Topological Corner Modes Induced by Dirac Vortices in Arbitrary Geometry

Xiaoxiao Wu, Yan Meng, Yiran Hao, Ruo-Yang Zhang, Jensen Li, and Xiang Zhang

Phys. Rev. Lett. 126, 226802 – Published 4 June 2021

Abstract

Recently, higher-order topologies have been experimentally realized, featuring topological corner modes (TCMs) between adjacent topologically distinct domains. However, they have to comply with specific spatial symmetries of underlying lattices, hence their TCMs only emerge in very limited geometries, which significantly impedes generic applications. Here, we report a general scheme of inducing TCMs in arbitrary geometry based on Dirac vortices from aperiodic Kekulé modulations. The TCMs can now be constructed and experimentally observed in square and pentagonal domains incompatible with underlying triangular lattices. Such bound modes at arbitrary corners do not require their boundaries to run along particular lattice directions. Our scheme allows an arbitrary specification of numbers and positions of TCMs, which will be important for future on-chip topological circuits. Moreover, the general scheme developed here can be extended to other classical wave systems. Our findings reveal rich physics of aperiodic modulations, and advance applications of TCMs in realistic scenarios.

Received 8 September 2020
Revised 2 May 2021
Accepted 7 May 2021

DOI:https://doi.org/10.1103/PhysRevLett.126.226802

Physics Subject Headings (PhySH)

Mechanical & acoustical properties Topological insulators Topological phases of matter

Topological materials

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Xiaoxiao Wu¹, Yan Meng², Yiran Hao², Ruo-Yang Zhang², Jensen Li ^2,*, and Xiang Zhang^1,†

¹Faculties of Sciences and Engineering, The University of Hong Kong, Hong Kong, China
²Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

^*To whom correspondence should be addressed. jensenli@ust.hk
^†To whom correspondence should be addressed. president@hku.hk

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Vol. 126, Iss. 22 — 4 June 2021

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Images

Figure 1
Initial phononic crystal and homogeneous Kekulé modulation. (a) Schematic of initial phononic crystal, which is a thin elastic plate with a triangular lattice of depicted identical slots. Green (red) shaded region denotes a primitive (enlarged) unit cell. (b) Schematic of enlarged unit cell. Initially, slot length $l_{0} = 7.0 mm$ , and uniform slot widths $d_{A} = d_{B 1} = d_{B 2} = d_{0} = 3.0 mm$ . Other geometric parameters $a_{0} = 12.0 mm$ , $h = 2.0 mm$ . (c) Band structure of flexural ( $A_{0}$ ) modes calculated using enlarged unit cell, with unperturbed uniform slot width $d_{0}$ . Inset: two inequivalent Dirac cones at $K_{+}$ and $K_{-}$ valleys folded back into $Γ$ point. (d) Band structures around $Γ$ point when we apply a homogeneous Kekulé modulation with $φ = 0$ and $π$ , respectively. The modulation strength $δ d = 1.4 mm$ . The frequency order of dipolelike ( $p$ ) and quadrupolelike ( $d$ ) modes suggests that $φ = 0$ ( $φ = π$ ) corresponds to an ordinary insulator (topological insulator), or OI (TI). Yellow shaded regions denote the common band gap. (e),(f) Size of the band gap with respect to modulation strength $δ d$ (e) and phase $φ$ (f).
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Figure 2
Topological corner modes (TCMs) induced by Dirac vortices of aperiodic Kekulé modulation. (a) Schematic of designed metastructure with TCMs at four corners, a square domain (blue region) with aperiodic Kekulé modulation [ $Δ φ (r) = 0$ ] cladded by OIs [ $φ (r) = 0$ , gray region]. The modulated domain is referred to as a Kekulé modulated insulator (KMI). Yellow dots represent Dirac vortices of $2 π$ -phase winding. (b) Generated pattern of metastructure in (a). The red dashed square ( $L_{sqr} = 320 mm$ ) delineate the OI and KMI. Inset: phase $φ (r)$ of KMI. Black scale bar: 48 mm. (c) Calculated flexural eigenmodes of metastructure. (d) Field profile [ $Re (w)$ ] of one of the midgap modes. Selected area (green square) around the corner is enlarged, with black arrows denoting in-plane displacement [ $Re (u)$ , $Re (v)$ ]. Black scale bar: 48 mm. (e) Response spectra when exciting the metastructure and a reference structure. Positions of source point ( $S$ ) and detection point ( $D$ ) are depicted in (d). The orange arrow indicates the resonance at 6.97 kHz in the band gap (yellow shaded region). The reference structure only comprises OIs. Maximum of each spectrum is normalized to unity. (f) Fourier spectra [ $| FT (w) |$ ] of experimentally imaged field of metastructure (meta.) and reference structure (ref.) at 6.90 kHz. Green dashed hexagon: FBZ of primitive unit cell shown in Fig. 1. (g) Frequencies of the four TCMs versus modulation strength ( $δ d$ ). Dashed lines indicate frequency range of the band gap.
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Figure 3
Direct observations of TCMs and their robustness. (a) Photograph of the 3D-printed metastructure sample with a square domain of the KMI. Red dashed lines denote boundaries between the domains of OI and KMI. Orange (cyan) point denotes where we excite (detect) the out-of-plane displacement. (b) Experimentally imaged field of out-of-plane displacement ( $| w |$ ) at 6.90 kHz. Green scale bar: 36 mm. (c) Detected response spectra of the metastructure when exciting at the corner, edge, and bulk, respectively. The orange arrow indicates the resonance at 6.90 kHz. When measuring each spectrum, the displacement between the source and detection points is always $r_{D} - r_{S} = (- 3 a_{0} / 2, \sqrt{3} a_{0} / 2)$ . Inset: colored dots indicate positions of detection points. (d) Fourier spectra ( $| FT (w) |$ ) of imaged fields of the metastructure. Green dashed hexagon: FBZ of primitive unit cell. (e) Detected response spectra at the upper-left corner for metastructure samples with varying strength $δ d$ . The resonance peak of the corner mode persists around 6.90 kHz in the broadening band gap (yellow shaded region), as indicated by orange arrows. The maximum of each spectrum is normalized to unity.
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Figure 4
Experimental observation of TCMs induced by Dirac vortices in pentagonal domains. (a) Schematic of a pentagonal KMI (blue region) cladded by OIs (gray region). The corners are labeled as $A$ to $E$ for convenience. Yellow dots represent Dirac vortices of $2 π$ -phase winding. (b) Pattern of the metastructure ( $L_{pen} = 200 mm$ ), denoted as pentagon I (pen. I). Blue dashed rectangle indicates the region imaged in experiments. Inset: distribution of phase $φ (r)$ in the pentagonal domain. Black scale bar: 48 mm. (c) Calculated eigenmode spectra of the metastructure. (d) Local response spectra at corner $A$ and corner $C$ . Source and detection points around corner $C$ are depicted in (e) as orange and cyan dots, respectively. (e) Imaged field map ( $| w |$ ) of the denoted region in (b) when exciting corner $C$ at 6.80 kHz. Green scale bar: 36 mm. (f) Fourier spectra $(| FT (w) |$ ) of the experimental field at 6.80 kHz. Green dashed hexagon: FBZ of primitive unit cell.
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