das, and B. Roy, Phys. Rev. B109, 174512 (2024)

    2024

  73. [82]

    Marsal, D

    Q. Marsal, D. Varjas, and A. G. Grushin, Phys. Rev. B 107, 045119 (2023)

    2023

  74. [83]

    A. J. Ur ´ ıa-´Alvarez and J. J. Palacios, arXiv:2410.16034

    arXiv

  75. [84]

    Alexandradinata, X

    A. Alexandradinata, X. Dai, and B. A. Bernevig, Phys. Rev. B89, 155114 (2014)

    2014

  76. [85]

    W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Science357, 61 (2017)

    2017

  77. [86]

    W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Phys. Rev. B96, 245115 (2017)

    2017

  78. [87]

    R. J. Zijderveld, I. A. Day, and A. R. Akhmerov, SciPost Phys.19, 058 (2025)

    2025

  79. [88]

    Cerjan, T

    A. Cerjan, T. A. Loring, and H. Schulz-Baldes, Phys. Rev. Lett.132, 073803 (2024)

    2024

  80. [89]

    Jezequel and P

    L. Jezequel and P. Delplace, SciPost Phys.17, 060 (2024)

    2024

  81. [90]

    Jezequel and P

    L. Jezequel and P. Delplace, SciPost Phys.18, 193 (2025)

    2025

  82. [91]

    C. L. Kane and T. C. Lubensky, Nat. Phys.10, 39 (2014)

    2014

  83. [92]

    Guzm´ an, D

    M. Guzm´ an, D. Bartolo, and D. Carpentier, SciPost Phys.12, 038 (2022)

    2022

  84. [93]

    H. Yang, L. Song, Y. Cao, and P. Yan, Nano Lett.22, 3125 (2022), publisher: American Chemical Society

    2022

  85. [94]

    Ringel, Y

    Z. Ringel, Y. E. Kraus, and A. Stern, Phys. Rev. B86, 045102 (2012)

    2012

  86. [95]

    Kobayashi, T

    K. Kobayashi, T. Ohtsuki, and K.-I. Imura, Phys. Rev. Lett.110, 236803 (2013)

    2013

  87. [96]

    I. C. Fulga, D. I. Pikulin, and T. A. Loring, Phys. Rev. Lett.116, 257002 (2016)

    2016

  88. [97]

    B. Yan, L. M¨ uchler, and C. Felser, Phys. Rev. Lett.109, 116406 (2012)

    2012

  89. [98]

    M. F. Mart ´ ınez, L. Jezequel, J. H. Bardarson, T. K. Kvorning, and J. D. Hannukainen, A one-particle density 10 matrix framework for mode-shell correspondence: Char- acterizing topology in amorphous higher-order topologi- cal insulators (2025)

    2025

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