Fractional corner charges in spin-orbit coupled crystals

Schindler, Frank; Brzezińska, Marta; Benalcazar, Wladimir A.; Iraola, Mikel; Bouhon, Adrien; Tsirkin, Stepan S.; Vergniory, Maia G.; Neupert, Titus

doi:10.1103/physrevresearch.1.033074

Cited by 124 publications

(120 citation statements)

References 91 publications

(176 reference statements)

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“…In recent years, higher-order topological phases have attracted high attention, where higher-order topological states appear due to nontrivial bulk topology. There are two types of higher-order topological phases: those supporting currents [4][5][6][7][8] and those supporting charges [9][10][11][12][13]. One of the realistic materials having helical hinge states is bismuth.…”

Section: Introductionmentioning

confidence: 99%

Higher-order topological crystalline insulating phase and quantized hinge charge in topological electride apatite

Hirayama

Takahashi

Matsuishi

et al. 2020

Phys. Rev. Research

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In some kind of three-dimensional higher-order topological insulators, bulk and surface electronic states are gapped, while there appear gapless hinge states protected by spatial symmetry. Here we show by ab initio calculations that the La apatite electride is a higher-order topological crystalline insulator. It is a one-dimensional electride, in which the one-dimensional interstitial hollows along the c axis support anionic electrons, and the electronic states in these one-dimensional channels are well approximated by the one-dimensional Su-Schrieffer-Heeger model. When the crystal is cleaved into a hexagonal prism, the 120 • hinges support gapless hinge states, with their filling quantized to be 2/3. This quantization of the filling comes from a topological origin. We find that the quantized value of the filling depends on the fundamental blocks that constitute the crystal. The apatite consists of the triangular blocks, which is crucial for giving nontrivial fractional charge at the hinge.

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Section: Introductionmentioning

confidence: 99%

Higher-order topological crystalline insulating phase and quantized hinge charge in topological electride apatite

Hirayama

Takahashi

Matsuishi

et al. 2020

Phys. Rev. Research

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“…A paradigmatic example is the family of topological insulators which manifest quantized dipole moments in their bulk and charge fractionalization at their boundaries [1][2][3], epitomized by the inversion-symmetric onedimensional Su-Schieffer-Hegger model [4]. This property of boundary charge fractionalization has recently been generalized through the discovery of higher-order topological insulators (HOTIs) whose topology is solely protected by crystalline symmetries and which can host corner fractional charges in 2D and 3D [5][6][7][8][9][10][11][12][13][14].…”

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confidence: 99%

Bound states in the continuum of higher-order topological insulators

2020

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We show that lattices with higher-order topology can support corner-localized bound states even in the absence of a bulk energy gap. Topological bound states in these phases thus constitute condensed matter realizations of bound states in the continuum (BICs). We propose a method for the direct identification of BICs in condensed matter settings and use it to demonstrate the existence of BICs in a concrete lattice model. Although the onset for these states is given by corner-induced filling anomalies in certain topological crystalline phases, additional symmetries are required to protect the BICs from hybridizing with their degenerate bulk states. We analytically demonstrate the protection mechanism for BICs in this model and numerically show how breaking this mechanism transforms the BICs into resonances. Our work shows that topological bulk-boundary correspondences are immune to the existence of interfering bulk bands, expanding the search space for crystalline topological phases.arXiv:1908.05687v2 [cond-mat.str-el]

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“…For instance, it was shown that, in a two-dimensional (2D) fragile TI protected by inversion symmetry, fractional corner charges appear at the boundary when the geometry of the system preserves inversion symmetry [20,48]. Moreover, many atomic insulators protected by rotation symmetry were shown to host fractional corner charges in 2D, which is similar to the case of 2D second-order TIs [36,37,40,45,60,61]. The origin of corner charges in fragile topological and atomic insulators has been attributed to the filling anomaly [60], which denotes an obstruction to fulfill the electron filling for charge neutrality when the relevant crystalline symmetry is preserved [20,40,48,60,61].…”

Section: Introductionmentioning

confidence: 97%

Fragile topology protected by inversion symmetry: Diagnosis, bulk-boundary correspondence, and Wilson loop

2019

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We study the bulk and boundary properties of fragile topological insulators (TIs) protected by inversion symmetry, mostly focusing on the class A of the Altland-Zirnbauer classification. First, we propose an efficient method for diagnosing fragile band topology by using the symmetry data in momentum space. Using this method, we show that among all the possible parity configurations of inversion-symmetric insulators, at least 17% of them have fragile topology in two dimensions while fragile TIs are less than 3% percent in three dimensions. Second, we study the bulk-boundary correspondence of fragile TIs protected by inversion symmetry. In particular, we generalize the notion of d-dimensional (dD) kth-order TIs, which is normally defined for 0 < k ≤ d, to the cases with k > d, and show that they all have fragile topology. In terms of the Dirac Hamiltonian, a dD kth-order TI has (k − 1) boundary mass terms. We show that a minimal fragile TI with the filling anomaly can be considered as the dD (d + 1)th-order TI, and all the other dD kth-order TIs with k > (d + 1) can be constructed by stacking dD (d + 1)th-order TIs. Although dD (d + 1)thorder TIs have no in-gap states, the boundary mass terms carry an odd winding number along the boundary, which induces localized charges on the boundary at the positions where the boundary mass terms change abruptly. In the cases with k > (d+1), we show that the net parity of the system with boundaries can distinguish topological insulators and trivial insulators. Also, by studying the Wilson loop and nested Wilson loop spectra, we show that all the spectral windings of the Wilson loop and nested Wilson loop should be unwound to resolve the Wannier obstruction of fragile TIs. By counting the minimal number of bands required to unwind the spectral winding of the Wilson loop and nested Wilson loop, we determine the minimal number of bands to resolve the Wannier obstruction, which is consistent with the prediction from our diagnosis method of fragile topology. Finally, we show that a (d + 1)D (k − 1)th-order TI can be obtained by an adiabatic pumping of dD kth-order TI, which generalizes the previous study of the 2D third-order TI.

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Fractional corner charges in spin-orbit coupled crystals

Cited by 124 publications

References 91 publications

Higher-order topological crystalline insulating phase and quantized hinge charge in topological electride apatite

Higher-order topological crystalline insulating phase and quantized hinge charge in topological electride apatite

Bound states in the continuum of higher-order topological insulators

Fragile topology protected by inversion symmetry: Diagnosis, bulk-boundary correspondence, and Wilson loop

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