Edge states at an intersection of edges of a topological material

Hashimoto, Koji; Wu, Xi; Kimura, Taro

doi:10.1103/physrevb.95.165443

Cited by 71 publications

(107 citation statements)

References 33 publications

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“…Since ν = 2 and L 1 − 1 = 2M 1 , from Eq. (20) and (21), we get n − (Γ)| λ=1 = 2M 1 +n − (Γ) = 2M 1 +1, andñ − (Γ)| λ=−1 = 2M 1 + n − (X) = 2M 1 + 1. Therefore, through a change from λ = 1 to λ = −1, the number of states with even (odd) parity which cross E F from above is equal to that from below as a whole.…”

Section: B Illustrative Examplesmentioning

confidence: 83%

Bulk-edge and bulk-hinge correspondence in inversion-symmetric insulators

Takahashi

Tanaka

Murakami

2020

Phys. Rev. Research

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We show that a slab of a three-dimensional inversion-symmetric higher-order topological insulator (HOTI) in class A is a 2D Chern insulator, and that in class AII is a 2D Z2 topological insulator. We prove it by considering a process of cutting the three-dimensional inversion-symmetric HOTI along a plane, and study the spectral flow in the cutting process. We show that the Z4 indicators, which characterize three-dimensional inversion-symmetric HOTIs in classes A and AII, are directly related to the Z2 indicators for the corresponding two-dimensional slabs with inversion symmetry, i.e. the Chern number parity and the Z2 topological invariant, for classes A and AII respectively. The existence of the gapless hinge states is understood from the conventional bulk-edge correspondence between the slab system and its edge states. Moreover, we also show that the spectral-flow analysis leads to another proof of the bulk-edge correspondence in one-and two-dimensional inversionsymmetric insulators. arXiv:1910.08290v1 [cond-mat.mes-hall]

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Section: B Illustrative Examplesmentioning

confidence: 83%

Bulk-edge and bulk-hinge correspondence in inversion-symmetric insulators

Takahashi

Tanaka

Murakami

2020

Phys. Rev. Research

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“…In the conventional approach with the manifold with boundary, this is impossible since the boundary of the boundary must be trivial as a consequence of the homology. Combining two domain-walls having different quantum numbers, however, we have already proposed such an interesting "doubly gapped" fermion system [27], where the edge-of-edge state [47] appears only at the junction of the domain-walls. Our one-loop level computation shows that the structure of the full set of anomaly descent equations is embedded in the fermion determinant.…”

Section: Summary and Discussionmentioning

confidence: 99%

Atiyah-Patodi-Singer index from the domain-wall fermion Dirac operator

2017

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The Atiyah-Patodi-Singer(APS) index theorem attracts attention for understanding physics on the surface of materials in topological phases. The mathematical set-up for this theorem is, however, not directly related to the physical fermion system, as it imposes on the fermion fields a non-local boundary condition known as the "APS boundary condition" by hand, which is unlikely to be realized in the materials. In this work, we attempt to reformulate the APS index in a "physicistfriendly" way for a simple set-up with U (1) or SU (N ) gauge group on a flat four-dimensional Euclidean space. We find that the same index as APS is obtained from the domain-wall fermion Dirac operator with a local boundary condition, which is naturally given by the kink structure in the mass term. As the boundary condition does not depend on the gauge fields, our new definition of the index is easy to compute with the standard Fujikawa method.

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“…Such is the case for the 3D phases like weak TI [1], the mirror Chern insulator [6,7], and the nonsymmorphic insulators featuring so-called hourglass [14,15] or wallpaper [17] fermions, which are respectively protected by lattice translation, reflection, and glide symmetries. While these conventional phases feature 2D gapless surface states on appropriate surfaces, we will also allow for more delicate, 1D "hinge" surface states, whose existence is globally guaranteed on suitably chosen sample geometries although any given crystal facet can be fully gapped [9,[18][19][20][21][22][23][31][32][33]. We will primarily focus on 3D time-reversal symmetric band structures with a bulk gap and significant spin-orbit coupling (class AII).…”

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confidence: 99%

Symmetry Indicators and Anomalous Surface States of Topological Crystalline Insulators

et al. 2018

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The rich variety of crystalline symmetries in solids leads to a plethora of topological crystalline insulators (TCIs) featuring distinct physical properties, which are conventionally understood in terms of bulk invariants specialized to the symmetries at hand. While isolated examples of TCI have been identified and studied, the same variety demands a unified theoretical framework. In this work, we show how the surfaces of TCIs can be analyzed within a general surface theory with multiple flavors of Dirac fermions, whose mass terms transform in specific ways under crystalline symmetries. We identify global obstructions to achieving a fully gapped surface, which typically lead to gapless domain walls on suitably chosen surface geometries. We perform this analysis for all 32 point groups, and subsequently for all 230 space groups, for spin-orbit-coupled electrons. We recover all previously discussed TCIs in this symmetry class, including those with "hinge" surface states. Finally, we make connections to the bulk band topology as diagnosed through symmetry-based indicators. We show that spin-orbit-coupled band insulators with nontrivial symmetry indicators are always accompanied by surface states that must be gapless somewhere on suitably chosen surfaces. We provide an explicit mapping between symmetry indicators, which can be readily calculated, and the characteristic surface states of the resulting TCIs. arXiv:1711.11589v3 [cond-mat.str-el]

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Edge states at an intersection of edges of a topological material

Cited by 71 publications

References 33 publications

Bulk-edge and bulk-hinge correspondence in inversion-symmetric insulators

Bulk-edge and bulk-hinge correspondence in inversion-symmetric insulators

Atiyah-Patodi-Singer index from the domain-wall fermion Dirac operator

Symmetry Indicators and Anomalous Surface States of Topological Crystalline Insulators

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