Hybrid-Order Topological Insulators in a Phononic Crystal

Topological phases, including the conventional first-order and higher-order topological insulators and semimetals, have emerged as a thriving topic in the fields of condensed-matter physics and materials science. Usually, a topological insulator is characterized by a fixed order topological invariant and exhibits associated bulk-boundary correspondence. Here, we realize a new type of topological insulator in a bilayer phononic crystal, which hosts simultaneously the first-order and second-order topologies, referred to here as the hybrid-order topological insulator. The one-dimensional gapless helical edge states, and zero-dimensional corner states coexist in the same system. The new hybrid-order topological phase may produce novel applications in topological acoustic devices.

Figure 1

Acoustic hybrid-order topological insulator with the bulk band dispersions. (a) The model based on the bilayer breathing kagome lattice. The blue and yellow lines denote the different intralayer interactions, and the dashed arrows denote the interlayer couplings from the lower layer to the upper one. (b) The unit cell of the bilayer PC, corresponding to the unit cell marked by the green shaded area in (a). (c) The simulated bulk band dispersions of the PC along the high symmetry lines. Two complete band gaps labeled in purple and cyan colors exhibit the topological properties of the first order and the second order TIs, respectively. Inset: the first Brillouin zone. (d) The pressure field distributions of the eigenmodes at the $K$ point below the two band gaps, marked as $S_1$ and $P_1$. They are derived from the monopole and dipole modes of the cylinder, respectively.

Figure 2

Acoustic helical edge states in the monopole band gap. (a) Photo of the parallelogram-shaped PC. Red and green boxes denote the enlarged views of the whole-cell and half-cell boundaries (marked by the dashed black lines). (b)–(c) The edge state dispersions on the whole-cell (b) and half-cell (c) boundaries. Color maps represent the measured results, the solid and dashed white lines denote the simulated ones with the opposite spin polarizations. The projected bulk bands are shaded in gray.

Figure 3

Acoustic corner states in the dipole band gap. (a) The lattice model (left panel) and PC sample (right panel) of a triangle-shaped structure. (b) The simulated eigenfrequencies of the triangle-shaped PC. Black, blue, and red circles represent the bulk, edge, and corner states, respectively. (c) The measured pressure field distributions of the bulk (6.60 and 7.45 kHz), edge (6.96 kHz), and corner (7.22 kHz) states. (d) The measured corner (top panel), bulk and edge (bottom panel) response spectra, normalized by their maximum values.

Figure 4

A pair of acoustic corner states localized at one corner in the dipole band gap. (a)–(b) The simulated eigenmodes at one corner with in-phase and antiphase distributions, respectively. (c)–(d) The measured acoustic field phases at the height $H$ in the cylinders for the upper and lower corners along the $z$ direction, which are consistent with (a) and (b). (e) The measured response spectra of the upper corners with respect to the in-phase and antiphase distributions.