Pseudospin induced topological corner state at intersecting sonic lattices

Inspired by the discoveries of electronic topological phases and topological insulators, topologically protected boundary states in classical wave-based systems have attracted considerable interest in the last decade. Most recently, acoustic higher-order topological insulators and Kekulé-distorted sonic lattices have been proposed to support topological corner states and zero-dimensional bound states. Here, we demonstrate a domain wall induced topological corner state that is bound at the crossing point among finite acoustic graphenelike crystals. The approach is based on designing multipolar pseudospin resonances, which give rise to topologically trivial and nontrivial transitions across the domain walls that flank this unusual corner excitation toward the crossing point. By deliberately adding a substantial amount of defects into the cavities of the sonic lattice, we find that the pseudospin induced topological corner state remains entirely unaffected and pinned spectrally to the complete audible band gap. Our findings may thus have the potential to broaden the possibilities for sound confinement and focusing.

Figure 1

(a) Schematic of the acoustic graphenelike honeycomb lattice, which is composed of identical cavity resonators (yellow) connected by variable coupling tubes (blue/red). (b) Enlarged view of the metamolecule. ${\mathit{t}}_{\text{intra}}$ (${\mathit{t}}_{\text{inter}}$) denote the intra- (inter-) metamolecular coupling strength that is controlled by the tube radius ${\mathit{r}}_{\text{intra}}$ (${\mathit{r}}_{\text{inter}}$). (c) Calculated band diagram of the lattice with ${\mathit{r}}_{\text{intra}}={\mathit{r}}_{\text{inter}}$. Inset: First BZ. Pressure field of the acoustic modes corresponding to the (d) second and (e) first double Dirac cone. The color scale represents the total pressure field.

Figure 2

(a) Band diagram of the honeycomb lattice where ${\mathit{r}}_{\text{intra}}/{\mathit{r}}_{\text{inter}}=1.92$. Inset: Schematic of the trivial metamolecule. (b) Corresponding acoustic eigenmodes near the band gap at the $\mathrm{\Gamma }$ point. $p_x$ and $p_y$ display pseudospin dipole modes, while $d_{xy}$ and $d_{x^2-y^2}$ mark pseudospin quadrupole modes, respectively. (c), (d) Same as (a) and (b), but for the nontrivial counterpart with ${\mathit{r}}_{\text{intra}}/{\mathit{r}}_{\text{inter}}=0.61$. (e) Relationship between the eigenfrequencies of the double twofold degenerated states and the ratio ${\mathit{r}}_{\text{intra}}/{\mathit{r}}_{\text{inter}}$. The band inversion effect is clearly observed as indicated by different colors.

Figure 3

(a) Schematic of the quadripartite sonic crystal comprising two different types of interfaces. The blue line denotes the zigzag-type interface while the magenta line corresponds to the broken armchair-type interface. (b) Enlarged view of the center region surrounding the dashed zone in (a). Projected band diagram simulations for ribbons having the interfaces of type (c) I and (d) II [see (a)]. The gray circles and lines represent the bulk and trivial edge states, respectively. The blue and magenta bands denote the topological edge states along their respective interfaces. The red dashed line marks the corresponding frequency of the corner mode.

Figure 4

(a) Calculated eigenfrequencies of the QSC. Black circles, blue dots, and red stars denote the bulk, edge, and corner states, respectively. (b) Detected pressure spectra for the bulk, edge, and corner states. Gray regions represent the bulk bands and the yellow one indicates the complete band gap. Corresponding eigenmodes of the (c) bulk, (d) edge, and (e) corner states as labeled in (a). Color represents the total pressure fields.

Figure 5

(a) Within the shaded green (defect) zone, we introduce a number of rigid perturbation cylinders into the cavities where $r_{\text{p}}$ and $H_{\text{p}}$ are their radius and height. (b) Evolution of the corner states with the number of inserted cylinders $N$. (c) Same as (b), but here the radius of the perturbation cylinders $r_{\text{p}}$ is increased. The red dashed line indicates the spectral location of the pristine corner state.