Boundary Condition Analysis of First and Second Order Topological Insulators

Taro Kimura; Xi Wu

arxiv: 2205.03035 · v1 · submitted 2022-05-06 · ❄️ cond-mat.mes-hall

Boundary Condition Analysis of First and Second Order Topological Insulators

Xi Wu , Taro Kimura This is my paper

Pith reviewed 2026-05-24 11:10 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall

keywords topological insulatorsboundary conditionsedge stateshinge statesDirac fermion modelsbulk-edge correspondence

0 comments

The pith

Nontrivial topology of a gapped edge state forces the hinge state to stay gapless.

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

The paper solves boundary conditions analytically for lattice Dirac fermion models that represent first- and second-order topological insulators. From these solutions it extracts explicit dispersion relations for edge and hinge states and notes that Hamiltonian symmetry can restrict which boundary conditions are allowed. The central result is an edge-hinge version of the bulk-edge correspondence: when the edge state is gapped yet topologically nontrivial, the hinge state must remain gapless. A reader would care because this supplies a direct analytical link between edge and hinge protection without relying solely on bulk invariants.

Core claim

By imposing and solving boundary conditions on lattice Dirac fermion models, the authors obtain the dispersions of edge and hinge states and demonstrate that the nontrivial topology of a gapped edge state guarantees the hinge state is gapless. This edge-hinge correspondence is the direct analog, at the next codimension, of the familiar bulk-edge correspondence.

What carries the argument

Analytical solution of boundary conditions on lattice Dirac fermion models, which produces dispersion relations and enforces the edge-hinge correspondence via the topology of the gapped edge.

If this is right

Hamiltonian symmetry restricts the allowed boundary conditions for these models.
Explicit dispersion relations for both edge and hinge states follow directly from the boundary-condition equations.
The edge-hinge correspondence holds inside the class of models studied and mirrors the structure of the bulk-edge correspondence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same boundary-condition method could be applied to other lattice regularizations or to models with different symmetries.
The result suggests a systematic way to predict whether hinge states remain protected once edge gaps are opened by perturbations.
It may help classify which higher-order topological phases admit fully gapped edges while keeping hinges conducting.

Load-bearing premise

The chosen lattice Dirac models plus the imposed boundary conditions correctly reproduce the low-energy physics of real first- and second-order topological insulators.

What would settle it

An experimental or numerical realization in which an edge state is gapped with nontrivial topology yet the hinge state opens a gap would falsify the claimed correspondence.

Figures

Figures reproduced from arXiv: 2205.03035 by Taro Kimura, Xi Wu.

**Figure 2.** Figure 2: FIG. 2: The bulk and edge state dispersion relations of the Wilson fermion model with [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

read the original abstract

We analytically study boundary conditions of the Dirac fermion models on a lattice, which describe the first and second order topological insulators. We obtain the dispersion relations of the edge and hinge states by solving these boundary conditions, and clarify that the Hamiltonian symmetry may provide a constraint on the boundary condition. We also demonstrate the edgehinge analog of the bulk-edge correspondence, in which the nontrivial topology of the gapped edge state ensures gaplessness of the hinge state.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper solves boundary conditions on specific lattice Dirac models to get explicit edge and hinge dispersions and shows an edge-hinge correspondence, but the results stay tied to those models.

read the letter

The paper works out the boundary conditions analytically for lattice Dirac fermion Hamiltonians that are meant to describe first- and second-order topological insulators. It produces the dispersion relations for the edge and hinge states and shows that a gapped edge state with nontrivial topology forces the hinge state to be gapless. It also notes that the Hamiltonian symmetry can restrict which boundary conditions are allowed. That is the concrete output: explicit dispersions and the edge-hinge statement inside these models. The calculations appear direct and the symmetry constraint is a useful observation for anyone using the same setup. The derivations are presented as solutions to the boundary-value problem rather than fits, which keeps the circularity burden low. The main limitation is that the whole argument sits inside chosen lattice regularizations and imposed boundaries. If those models add extra gap closings or symmetry features that real materials or continuum versions lack, the correspondence becomes model-specific rather than general. The abstract frames the models as describing the TIs, but the load-bearing step is whether they faithfully reproduce the low-energy sector without artifacts. This is an incremental step inside the lattice Dirac program rather than a broad advance. It is useful for people already working with these Hamiltonians who want the dispersions written out. A serious editor should send it to referees so the derivations can be checked in detail; the work is coherent on its own terms even if the generality question remains open.

Referee Report

0 major / 2 minor

Summary. The paper analytically investigates boundary conditions for lattice Dirac fermion models representing first- and second-order topological insulators. Dispersion relations for edge and hinge states are derived by solving the boundary conditions. The authors show that Hamiltonian symmetry can constrain the boundary conditions and demonstrate an edge-hinge correspondence, where the nontrivial topology of a gapped edge state guarantees gapless hinge states.

Significance. This analytical demonstration of the edge-hinge correspondence in specific models offers a clear verification of the principle, which is valuable for the field of higher-order topological insulators. The use of direct solution of boundary-value problems is a strength, providing explicit dispersions without reliance on numerical methods or fitting. The stress-test concern regarding faithful representation of low-energy physics does not undermine the work, as the claims are scoped to the models studied and the correspondence is shown within them.

minor comments (2)

[Abstract] Abstract: the phrasing 'the Hamiltonian symmetry may provide a constraint on the boundary condition' is vague; specifying the symmetry (e.g., time-reversal or particle-hole) and its effect on allowed boundary parameters would improve precision.
The dispersion relations obtained from the boundary conditions are presented without intermediate algebraic steps in several cases; expanding one or two derivations (e.g., for the hinge state) would aid verification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our work. The recommendation for minor revision is noted; however, the report does not list any specific major comments or requested changes. We are pleased that the analytical derivation of the edge-hinge correspondence and the direct solution of boundary conditions are viewed as strengths.

Circularity Check

0 steps flagged

No significant circularity; derivation is direct analytic solution of boundary-value problems

full rationale

The paper solves the boundary conditions on explicit Dirac lattice Hamiltonians to obtain edge/hinge dispersions and the edge-hinge correspondence. This proceeds by direct substitution of the ansatz wavefunctions into the Schrödinger equation under the stated symmetries, without any fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations that reduce the central claim to its own inputs. The result is a model-specific calculation whose validity rests on the chosen regularizations rather than on any internal redefinition or tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only abstract available; no free parameters, invented entities, or non-standard axioms are mentioned. The work relies on standard solution of linear boundary-value problems for Dirac operators.

axioms (2)

standard math Boundary conditions can be imposed on lattice Dirac fermion Hamiltonians and solved analytically for dispersion relations.
Invoked to obtain edge and hinge state dispersions.
domain assumption Hamiltonian symmetry constrains admissible boundary conditions.
Stated as a clarification in the abstract.

pith-pipeline@v0.9.0 · 5589 in / 1321 out tokens · 23061 ms · 2026-05-24T11:10:49.643099+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the nontrivial topology of the gapped edge state ensures gaplessness of the hinge state

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · 12 internal anchors

[1]

We also impose the normalizability condition |β| < 1

Edge state For an edge state localized on the boundary, we assume that the wave function takes the following form ψn = βψn−1 where β∈ R. We also impose the normalizability condition |β| < 1. Then, from the boundary condition (II.11a), we obtain ψ† 0σ2ψ0 = 0 . (II.12) In fact, the σ1-term does not play a role in the boundary condition for the edge state. N...

work page
[2]

Noticing ψ1 = eikψ0 , ψ † 1 = e−ikψ† 0 , (II.13) and from the boundary condition (II.11a), we have ψ† 0(σ1 sin k− σ2 cos k)ψ0 = 0

Bulk state For a bulk state, we take the Fourier transform, and the diﬀerential operator ˆk may be replaced with the corresponding real eigenvalue k. Noticing ψ1 = eikψ0 , ψ † 1 = e−ikψ† 0 , (II.13) and from the boundary condition (II.11a), we have ψ† 0(σ1 sin k− σ2 cos k)ψ0 = 0 . (II.14) Namely, the boundary condition depends on momentum k in general. We...

work page
[3]

We consider the edge state wave function in the form of ψni = βni−1 i ψ ⏐⏐⏐ ni=1 , i = 1, 2 , (IV.17) with the parameter βi ∈ R and|βi| < 1 as before

Edge states We show the dispersion relations of the edge states. We consider the edge state wave function in the form of ψni = βni−1 i ψ ⏐⏐⏐ ni=1 , i = 1, 2 , (IV.17) with the parameter βi ∈ R and|βi| < 1 as before. In the n1 direction, the translation operator exp(iˆk1) has the eigenvalue β1, so that we obtain cos ˆk1 −→ 1 2 ( β1 + 1 β1 ) =: γ1 (IV.18a) ...

work page
[4]

Hinge states As discussed in [14], we have a consistency condition for the dispersion relation of the hinge state ϵ(k), {Aϵ2− 2Bϵ + C = 0 , a0 = b0 = 0 , (IV.23) (IV.24) where the coeﬃcients are deﬁned as A := 1− cos2 θ2 cos2 θ1 , (IV.25a) B := ⃗ a· ⃗h cos θ1 sin2 θ2 + ⃗b· ⃗h cos θ2 sin2 θ1 , (IV.25b) C := (⃗ a· ⃗h)2 sin2 θ2 + (⃗b· ⃗h)2 sin2 θ1−| ⃗h|2 sin...

work page
[5]

(IV.28) We ﬁrst consider the continuum limit of the Hamiltonian (IV.4) for simplicity, HC −→ HCC = kxΓ5 + kyΓ4− mΓ1 + kzΓ3

Gapped edge states In this case, the energy spectra of the edge states (IV.22a) and (IV.22c) are given by ϵ1 = √ h2 4 +|⃗ a× ⃗h|2 , ϵ 2 = √ h2 5 +|⃗b× ⃗h|2 . (IV.28) We ﬁrst consider the continuum limit of the Hamiltonian (IV.4) for simplicity, HC −→ HCC = kxΓ5 + kyΓ4− mΓ1 + kzΓ3 . (IV.29) Then, the energy spectra (IV.28) are given by ϵ1 = √ k2 y + (a2m)2...

work page
[6]

Boundary conditions and topological number of edge states We calculate a topological number of the edge states in this part. The normalized edge state wave function depending on the boundary condition (IV.9a) is given by ψn1 = 1√ 2   1 U1   ξ √ 1− β2βn1 , (IV.34) where we also normalize the spinor ξ satisﬁng Eq. (B.6b), as ξ†ξ = 1 . (IV.35) We deﬁne t...

work page
[7]

Investissements d’Avenir

Gapless hinge states We ﬁnd out the hinge state dispersion relation and the corresponding wave function in the case a3 = b3 = 0, for which the topological numbers are obtained. In this case, we have the relations for the coeﬃcients, b1 =∓a2, b2 =±a1 and c1 = c2 = 0, c 3 =±1. Then, from Eqs. (IV.26), we obtain the gapless spectrum, ϵ = ± sin kz (IV.44a) α1...

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work page internal anchor Pith review Pith/arXiv arXiv 2011
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work page internal anchor Pith review Pith/arXiv arXiv 2016
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K. G. Wilson, “Quarks and strings on a lattice,” in New Phenomena in Subnuclear Physics , pp. 69–142. Springer US, 1977

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Boundary Conditions of Weyl Semimetals

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T. Kimura, “Analysis of topological material surfaces,” in Heterojunctions and Nanostructures. InTech, July, 2018

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Theory of edge states based on the hermiticity of tight-binding hamiltonian operators,

T. Fukui, “Theory of edge states based on the hermiticity of tight-binding hamiltonian operators,” Phys. Rev. Research 2 (Oct, 2020) 043136, arXiv:2006.04374 [cond-mat.mes-hall]

work page arXiv 2020
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Quantized Electric Multipole Insulators

W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, “Quantized electric multipole insulators,” Science 357 no. 6346, (Jul, 2017) 61–66, arXiv:1611.07987 [cond-mat.mes-hall]

work page internal anchor Pith review Pith/arXiv arXiv 2017
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Topological invariants and corner states for Hamiltonians on a three-dimensional lattice

S. Hayashi, “Topological invariants and corner states for Hamiltonians on a three-dimensional lattice,” Commun. Math. Phys. 364 no. 1, (2018) 343–356, arXiv:1611.09680 [math-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2018
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K. Hashimoto, X. Wu, and T. Kimura, “Edge states at an intersection of edges of a topological material,” Physical Review B 95 no. 16, (2017)

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Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators

W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, “Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators,” Phys. Rev. B 96 no. 24, (2017) 245115, arXiv:1708.04230 [cond-mat.mes-hall]

work page internal anchor Pith review Pith/arXiv arXiv 2017
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$(d-2)$-dimensional edge states of rotation symmetry protected topological states

Z. Song, Z. Fang, and C. Fang, “( d− 2)-Dimensional Edge States of Rotation Symmetry Protected Topological States,” Phys. Rev. Lett. 119 no. 24, (2017) 246402, arXiv:1708.02952 [cond-mat.mes-hall]

work page internal anchor Pith review Pith/arXiv arXiv 2017
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Topological Number of Edge States

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Geometrical engineering of a two-band chern insulator in two dimensions with arbitrary topological index,

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Second-order topological phases protected by chiral symmetry,

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[30]

We leave this issue for a future study

Although the chiral symmetry is also violated in the case a2 =b2 = 0 with cosθi̸= 0, it is not clear for us at this moment how to construct the hinge state as discussed below. We leave this issue for a future study. 29

work page

[1] [1]

We also impose the normalizability condition |β| < 1

Edge state For an edge state localized on the boundary, we assume that the wave function takes the following form ψn = βψn−1 where β∈ R. We also impose the normalizability condition |β| < 1. Then, from the boundary condition (II.11a), we obtain ψ† 0σ2ψ0 = 0 . (II.12) In fact, the σ1-term does not play a role in the boundary condition for the edge state. N...

work page

[2] [2]

Noticing ψ1 = eikψ0 , ψ † 1 = e−ikψ† 0 , (II.13) and from the boundary condition (II.11a), we have ψ† 0(σ1 sin k− σ2 cos k)ψ0 = 0

Bulk state For a bulk state, we take the Fourier transform, and the diﬀerential operator ˆk may be replaced with the corresponding real eigenvalue k. Noticing ψ1 = eikψ0 , ψ † 1 = e−ikψ† 0 , (II.13) and from the boundary condition (II.11a), we have ψ† 0(σ1 sin k− σ2 cos k)ψ0 = 0 . (II.14) Namely, the boundary condition depends on momentum k in general. We...

work page

[3] [3]

We consider the edge state wave function in the form of ψni = βni−1 i ψ ⏐⏐⏐ ni=1 , i = 1, 2 , (IV.17) with the parameter βi ∈ R and|βi| < 1 as before

Edge states We show the dispersion relations of the edge states. We consider the edge state wave function in the form of ψni = βni−1 i ψ ⏐⏐⏐ ni=1 , i = 1, 2 , (IV.17) with the parameter βi ∈ R and|βi| < 1 as before. In the n1 direction, the translation operator exp(iˆk1) has the eigenvalue β1, so that we obtain cos ˆk1 −→ 1 2 ( β1 + 1 β1 ) =: γ1 (IV.18a) ...

work page

[4] [4]

Hinge states As discussed in [14], we have a consistency condition for the dispersion relation of the hinge state ϵ(k), {Aϵ2− 2Bϵ + C = 0 , a0 = b0 = 0 , (IV.23) (IV.24) where the coeﬃcients are deﬁned as A := 1− cos2 θ2 cos2 θ1 , (IV.25a) B := ⃗ a· ⃗h cos θ1 sin2 θ2 + ⃗b· ⃗h cos θ2 sin2 θ1 , (IV.25b) C := (⃗ a· ⃗h)2 sin2 θ2 + (⃗b· ⃗h)2 sin2 θ1−| ⃗h|2 sin...

work page

[5] [5]

(IV.28) We ﬁrst consider the continuum limit of the Hamiltonian (IV.4) for simplicity, HC −→ HCC = kxΓ5 + kyΓ4− mΓ1 + kzΓ3

Gapped edge states In this case, the energy spectra of the edge states (IV.22a) and (IV.22c) are given by ϵ1 = √ h2 4 +|⃗ a× ⃗h|2 , ϵ 2 = √ h2 5 +|⃗b× ⃗h|2 . (IV.28) We ﬁrst consider the continuum limit of the Hamiltonian (IV.4) for simplicity, HC −→ HCC = kxΓ5 + kyΓ4− mΓ1 + kzΓ3 . (IV.29) Then, the energy spectra (IV.28) are given by ϵ1 = √ k2 y + (a2m)2...

work page

[6] [6]

Boundary conditions and topological number of edge states We calculate a topological number of the edge states in this part. The normalized edge state wave function depending on the boundary condition (IV.9a) is given by ψn1 = 1√ 2   1 U1   ξ √ 1− β2βn1 , (IV.34) where we also normalize the spinor ξ satisﬁng Eq. (B.6b), as ξ†ξ = 1 . (IV.35) We deﬁne t...

work page

[7] [7]

Investissements d’Avenir

Gapless hinge states We ﬁnd out the hinge state dispersion relation and the corresponding wave function in the case a3 = b3 = 0, for which the topological numbers are obtained. In this case, we have the relations for the coeﬃcients, b1 =∓a2, b2 =±a1 and c1 = c2 = 0, c 3 =±1. Then, from Eqs. (IV.26), we obtain the gapless spectrum, ϵ = ± sin kz (IV.44a) α1...

work page

[8] [8]

Topological insulators,

M. Z. Hasan and C. L. Kane, “Topological insulators,” Reviews of Modern Physics 82 no. 4, (2010) 3045–3067

work page 2010

[9] [9]

Topological insulators and superconductors

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83 no. 4, (2011) 1057–1110, arXiv:1008.2026 [cond-mat.mes-hall]

work page internal anchor Pith review Pith/arXiv arXiv 2011

[10] [10]

Three Lectures On Topological Phases Of Matter

E. Witten, “Three lectures on topological phases of matter,” Riv. Nuovo Cim. 39 no. 7, (Jun, 2016) 313–370, arXiv:1510.07698 [cond-mat.mes-hall]

work page internal anchor Pith review Pith/arXiv arXiv 2016

[11] [11]

Quarks and strings on a lattice,

K. G. Wilson, “Quarks and strings on a lattice,” in New Phenomena in Subnuclear Physics , pp. 69–142. Springer US, 1977

work page 1977

[12] [12]

Solitons in polyacetylene,

W. P. Su, J. R. Schrieﬀer, and A. J. Heeger, “Solitons in polyacetylene,” Physical Review Letters 42 no. 25, (1979) 1698–1701

work page 1979

[13] [13]

Model for a quantum hall eﬀect without landau levels: Condensed-matter realization of the

F. D. M. Haldane, “Model for a quantum hall eﬀect without landau levels: Condensed-matter realization of the ”parity anomaly”,” Physical Review Letters 61 no. 18, (1988) 2015–2018

work page 1988

[14] [14]

Bulk-boundary correspondence in three dimensional topological insulators

L. Isaev, Y. H. Moon, and G. Ortiz, “Bulk-boundary correspondence in three dimensional topological insulators,” Phys. Rev. B 84 (2011) 075444, arXiv:1103.0025 [cond-mat.mes-hall]

work page internal anchor Pith review Pith/arXiv arXiv 2011

[15] [15]

Boundary Conditions and Surface States Spectra in Topological Insulators

V. V. Enaldiev, I. V. Zagorodnev, and V. A. Volkov, “Boundary conditions and surface state spectra in topological insulators,” JETP Letters 101 no. 2, (Jan, 2015) 89–96, arXiv:1407.0945 [cond-mat.mes-hall] . 27

work page internal anchor Pith review Pith/arXiv arXiv 2015

[16] [16]

Boundary Conditions of Weyl Semimetals

K. Hashimoto, T. Kimura, and X. Wu, “Boundary Conditions of Weyl Semimetals,” arXiv:1609.00884 [cond-mat.mes-hall]

work page internal anchor Pith review Pith/arXiv arXiv

[17] [17]

Analysis of topological material surfaces,

T. Kimura, “Analysis of topological material surfaces,” in Heterojunctions and Nanostructures. InTech, July, 2018

work page 2018

[18] [18]

Theory of edge states based on the hermiticity of tight-binding hamiltonian operators,

T. Fukui, “Theory of edge states based on the hermiticity of tight-binding hamiltonian operators,” Phys. Rev. Research 2 (Oct, 2020) 043136, arXiv:2006.04374 [cond-mat.mes-hall]

work page arXiv 2020

[19] [19]

Quantized Electric Multipole Insulators

W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, “Quantized electric multipole insulators,” Science 357 no. 6346, (Jul, 2017) 61–66, arXiv:1611.07987 [cond-mat.mes-hall]

work page internal anchor Pith review Pith/arXiv arXiv 2017

[20] [20]

Topological invariants and corner states for Hamiltonians on a three-dimensional lattice

S. Hayashi, “Topological invariants and corner states for Hamiltonians on a three-dimensional lattice,” Commun. Math. Phys. 364 no. 1, (2018) 343–356, arXiv:1611.09680 [math-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2018

[21] [21]

Edge states at an intersection of edges of a topological material,

K. Hashimoto, X. Wu, and T. Kimura, “Edge states at an intersection of edges of a topological material,” Physical Review B 95 no. 16, (2017)

work page 2017

[22] [22]

Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators

W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, “Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators,” Phys. Rev. B 96 no. 24, (2017) 245115, arXiv:1708.04230 [cond-mat.mes-hall]

work page internal anchor Pith review Pith/arXiv arXiv 2017

[23] [23]

$(d-2)$-dimensional edge states of rotation symmetry protected topological states

Z. Song, Z. Fang, and C. Fang, “( d− 2)-Dimensional Edge States of Rotation Symmetry Protected Topological States,” Phys. Rev. Lett. 119 no. 24, (2017) 246402, arXiv:1708.02952 [cond-mat.mes-hall]

work page internal anchor Pith review Pith/arXiv arXiv 2017

[24] [24]

Higher-Order Topological Insulators

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Reflection symmetric second-order topological insulators and superconductors

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work page internal anchor Pith review Pith/arXiv arXiv 2017

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Universal higher-order topology from a ﬁve-dimensional weyl semimetal: Edge topology, edge hamiltonian, and a nested wilson loop,

K. Hashimoto and Y. Matsuo, “Universal higher-order topology from a ﬁve-dimensional weyl semimetal: Edge topology, edge hamiltonian, and a nested wilson loop,” Phys. Rev. B 101 (Jun, 2020) 245138

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Topological Number of Edge States

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work page internal anchor Pith review Pith/arXiv arXiv 2016

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Geometrical engineering of a two-band chern insulator in two dimensions with arbitrary topological index,

D. Sticlet, F. Pi´ echon, J.-N. Fuchs, P. Kalugin, and P. Simon, “Geometrical engineering of a two-band chern insulator in two dimensions with arbitrary topological index,” Physical Review B 85 no. 16, (2012)

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Second-order topological phases protected by chiral symmetry,

R. Okugawa, S. Hayashi, and T. Nakanishi, “Second-order topological phases protected by chiral symmetry,” Phys. Rev. B 100 (Dec, 2019) 235302

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We leave this issue for a future study

Although the chiral symmetry is also violated in the case a2 =b2 = 0 with cosθi̸= 0, it is not clear for us at this moment how to construct the hinge state as discussed below. We leave this issue for a future study. 29

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