Valley-Selective Topological Corner States in Sonic Crystals

Xiujuan Zhang, Le Liu, Ming-Hui Lu, and Yan-Feng Chen

Phys. Rev. Lett. 126, 156401 – Published 12 April 2021

Abstract

Higher-order topological insulators (HOTIs), a new horizon of topological phases of matter, host lower-dimensional corner or hinge states, providing important stepping stones to the realization of robust topological waveguides in higher dimensions. The nontrivial band topology that gives rise to the corner or hinge states is usually enabled by certain crystalline symmetries. As a result, higher-order topological boundary states are tied to specific corners or hinges, lacking the flexibility of switching and selecting. Here, we report the experimental realization of topologically switchable and valley-selective corner states in a two-dimensional sonic crystal. Such intriguing properties are enabled by exploiting the higher-order topology assisted with the valley degree of freedom. For this purpose, we realize a valley HOTI of second-order topology characterized by the nontrivial bulk polarization. Interestingly, the hosted corner states are found to be valley dependent and therefore enable flexible control and manipulation on the wave localization. Topological switch on or off and valley selection of the corner states are directly observed through spatial scanning of the sound field. We further design an arbitrary structure of complex patterns containing corners with various intersection angles, among which selected corners can be illuminated or darkened upon valley selection. The reported valley HOTI and the valley-selective corner states provide fundamental understanding on the interplay between higher-order topology and valley degree of freedom and pave the way for lower-dimensional valleytronics, which may find potential applications in integrated acoustics and photonics.

Received 18 December 2020
Accepted 25 February 2021

DOI:https://doi.org/10.1103/PhysRevLett.126.156401

Physics Subject Headings (PhySH)

Acoustic metamaterials Topological insulators Topological phases of matter

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Xiujuan Zhang^1,*, Le Liu^1,2,*, Ming-Hui Lu ^1,3,4,†, and Yan-Feng Chen^1,4,‡

¹National Laboratory of Solid State Microstructures and Department of Materials Science and Engineering, Nanjing University, Nanjing 210093, China
²School of Physics, Nanjing University, Nanjing 210093, China
³Jiangsu Key Laboratory of Artificial Functional Materials, Nanjing University, Nanjing, 210093, China
⁴Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

^*These authors contributed equally to this work.
^†luminghui@nju.edu.cn
^‡yfchen@nju.edu.cn

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Vol. 126, Iss. 15 — 16 April 2021

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Images

Figure 1
(a) Schematic of the valley SC, with the scatterers indicated by blue color and air by white. ${\vec{a}}_{1}$ and ${\vec{a}}_{2}$ denote the lattice vectors. The geometric parameters are taken as $a = 20 mm$ , $r = 3 mm$ , and $R = 9.5 mm$ . (b) Acoustic band structures of the valley SC with $θ = - 30 °$ , 0°, and 30° (inset shows the Brillouin zone). The sound pressure profiles (and the Poynting vectors represented by the arrows) of the eigenstates at the band edges of $θ = - 30 °$ and 30° reveal opposite vortex chirality, which is associated with the Dirac mass $m$ of opposite signs and therefore indicates a topological phase transition. (c) Topological phase diagram of the valley SC. The frequencies of the $q$ and $p$ states at the $K$ point are plotted as the red and blue curves, respectively. The eigenstates presented in (b) are marked in (c).
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Figure 2
(a) The adopted rhombic Brillouin zone in the calculations of the bulk polarization, which shares the same area with the original hexagonal Brillouin zone. The calculated Berry phase $ν_{1}$ as a function of $k_{2}$ is presented, respectively, for (b) the USC case and for (c) the DSC case. The bulk polarization is accordingly obtained by integration of $ν_{1}$ over $k_{2}$ and is shown by the dotted lines (for illustration) with the exact values of $\frac{1}{3}$ and $- \frac{1}{3}$ . The corresponding Wannier centers are marked by the green and oranges dots, which coincide with the Wyckoff positions.
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Figure 3
(a),(b) Sketched triangular and (c),(d) hexagonal corner structures. The Wannier center representations, illustrated by the thin green and orange dots, reveal the selectivity of corner states (indicated by the red thick dots). Calculated eigenspectra for the structures in (a)–(d) are presented in the upper panels of (e)–(h), successively. Experimental characterizations are presented in the corresponding lower panels, where the normalized transmission spectra for the bulk (gray), edge (blue), and corner (red) states are measured (see the unnormalized spectra in Ref. [33]). The selective excitation of corner states (with the excitation frequency denoted by the red or blue thick arrows) is also performed and the measured sound field maps are inserted for direct visualizations, where the green lines highlight the corner structures.
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Figure 4
(a),(b) Sketches of the complex corner structures made of DSC and USC, respectively. Eleven corners are created, which are labeled by numbers. The Wannier center representations suggest nine corner states in the DSC structure and five corner states in the USC structure. The corresponding normalized transmission spectra (see the unnormalized spectra in Ref. [33]) for all 11 corners are presented in (c) and (d). For clarification, the spectrum of each corner is indicated by a colored arrow and the corresponding corner number. The corner numbers enclosed by dashed boxes represent the deaf corner states while those enclosed by the blue solid boxes are associated with the edge states that appear around the corner frequencies. The measured sound field maps (with excitation frequencies indicated by the red thick arrows in the spectra) are presented in the right-hand panels of (c) and (d), respectively, for the DSC and USC structures. The edge states excited in the USC structure are marked in the sound field maps.
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