Majorana-like Zero Modes in Kekulé Distorted Sonic Lattices

Topological phases have recently been realized in bosonic systems. The associated boundary modes between regions of distinct topology have been used to demonstrate robust waveguiding, protected from defects by the topology of the surrounding bulk. A related type of topologically protected state that is not propagating but is bound to a defect has not been demonstrated to date in a bosonic setting. Here we demonstrate numerically and experimentally that an acoustic mode can be topologically bound to a vortex fabricated in a two-dimensional, Kekulé-distorted triangular acoustic lattice. Such lattice realizes an acoustic analog of the Jackiw-Rossi mechanism that topologically binds a bound state in a $p$-wave superconductor vortex. The acoustic bound state is thus a bosonic analog of a Majorana bound state, where the two valleys replace particle and hole components. We numerically show that it is topologically protected against arbitrary symmetry-preserving local perturbations, and remains pinned to the Dirac frequency of the unperturbed lattice regardless of parameter variations. We demonstrate our prediction experimentally by 3D printing the vortex pattern in a plastic matrix and measuring the spectrum of the acoustic response of the device. Despite viscothermal losses, the measured topological resonance remains robust, with its frequency closely matching our simulations.

Figure 1

An acoustic analog of a topological superconductor vortex made of rigid cylinders of varying radii, arranged in a finite triangular sonic lattice. The color of the cylinders illustrates the radius distortion $R_0\pm \delta R$, as given by Eq. (2). The Kekulé phase $\varphi$, together with the local intervalley gap as one moving around the vortex, are depicted around the perimeter.

Figure 2

(a) Schematic of the cluster with a magnification of its core showing the locations of sound excitation ($S$) and probing ($P$). (b) Calculated pressure $|P|$ spectra for a topological trivial ($n=0$) and nontrivial ($n=1$) cluster, with $\delta R/d=0.15$. The shaded background represents the complete band gap. Frequency $\omega$ is normalized as $\mathrm{\Omega }=\omega a/2\pi c$, with $c$ being the speed of sound and $a$ the lattice period. (c) Evolution of the localized states as a function of the distortion $\delta R/d$. The horizontally dashed line indicates the Dirac frequency, ${\mathrm{\Omega }}_D=0.88$. (d),(e) Pressure amplitudes $|P|$ in real space for the two peaks $P_1$ and $P_2$ of (b).

Figure 3

Response spectra $|P(\mathrm{\Omega })|$ within the band gap of the topological $n=1$ vortex cluster ($\delta R/d=0.15$) with defect areas introduced (a) at the vortex core and (b) slightly away from it. The defects perturb the vortex by making $\delta R=0$ inside the dashed circles. The different datasets to the left of each panel correspond to an increasing normalized radius $R_D/d=m$ of the circular defect area, while the spatial map to the right shows in red the mode profile of state $P_1$, which is found to remain pinned at the Dirac frequency ${\mathrm{\Omega }}_D=0.88$ for all defects.

Figure 4

(a) Photograph of the 3D printed Kekulé distorted sonic lattice, with parameters $n=1$, $d=9\mathrm{mm}$, $R_0=0.35d$, $\delta R=0.15d$, $\xi =2a$, and $a=\sqrt{3}d$. (b) A rigid lid was mounted above the cluster into which holes were drilled to fasten the speaker and the microphone. (c) The spectrally measured pressure at point $P$ (black) is compared to both inviscid (red) and viscid (blue) numerical computations in near proximity to the edge of the band gap.