Acoustic Realization of Quadrupole Topological Insulators

A quadrupole topological insulator, being one higher-order topological insulator with nontrivial quadrupole quantization, has been intensely investigated very recently. However, the tight-binding model proposed for such emergent topological insulators demands both positive and negative hopping coefficients, which imposes an obstacle in practical realizations. Here, we introduce a feasible approach to design the sign of hopping in acoustics, and construct the first acoustic quadrupole topological insulator that stringently emulates the tight-binding model. The inherent hierarchy quadrupole topology has been experimentally confirmed by detecting the acoustic responses at the bulk, edge, and corner of the sample. Potential applications can be anticipated for the topologically robust in-gap states, such as acoustic sensing and energy trapping.

Figure 1

(a),(b) Sketches of the split energy levels for two coupled $P_z$ dipoles with hoppings $\kappa <0$ and $\kappa >0$, respectively. For the case of $\kappa <0$ ($\kappa >0$), the in-phase coupled mode has an energy lower (higher) than that of out-of-phase one. (c),(d) Acoustic realizations of the negative and positive hoppings with coupled resonators, respectively. Left: Double cavity structures coupled with different connectivity. The resonance cavities (yellow) and narrow tubes (red or blue) are filled with air and bounded with hard boundaries. Right: Pressure distributions at given eigenfrequencies.

Figure 2

(a),(b) Experimental setups for testing the hopping signs of the straight-linked and cross-linked double cavity systems, respectively. The acoustic signal is injected from the backside of the sample, and four typical positions (labeled with color circles) in the front are used for detection. (c),(d) Measured (circles) and simulated (lines) pressure amplitude responses at the four locations (plotted with consistent color). The green arrows indicate the eigenfrequencies of the coupled $P_z$ modes. (e),(f) The associated phase spectra.

Figure 3

(a) Square-lattice TB model with positive (red) and negative (blue) nearest-neighbor hoppings ${\lambda }_{1,2}$ and ${\gamma }_{1,2}$, where the shaded square sketches a four-site unit cell. This hoping distribution results in a $\pi$ flux per plaquette. (b) Unit structure of our acoustic system that emulates the TB model: yellow for resonant cavities, and blue and red for coupling tubes. (c) Band structure. The $P_z$ bands (black solid lines) are captured well by the TB model (green dashed lines). Inset: the first Brillouin zone of the square lattice. (d) Eigenvalue spectrum for a finite-sized sample constructed with the topologically distinct quadruple phases $\mathbf{A}$ and $\mathbf{B}$ (inset), identified with bulk, edge, and corner states through inspecting eigenfields.

Figure 4

(a) Experimental setup for identifying the hierarchy quadrupole topology. Upper panel: experimental sample. Lower panel: an enlarged picture for demonstrating the pump-probe measurement of the corner response. The color stars and squares label, respectively, the positions of the source and detector when measuring the bulk (black), edge (purple), and corner (blue) responses. (b) Experimentally measured bulk (black), edge (purple), and corner (blue) spectra. (c) Numerical comparison of (b). (d)–(f) Measured and simulated pressure amplitude distributions for the bulk, edge, and corner states, respectively, plotted in logarithmic scale with bright (dark) for strong (weak) field. The frequencies correspond to the peaks labeled in (b) and (c). In each case, the white star indicates the position of the acoustic source.