Anomalous quadrupole topological insulators in two-dimensional nonsymmorphic sonic crystals

Zhi-Kang Lin, Hai-Xiao Wang, Zhan Xiong, Ming-Hui Lu, and Jian-Hua Jiang

Phys. Rev. B 102, 035105 – Published 2 July 2020

Abstract

The discovery of quadrupole topology opens a new horizon in the study of topological phenomena. However, the existing experimental realizations of quadrupole topological insulators in symmorphic lattices with $π$ fluxes often break the protective mirror symmetry. Here, we present a theory for anomalous quadrupole topological insulators in nonsymmorphic crystals without flux using two-dimensional sonic crystals with $p 4 g m$ and $p 2 g g$ symmetry groups as concrete examples. We reveal that the anomalous quadrupole topology is protected by two orthogonal glide symmetries in square or rectangular lattices. The distinctive features of the anomalous quadrupole topological insulators include: (i) minimal four bands below the topological band gap, (ii) nondegenerate gapped Wannier bands and special Wannier sectors with gapped composite Wannier bands, and (iii) quantized Wannier band polarizations in these Wannier sectors. With no need for flux insertion, the protective glide symmetries are well preserved in the sonic-crystal realizations where higher-order topological transitions can be triggered by symmetry or geometry engineering.

Received 18 January 2020
Revised 16 June 2020
Accepted 18 June 2020

DOI:https://doi.org/10.1103/PhysRevB.102.035105

Physics Subject Headings (PhySH)

Topological insulators

Metamaterials

Wannier function methods

Nonlinear DynamicsInterdisciplinary PhysicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Zhi-Kang Lin¹, Hai-Xiao Wang^2,1, Zhan Xiong¹, Ming-Hui Lu³, and Jian-Hua Jiang ^1,*

¹School of Physical Science and Technology, Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, 1 Shizi Street, Suzhou 215006, China
²College of Physics and Technology, Guangxi Normal University, Guilin 541004, China
³National Laboratory of Solid State Microstructures and Department of Materials Science and Engineering, Nanjing University, Nanjing, 210093, China and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

^*Corresponding author. jianhuajiang@suda.edu.cn

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Issue

Vol. 102, Iss. 3 — 15 July 2020

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Images

Figure 1
(a) A $p 4 g m$ SC that realizes the AQTI. The inset depicts the bulk quadrupole moment. (b) Acoustic band structures for $l = 0.42 a, h = 0.21 a, w = 0.1 a$ , and $a = 2 cm$ . Symbols $\pm$ label parities at the $Γ$ and $X$ points. Right panel: Unit-cell (brown shapes denote the epoxy scatterers, whereas dashed lines depict the glide invariant lines) and Brillouin zone of the SC. (c) Wannier bands for the four acoustic bands below the gap (left) and their combinations (right). (d) Nested Wannier bands and their combinations in the “1 + 3” and “2 + 4” Wannier sectors. Sound velocity and mass density of air are set as 343 m/s and $1.29 {kg / m}^{3}$ , respectively.
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Figure 2
(a) Schematic of four types of SCs with $p 4 g m, p 2 g g, p 2 m g$ , and $p 4 m m$ symmetries, separately. The brown shapes denote the epoxy scatterers, whereas the dashed lines depict the glide invariant lines. We use the check marks to denote whether the Wannier bands $ν_{x}$ and $ν_{y}$ are gapped and nondegenerate or not. (b) and (c) Acoustic band structures for the (b) $p 2 g g$ and (c) $p 4 m m$ SCs. The cyan regions denote the bulk band gaps of concern. Geometry parameters for (b) $l = 0.5 a, h = 0.25 a, w = 0.05 a$ , and $δ = 0.1 a$ ( $a$ is the lattice constant), and for (c) $l = 0.35 a, h = 0$ , and $w = 0.25 a$ . Here, the $p 2 g g$ SC is derived from the $p 4 g m$ SC by removing the gray regions of length $δ$ for the epoxy scatterers [see (a)].
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Figure 3
(a) Eigenstates spectrum for a square-shaped finite-sized $p 4 g m$ SC with hard-wall boundary conditions, showing the coexistence of the bulk, edge, and corner states. (b) and (c) Acoustic wave functions (the acoustic pressure field $p$ ) for the (b) corner and (c) edge states. (d) Evolution of the bulk, edge, and corner spectra with the truncation length $δ$ for the $p 2 g g$ SC. The topological band gap is tuned to close and reopen with increasing $δ$ . The bulk band gap closing at $δ = 0.3 a$ is indicated by the dashed line.
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Figure 4
(a) Evolution of the bulk, edge, and corner spectra during a geometry transformation from the $p 4 g m$ group to the $p 4 m m$ group: The starting point has $l = 0.5 a, h = 0.25 a$ , and $w = 0.05 a$ . First, $h$ goes from $0.25 a$ to 0. Then, $l$ goes from $0.5 a$ to $0.35 a$ . Finally, $w$ goes from $0.05 a$ to $0.25 a$ . The vertical dashed lines indicate the conditions where the bulk band gap closes or opens. The left, middle, and right regions separated by the dashed lines correspond to the QTI phase, the gapless phase, and the trivial phase, separately. Several representative unit-cell geometries are illustrated as the insets. (b) and (c) Evolution of the Wannier bands (b) and the quadrupole topological index $q_{x y}$ (c) during the geometry transformation process.
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