Topological interface modes in local resonant acoustic systems

Degang Zhao, Meng Xiao, C. W. Ling, C. T. Chan, and Kin Hung Fung

Phys. Rev. B 98, 014110 – Published 27 July 2018

Abstract

Topological phononic crystals are artificial periodic structures that can support nontrivial acoustic topological bands, and their topological properties are linked to the existence of topological edge modes. Most previous studies have been focused on the topological edge modes in Bragg gaps, which are induced by lattice scattering. While local resonant gaps would be of great use in subwavelength control of acoustic waves, whether it is possible to achieve topological interface states in local resonant gaps is a question. In this paper, we study the topological properties of subwavelength bands in a local resonant acoustic system and elaborate the band-structure evolution using a spring-mass model. Our acoustic structure can produce three band gaps in the subwavelength region: one originates from the local resonance of unit cell and the other two stem from band folding. It is found that the topological interface states can only exist in the band-folding-induced band gaps, but never appear in the local resonant band gap. In addition, the numerical simulation in a practical system perfectly agrees with the theoretical results. Our study provides an effective approach of producing robust acoustic topological interface states in the subwavelength region.

Received 12 December 2017
Revised 11 July 2018

DOI:https://doi.org/10.1103/PhysRevB.98.014110

Physics Subject Headings (PhySH)

Acoustic metamaterials

1-dimensional systems

Bloch-Floquet theorem

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Degang Zhao^1,2, Meng Xiao², C. W. Ling³, C. T. Chan², and Kin Hung Fung^3,*

¹Department of Physics, Huazhong University of Science and Technology, Wuhan, 430074, China
²Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
³Department of Applied Physics, The Hong Kong Polytechnic University, Hong Kong, China

^*khfung@polyu.edu.hk

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Issue

Vol. 98, Iss. 1 — 1 July 2018

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Images

Figure 1
Schematics of the spring-mass model of a 1D array of local resonant unit cells in (a) monatomic chain, (b) a chain identical to that in (a) except that the new unit cell consists of two unit cells in (a), and (c) diatomic chain with $K_{1} \neq K_{2}$ . The dashed frames mark the unit cells for these three distinct chains. $L$ denotes the lattice constant. (d), (e), and (f) are the band structures for the systems depicted in (a), (b), and (c), respectively. The cyan and magenta strips represent the local resonant gaps and band-folding-induced gaps, respectively. The three gaps in (f) are indicated by Roman numerals I, II, and III. S1–S6 in (f) denotes the band edges of all gaps.
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Figure 2
(a) Topological interface states (red lines) of a supercell system composed of a SSH chain with $K_{1} < K_{2}$ on the left side and a SSH chain with $K_{1} > K_{2}$ on the right side. The periods of both chains in one supercell unit are 20. (b) Dependence of the interface states (red lines) on $K_{1} / K_{2}$ for the right SSH chain, while the left SSH chain $(K_{1} < K_{2})$ remains unchanged. For simplicity, $K_{1} + K_{2}$ is held constant in the calculations. The gray-shaded regions in both (a) and (b) denote the projection bulk bands.
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Figure 3
(a) Configuration of a practical structural model of the connected SSH chain with local resonant unit cells. Each scatterer is a cylinder with an epoxy core wrapped in soft rubber, and all cylinders are immersed in water. The dashed black frames denote the unit cell. The separation between two individual cylinders in a unit cell is $d_{L}$ for the left lattice and $d_{R}$ for the right lattice. $L$ is the lattice constant. The inner and outer radii of the cylinders are $0.165 L$ and $0.2 L$ . The red dashed line denotes the interface between the two lattices. (b) Band structure of the left or right lattice. The frequencies are in units of $2 π c / L$ , where $c$ is the acoustic wave velocity in water. All of the symbols have the same meanings as in Fig. 1. (c) Pressure-field distributions of band edges for the left and right lattices, where only the pressure fields in the water are shown.
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Figure 4
(a) Transmission spectrum of a finite-size lattice shown in Fig. 3, where both the left and right lattices have eight periods. A and B denote the transmission peaks in the band gaps. (b) Absolute values of the pressure-field distributions (only in the background) of peaks A and B.
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Figure 5
Solutions of $x$ in Eq. (B1) at all of the band edges.
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Figure 6
(a) Band structure of SSH chain. All bands are distinguished by different colors. (b) Eigenvalues for all of the bands. The colors have one-to-one correspondence to those in (a).
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