Observation of topological valley modes in an elastic hexagonal lattice

We report on the experimental observation of topologically protected edge waves in a two-dimensional elastic hexagonal lattice. The lattice is designed to feature $K$-point Dirac cones that are well separated from the other numerous elastic wave modes characterizing this continuous structure. We exploit the arrangement of localized masses at the nodes to break mirror symmetry at the unit-cell level, which opens a frequency band gap. This produces a nontrivial band structure that supports topologically protected edge states along the interface between two realizations of the lattice obtained through mirror symmetry. Detailed numerical models support the investigations of the occurrence of the edge states, while their existence is verified through full-field experimental measurements. The test results show the confinement of the topologically protected edge states along predefined interfaces and illustrate the lack of significant backscattering at sharp corners. Experiments conducted on a trivial waveguide in an otherwise uniformly periodic lattice reveal the inability of a perturbation to propagate and its sensitivity to backscattering, which suggests the superior waveguiding performance of the class of nontrivial interfaces investigated herein.

Figure 1

(a) Hexagonal discrete lattice emulating the QVHE. (b) Dispersion surfaces and contours of the first surface for $\gamma =0$, highlighting the presence of Dirac cones, the first irreducible Brillouin zone (solid red line), and the location of the $K$ and $K^{\prime }$ points. (c) Band diagrams reporting the values of the valley Chern numbers at the $K$ and $K^{\prime }$ points for $\gamma =0$ (dashed black line) and $\gamma =+0.2$ (solid red and blue lines). (d) Inversion of the bounding frequencies of the band gap as a function of the parameter $\gamma$.

Figure 2

Experimental lattice with added masses at the sublattice sites. Insets show the FE discretization of the unit cell for numerical study for configurations defined by $\gamma =0$ and $\gamma =1$, as well as the first and irreducible Brillouin zone (red solid line) for the lattice.

Figure 3

Dispersion diagrams: (a) lattice without masses, $\gamma =0$; (b) lattice with masses, $\gamma =1$ (thin dashed black line: location of the Dirac point, thick blue lines: out-of-plane wave mode, thin red lines: shear polarized modes). (c) Variation of band gap bounding frequencies ${\omega }^{+}$ (red line) and ${\omega }^{-}$ (blue line) at $K$ as a function of $\gamma$ showing band inversion and (d) phase of corresponding eigenfunctions of the first two out-of-plane modes at $K$ for $\gamma =-1$ and $\gamma =1$.

Figure 4

(a) Finite strip and close-up of the topological interface. (b) Dispersion diagram: edge mode (black line with dots), out-of-plane modes (thick blue lines), and in-plane polarized modes (thin red lines). (c) Eigenvector corresponding to the edge mode evaluated for wave number ${\kappa }_x=0.7\pi$ at $f\simeq 3$ kHz.

Figure 5

Experimental 3D-FTs. (a) Lattice with $\gamma =0$: cross section $|\widehat{w}({\kappa }_x,{\kappa }_y,{\omega }_0)|$ at frequency $f_0\approx 3.75$ kHz close to the numerically predicted Dirac cone. Frequency/wave-number representation $|\widehat{w}(\kappa {|}_{\mathcal{C}},\omega )|$ along the path $\mathcal{C}:\mathrm{\Gamma }-K-\mathrm{\Gamma }$ and comparison with comsol predictions (solid red line): lattice with (b) $\gamma =0$ and (c) $\gamma =1$.

Figure 6

(a) Line interface (red dots indicate two masses attached at the sublattice sites, and cyan dots denote locations where no masses are added) with $\gamma =1$ ($-1$) on the left (right). (b) and (c) Snapshots of measured wave motion at two instants of time. Excitation is an 11-cycle tone burst at 3 kHz (contours are normalized by the displacement amplitude of $3\times {10}^{-8}$ m).

Figure 7

(a) A nontrivial $\mathsf{N}$-shaped interface (red dots indicate two masses attached at the sublattice sites, and cyan dots denote locations where no masses are added) with $\gamma =1$ ($-1$) on the left (right). (b) and (c) Snapshots of measured wave motion at two instants of time. Excitation is an 11-cycle tone burst at 3 kHz (contours are normalized by the displacement amplitude of $1.5\times {10}^{-8}$ m).

Figure 8

(a) $\mathsf{N}$-shaped trivial interface (red dots indicate two masses attached at the sublattice sites, and cyan dots denote locations where no masses are added) with $\gamma =1$ everywhere. (b) and (c) Snapshots of measured wave motion at two instants of time. Excitation is an 11-cycle tone burst at 3 kHz (contours are normalized by the displacement amplitude of $1.5\times {10}^{-8}$ m).