Chiral Symmetry Breaking of Tight-Binding Models in Coupled Acoustic-Cavity Systems

Ze-Guo Chen, Licheng Wang, Guanqing Zhang, and Guancong Ma

Phys. Rev. Applied 14, 024023 – Published 11 August 2020

Abstract

A finite one-dimensional Su-Schrieffer-Heeger (SSH) chain exhibits “zero-energy” boundary-mode solutions that are protected by chiral symmetry. The breaking of chiral symmetry leads to several important consequences, including a shift of the boundary mode energies. Here, we systematically study the coupled acoustic-cavity system (CACS), which is an important acoustic platform for realizing tight-binding models (TBMs). We find that the length and number of coupling waveguides not only affect hopping, but also induce a perturbation to the onsite eigenfrequency, which can be attributed to the breaking of chiral symmetry in the TBM. The acoustic origin of these phenomena is discussed, and the conditions of the exact realization of TBMs are identified. Meanwhile, we build an acoustic second-order topological insulator by extending the SSH model to two dimensions and show that the frequency of the topological corner modes is tunable by the same chiral-symmetry-breaking term. This finding is experimentally validated through the demonstration of in-gap and in-band topological corner modes. Our study provides a detailed and accurate understanding of the CACS and clarifies several important nuances for realizing tight-binding systems in acoustics. These results solidify CACS as a foundation for future studies of topological acoustics and non-Hermitian acoustics.

Received 31 March 2020
Revised 2 July 2020
Accepted 6 July 2020

DOI:https://doi.org/10.1103/PhysRevApplied.14.024023

Physics Subject Headings (PhySH)

Acoustic metamaterials Topological phases of matter

Coupled oscillators Topological insulators

Chiral symmetry

Cavity resonators Tight-binding model

Fluid DynamicsCondensed Matter, Materials & Applied PhysicsInterdisciplinary Physics

Authors & Affiliations

Ze-Guo Chen¹, Licheng Wang^1,2, Guanqing Zhang¹, and Guancong Ma ^1,*

¹Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
²Department of Applied Physics, Guangdong University of Technology, Guangzhou 510090, China

^*phgcma@hkbu.edu.hk

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Vol. 14, Iss. 2 — August 2020

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Images

Figure 1
(a) Schematic of a 2D CACS that can be described by a two-level TBM. Profiles of the coupled modes are shown, with red, blue, and white indicating negative, positive, and zero pressure, respectively. (b) Calculated eigenfrequencies of the system near the first resonant frequency, $f_{0} = c / 2 L_{0}$ , as functions of $η$ and $ξ$ . Frequency of the symmetric (antisymmetric) mode is plotted as the red (blue) surface. Average of the two is plotted in green. Gray plane represents $f_{0}$ . Green and gray surfaces intersect at $η = 0.5.$ Two (c), three (d), four (e), and five (f) coupled cavities. Eigenfrequencies of each system are plotted in the lower panels as functions of $η$ . Blue dots are the results obtained from finite-element simulations and red dots are computed using the TBM [Eq. (3)]. (g) Onsite perturbations as functions of $η$ . Number of coupling waveguides on one cavity also affects such perturbations. (h) Perturbations follow linear fits with slopes proportional to the number of waveguides m.
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Figure 2
Sign of hopping can be tuned in CACS by tuning the length of the coupling waveguide. (a) In a two-cavity system, symmetry types of the two modes near $f_{0}$ switch as $η$ crosses unity. (b) Four-cavity system with three short coupling waveguides $(L_{t} = 0.5 L_{0})$ and one long waveguide $(L_{t} = 1.5 L_{0})$ . Four eigenmodes are doubly degenerate at $f_{0} \pm \sqrt{2} w$ .
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Figure 3
(a) Schematic of 1D CACS chain, wherein the coupling waveguides have staggered widths $D_{w} = 3 mm$ and $D_{v} = 9 mm$ . System is an acoustic SSH model. Right panel schematically shows the boundary mode’s position in the band gap is tunable by $η$ . (b) Eigenspectrum of a 12-cavity acoustic SSH chain as functions of $η$ with $L_{0} = 120 mm$ (blue dots), with excellent agreement with the results from TBM theory (red dots). (c) Schematic of 2D CACS array, realizing a 2D SSH model. Position of the second-order topological corner modes is tunable by $η$ . (d) Eigenspectra of a 12 × 12 CACS array, wherein blue (red) dots are the results from finite-element simulation (TBM theory).
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Figure 4
(a) Photograph of a 2D CACS array with 11 × 11 cavities coupled by two types of coupling waveguides, one with a cross-section width of $D_{v} = 9 mm$ , another has a side width $D_{w} = 3 mm$ , as illustrated in the inset. (b) Eigenspectrum of configuration A with a cavity height $L_{A} = 42 mm$ . Topological corner mode is marked by blue dots. Bulk modes and edge modes are plotted in gray and orange dots, respectively. Results show a largely symmetric distribution of eigenmodes near the frequency $f_{A 0} = c / 2 L_{A}$ , indicating the existence of chiral symmetry. (c) Measured pressure response spectra. Gray area represents bulk response, orange area represents edge response, and blue curve is the corner response. Pressure fields are measured when the system is excited by a source located at bulk (d); edge (e),(f); and corner (g), where blue stars indicate the respective source locations. Field distributions indicate bulk mode (d); edge modes (e),(f); and corner modes (g). Excitation frequency in (d) and (g) is the same, indicating the corner mode in (g) is a bound mode in the continuum.
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Figure 5
(a) Simulated eigenspectrum of configuration B, with cavity height $L_{B} = 80 mm$ . Topological corner modes (blue dots) are inside the first band gap. (b) Measured corner response (blue curve) has a peak localized in the bulk gap. Bulk (edge) response is the gray (orange) shading. Pressure field distributions of the bulk (c); edges (d),(e); and corner (f). Blue stars indicate the source locations.
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