Helical Higher-Order Topological States in an Acoustic Crystalline Insulator

The topological states in quantum Hall insulators and quantum spin Hall insulators that emerge helical are considered nondissipative. However, in crystalline systems without spin-orbit couplings, the existing higher-order topological states are considered not helical, and the energy suffers from dissipation during propagation. In this work, by introducing the intrinsic pseudospin degree of freedom, we theoretically and experimentally present the existence of the helical higher-order topological states in the $C_6$-symmetric topological crystalline insulators based on the acoustic samples. Crucially, rather than considering the global interaction of the large bulk, we further intuitively reveal the impacts of the geometries of the crystal on the generation mechanisms and natural behaviors of these states based on the simple equivalent models. These results provide a versatile way for guiding the design of the desired topological materials.

Figure 1

(a) Images of the experimental hexagonal acoustic TCI sample with armchair boundaries. The inserted green hexagonal line and the black dashed line indicate the lattice and the armchair ribbon, respectively. $w_0=10$, $w=17\mathrm{mm}$ and the lattice constant $a=31.12\mathrm{mm}$ in this sample. (b) Schematic of the hoppings connecting the virtual atoms in the sample. (c) Calculated energy spectrum of the Hamiltonian of the sample. (d)–(f) Energy field distributions of the topological corner states at (d) ${\mathrm{CS}}_p$, (e) zero-energy state, and (f) ${\mathrm{CS}}_d$.

Figure 2

(a) Energy band structure of the armchair ribbon of the acoustic TCI with helical gapped edge states. (b) Numerically calculated eigenfrequency spectrum of the sample in Fig. 1. (c) Measured intensity spectra of the sample in Fig. 1. (d) Sound pressure field distributions of the helical edge states of the armchair ribbon with $k_A=\pm 0.46\times 2\pi /a_A$. (e) Sound pressure field distributions of the helical corner state at 10.3 kHz in the present sample. Insets: The inserted clockwise and anticlockwise arrows represent the spin-down and spin-up pseudospins, respectively. The “$+/-$” signs represent the opposite sound phase on the corner lattice.

Figure 3

(a)–(c) Schematics of the equivalent models with (a) one, (b) two, and (c) three interlattice hoppings in the corner lattices, respectively. (d)–(f) The corresponding energy spectra of the three models. Insets: The $+/-$ here represents the topological charge on the corners. The corresponding energy field distributions of these modes are shown in Fig. S8 of the Supplemental Material [56].

Figure 4

(a) Schematic of the rectangular TCI with mixed boundaries. (b) Numerically calculated eigenfrequencies of (a). (c),(d) Simulated pressure field distributions of (c) the ordinary resonant mode and (d) the helical topological corner state, respectively.