Corner states in a second-order acoustic topological insulator as bound states in the continuum

A second-order topological insulator is designed on a platform of a two-dimensional (2D) square lattice with all coupling coefficients having the same sign. Simulated results show the existence of two types of nontrivial corner states in this system, with one type being identified as bound states in the continuum (BIC). The non-BIC corner states are also found by surrounding a nontrivial sample by a trivial one, and interestingly, these perfectly confined corner states can be gradually delocalized and merge into edge states by tuning the intersystem coupling coefficient. Both BIC and non-BIC corner states originate from bulk dipole moments rather than quantized quadrupole moments, with the corresponding topological invariant being the 2D Zak phase. Full wave simulations based on realistic acoustic waveguide structures are demonstrated. Our proposal provides an experimentally feasible platform for the study of the interplay between BIC and a high-order topological insulator, and the evolution from corner states to edge states.

Figure 1

(a) Schematic of the acoustic ring-waveguide system that can be described by the Hamiltonian model, i.e., Eq. (1). Each unit cell contains four identical hollow rings, whose inner and outer radii are fixed, connected by rectangular waveguides. Two different unit cells with inversed intercell and intracell hoppings are circled out. The hopping is characterized by the rectangular waveguides, whose width represents the hopping strength. (b) The calculated band structure using the finite-element method (black dots), compared with the results obtained from the tight-binding model (red line), where the fitting parameter is ${\varepsilon }_s=26.375$, $w=5,v=1$, and all of the fitting parameters have the unit $\mathrm{Hz}$. The symbol “±" means the parity of the states under $\pi$ rotation, which is derived from the eigenfield distributions and eigenvectors shown in the right inset. Red (blue) in the right inset indicates the normalized maximum (minimum) acoustic pressure field, respectively.

Figure 2

The corner states in continuum. (a) Illustration of a $3\times 3$ lattice with $w=0$, where the corner states naturally appear. (b) The eigenmodes of a $10\times 10$ sample calculated from both the tight-binding model (red point) and the numerical simulation result (black point). In the tight-binding model calculations, we introduce small random perturbations to all of the site energies and coupling parameters. The energy spectrum is plotted after an ensemble average of 10 configurations. The result shows four corner states in the bulk frequency region marked as purple stars. The field distribution of the four corner states from the tight-binding model and the finite-element simulation are shown in (c) and (d), respectively. The sample size we calculated in (d) is $3\times 3$ for simplicity and clarity.

Figure 3

The corner states in a gap of topological phononic clusters. (a) The sample consists of a topologically nontrivial system B surrounded by a topologically trivial system A. (b) The eigenmodes of the sample calculated from both the tight-binding model (red point) and the finite-element simulation result. We also set the coupling between systems A and B as $t=5$. The result shows four corner states in the band-gap region. (c) and (d) The field distribution of the four corner states from the tight-binding model (c) and the finite-element simulation (d), respectively.

Figure 4

The corner states evolution against intersystem hopping $t$. The sample consists of $27\times 27$ unit cells. (a) The schematic of a structure that contains two interfaces between systems A and B, where the coupling parameters are undetermined. (b) The estimated band-gap size of the edge states as a function of hopping parameters $t$ in the tight-binding model. The result indicates there would be a gapless edge state when $t=t_c=10.8$. (c) The eigenfrequency spectrum of the sample when we change intersystem hopping $t$. The red points are corner states and the blue points are edge states. (d) The corner states delocalize when the gap of the edge states disappears at certain intersystem hopping $t=t_c=10.8$.

Figure 5

The projected band structure from the tight-binding calculations [(a) and (b)] and full wave simulations [(c)–(h)]. Only the system with boundary type III exhibits an edge state predicted by the tight-binding model.

Figure 6

Gapless edge state by tuning the width of the coupling waveguide. (a) The edge-state dispersion turns out to be gapless when we take $d=d_c=8.18\mathrm{cm}$ compared to the results in (b) $d=7\mathrm{cm}$ and (c)$d=9\mathrm{cm}$. (d) The gapless behavior is also captured by measuring the amplitude of acoustic pressure in the output channel.

Figure 7

The wave propagates along the interface between topological trivial system A and nontrivial system B. (a) The sample takes a boundary along the $x$ and $y$ directions. (b) The sample takes a boundary along $x$ and the $x=y$ direction. The results show the wave propagation along the interface.