Engineering negative coupling and corner modes in a three-dimensional acoustic topological network

Higher-order topological insulators have aroused massive attention due to their fancy topological properties. Artificial metamaterials, with their exquisite fabrications and measurements, offer an ideal approach to investigate topological properties in classical systems. Therein lower-dimensional corner states obtained in the three-dimensional (3D) Su-Schrieffer-Heeger (SSH) model and the octupole insulator have been extensively explored in acoustic systems. However, the existing acoustic negative couplings in the quantized multipole topological phases are mostly formed by shifting the connecting waveguides between every two resonators, which support a restricted dipole resonance mode. Here we establish subwavelength acoustic third-order topological insulators based on the theoretical frame of acoustic transmission lines and provide an innovative way for freely engineering acoustic negative couplings that work in arbitrary resonance modes. Furthermore, topological invariant calculations and numerical simulations in analogous acoustic networks validate the occurrence of topological corner states in both the 3D SSH model and octupole insulator. We foresee that the proposed methodology will broaden future avenues for studying atypical topological quantum effects with negative couplings in the classical macroscale context.

Figure 1

Modeling transmission-line networks for the 3D SSH and octupole insulator. (a) Tight-binding model of the building block of 3D SSH and octupole insulator. Middle insets: each unit consists of eight sites with different polarizations. Only positive coupling exists between the eight sites in the 3D SSH model, with zero gauge flux in each unit. In contrast, after introducing four pairs of negative couplings, the system turns into an octupole insulator with a flux of $\pi$ on each surface of the unit. (b), (c) Corresponding equivalent circuit models and acoustic network implementations. The nearest-neighboring coupling parameters $t_1$ and $t_2$ are determined by the ratio of the inductance $L_1/L_2$ (or capacitance $C_1/C_2$), which depends on the geometric tube length ratio $l_1/l_2$ (or cavity height ratio $h_1/h_2$). The width of acoustic tubes and cavities are identical. Insets: the grounding cavities $C_g$ on each site take the value $C_g^0=3(t_1+t_2)C_1, C_g^1=5(t_1+t_2)C_1, C_g^2=C_g^0$, and $C_g^3=7(t_1+t_2)C_1$.

Figure 2

Dispersion relation of a periodic 3D SSH unit. (a) Phase transition occurs when $t_1=t_2$. (b) Band gap opens within the concerned frequency range in trivial $(t_1>t_2)$ and nontrivial cases $(t_1<t_2)$. (c) Corresponding eigenstates with odd and even parities at high symmetry point $X$.

Figure 3

(a) Dispersion relation of the semi-infinite 3D SSH model structure. The coupling strengths take the value $t_2/t_1=8$ along each direction. (b) Eigenspectra at surface, hinge, and corner sites calculated from the eigenstates consisting of the finite-sized 3D SSH structure with $4\times 4\times 4$ units. Corner sound pressure peaks appear within the band gap that opens around the resonant frequency. (c)–(e) Pressure field distributions of (c) corner, (d) surface, and (e) hinge states.

Figure 4

Dispersion relation of octupole insulator. (a) Phase transition occurs at $t_1=t_2$. (b) Band gap opens within the concerned frequency range in trivial and nontrivial cases $t_1>t_2$ and $t_1<t_2$. (c) Corresponding eigenstates with odd and even parities at high symmetry point $X$.

Figure 5

(a) Dispersion relation of the semi-infinite octupole insulator. The coupling strength takes the value $t_2/t_1=8$ along each direction. (b) Eigenspectra at surface, hinge, and corner sites calculated from the eigenstates for the finite-sized 3D octupole insulator consisting of $4\times 4\times 4$ units. Corner sound pressure peaks appear within the band gap that opens around the resonant frequency. (c)–(e) Pressure field distributions of (c) corner, (d) surface, and (e) hinge states.

Figure 6

(a), (c) Eigenspectra of the finite-sized 3D SSH and octupole insulator model with finite size in $x, y$, and $z$ directions. Each side consists of 4.5 units. (b), (d) The only corner states localized at the base point of the cubic structure, the upper-left corner.