Topological Creation of Acoustic Pseudospin Multipoles in a Flow-Free Symmetry-Broken Metamaterial Lattice

The discovery of topological acoustics has revolutionized fundamental concepts of sound propagation, giving rise to strikingly unconventional acoustic edge modes immune to scattering. Because of the spinless nature of sound, the “spinlike” degree of freedom crucial to topological states in acoustic systems is commonly realized with circulating background flow or preset coupled resonator ring waveguides, which drastically increases the engineering complexity. Here we realize the acoustic pseudospin multipolar states in a simple flow-free symmetry-broken metamaterial lattice, where the clockwise (anticlockwise) sound propagation within each metamolecule emulates pseudospin down (pseudospin up). We demonstrate that tuning the strength of intermolecular coupling by simply contracting or expanding the metamolecule can induce the band inversion effect between the pseudospin dipole and quadrupole, which leads to a topological phase transition. Topologically protected edge states and reconfigurable topological one-way transmission for sound are further demonstrated. These results provide diverse routes to construct novel acoustic topological insulators with versatile applications.

Figure 1

(a) Schematic of the triangular acoustic metamaterial lattice composed of artificial “metamolecules.” Inset: enlarged view of the metamolecule. (b) Dispersion relation of the lattice based on the original unit cell $C$. (Inset: single Dirac cones at the 1st BZ corners).

Figure 2

(a) The 1st BZs of the original unit cell $C$ (gray, marked as $Z$) and $C_s$ (blue, marked as $Z_s$). (${\mathbf{b}}_1$, ${\mathbf{b}}_2$) and (${\mathbf{b}}_{s1}$, ${\mathbf{b}}_{s2}$) are the corresponding reciprocal lattice vectors. (b) Folding mechanism and procedure from $Z$ onto $Z_s$. (c) Step-by-step constructing process and (d) the final dispersion relation of the lattice based on the unit cell $C_s$ (Inset: double Dirac cone at the 1st BZ center).

Figure 3

(a) Topological transition as the coupling strength is tuned from weak (left: $R/a=0.320$) to strong (right: $R/a=0.345$). Critical coupling appears at $R/a=1/3$ [Fig. 2] with double Dirac cone. (b) Topological modes inversion underlying the transition. $p_x$ and $p_y$ show pseudospin dipole modes, while $d_{xy}$ and $d_{x^2-y^2}$ show pseudospin quadrupole modes. Sound intensity $I$ associated with pseudospin-down and pseudospin-up modes at the ${\mathrm{\Gamma }}_s$ point below the band gap in the (c) trivial regime ($R/a=0.320$) and (d) nontrivial regime ($R/a=0.345$). The rainbow color represents the amplitude of acoustic pressure, and the arrows show the direction and amplitude of the intensity.

Figure 4

(a) Schematic of a ribbon-shaped supercell composed of 15 and 24 metamolecules for the topologically nontrivial and trivial regions respectively. (b) Topological edge modes shown by the thick curves. (c) Acoustic pressure fields and intensity of the pseudospin-dependent one-way edge modes localized at the interface of the supercell, corresponding to the points $A$ and $B$ indicated in (b).

Figure 5

Topologically protected one-way edge waveguide for airborne sound and the robustness against defects. The pseudospin-down mode is excited at the (a) lower and (b) upper edge. (c) Topological edge states with three kinds of defects: disorder, bends, and cavity. Bottom panels: the corresponding distributions of acoustic intensity fields.