Filling anomaly for general two- and three-dimensional 
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 symmetric lattices

Fang, Yuan; Cano, Jennifer

doi:10.1103/physrevb.103.165109

Cited by 39 publications

(31 citation statements)

References 64 publications

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“…We obtain a formula for the hinge charge written in terms of the irreducible representations (irreps) at high-symmetry momenta in the Brillouin zone (BZ) based on the elementary band representation (EBR) matrix method developed in Refs. [21,31]. However, we find that this approach fails to determine the corner charge because the information on the irreps at high-symmetry momenta is not sufficient to fix the electronic Wannier centers of occupied bands.…”

Section: Introductionmentioning

confidence: 81%

“…Three-dimensional topological insulators usually feature gapless modes on their two-dimensional surfaces [7][8][9][10] . When all the surfaces are gapped but hinges or corners are gapless, the insulator is said to possess a higher-order topology [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] .…”

Section: Introductionmentioning

confidence: 99%

“…This charge imbalance is called filling anomaly. The fractional charge localized at a corner can be deduced by dividing the filling anomaly by the number of corners, which are related by symmetry [18][19][20][21][22][23][24][25]27 .…”

Section: Introductionmentioning

confidence: 99%

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Fractional hinge and corner charges in various crystal shapes with cubic symmetry

Naito,

Takahashi,

Watanabe

et al. 2021

Preprint

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Higher-order topological insulators host gapless states on hinges or corners of three-dimensional crystals. Recent studies suggested that even topologically trivial insulators may exhibit fractionally quantized charges localized at hinges or corners. Although most of the previous studies focused on two-dimensional systems, in this work, we take the initial step toward the systematic understanding of hinge and corner charges in three-dimensional insulators. We consider five crystal shapes of vertextransitive polyhedra with the cubic symmetry such as a cube, an octahedron and a cuboctahedron. We derive real-space formulas for the hinge and corner charges in terms of the electric charges associated with bulk Wyckoff positions. We find that both the hinge and corner charges can be predicted from the bulk perspective only modulo certain fractions depending on the crystal shape, because the relaxation near boundaries of the crystal may affect the fractional parts. In particular, we show that a fractionally quantized charge 1/24 mod 1/12 in the unit of elementary charge can appear in a crystal with a shape of a truncated cube or a truncated octahedron. We also investigate momentum-space formulas for the hinge and corner charges. It turns out that the irreducible representations of filled bands at high-symmetry momenta are not sufficient to determine the corner charge. We introduce an additional Wilson-loop invariant to resolve this issue.

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Section: Introductionmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 99%

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Fractional hinge and corner charges in various crystal shapes with cubic symmetry

Naito,

Takahashi,

Watanabe

et al. 2021

Preprint

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“…Since the two modes are related by inversion symmetry, however, this cannot be -instead each defect mode consists of a superposition of states originating in the lower and upper frequency band. This represents a photonic realization of the "fill-ing anomaly" from condensed matter physics [18,21,[33][34][35]68]. In a system with N unit cells and no defects, if we "fill" m bands, then we would have m × N photons in the system.…”

Section: Defect Bound States and The Jackiw-rebbi Modelmentioning

confidence: 99%

“…The non-Abelian Berry phase gives the centers of photonic hybrid Wannier functions, localized along one direction of the photonic crystal; the location and flow of these centers is connected to the presence of localized corner and edge states in topologically nontrivial systems. Similarly, filling anomalies in one-and two-dimensional insulators can be understood in terms of the centers of maximally localized Wannier functions relative to symmetry centers of the crystal [33][34][35].…”

Section: Introductionmentioning

confidence: 99%

Wannier Function Methods for Topological Modes in 1D Photonic Crystals

Gupta,

Bradlyn

2022

Preprint

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In this work, we use Wannier functions to analyze topological phase transitions in one dimensional photonic crystals. We first review the construction of exponentially localized Wannier functions in one dimension, and show how to numerically construct them for photonic systems. We then apply these tools to study a photonic analog of the Su-Schrieffer-Heeger model. We use photonic Wannier functions to construct a quantitatively accurate approximate model for the topological phase transition, and compute the localization of topological defect states. Finally, we discuss the implications of our work for the study of band representations for photonic crystals.

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Topological Corner Modes by Composite Wannier States in Glide‐Symmetric Photonic Crystal

Liu,

Zhou,

Wei

et al. 2024

Laser & Photonics Reviews

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Second‐order topological insulators can be characterized by their bulk polarization, which is believed to be intrinsically connected to the center of the Wannier function. In this study, the existence of second‐order topological insulators is demonstrated that feature a pair of partially degenerate photonic bands. These arise from the nonsymmorphic glide symmetry in an all‐dielectric photonic crystal. The center of the maximally localized Wannier function (MLWF) is consistently located at the origin but is not equivalent with respect to the sum of constituent polarizations. As a result, topological corner modes can be identified by the distinctly hybridized MLWFs that truncate at the sample boundary. Through full‐wave numerical simulations paired with microwave experiments, the second‐order topology is clearly confirmed and characterized. These topological corner states exhibit notably unique modal symmetries, which are made possible by the inversion of the Wannier bands. These results provide an alternative approach to explore higher‐order topological physics with significant potential for applications in integrated and quantum photonics.

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Filling anomaly for general two- and three-dimensional C4 symmetric lattices

Cited by 39 publications

References 64 publications

Fractional hinge and corner charges in various crystal shapes with cubic symmetry

Fractional hinge and corner charges in various crystal shapes with cubic symmetry

Wannier Function Methods for Topological Modes in 1D Photonic Crystals

Topological Corner Modes by Composite Wannier States in Glide‐Symmetric Photonic Crystal

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