Observation of Dirac Hierarchy in Three-Dimensional Acoustic Topological Insulators

Dirac cones (DCs) play a pivotal role in various unique phenomena ranging from massless electrons in graphene to robust surface states in topological insulators (TIs). Recent studies have theoretically revealed a full Dirac hierarchy comprising an eightfold bulk DC, a fourfold surface DC, and a twofold hinge DC, associated with a hierarchy of topological phases including first-order to third-order three-dimensional (3D) topological insulators, using the same 3D base lattice. Here, we report the first experimental observation of the Dirac hierarchy in 3D acoustic TIs. Using acoustic measurements, we unambiguously reveal that lifting of multifold DCs in each hierarchy can induce two-dimensional topological surface states with a fourfold DC in a first-order 3D TI, one-dimensional topological hinge states with a twofold DC in a second-order 3D TI, and zero-dimensional topological corner states in a third-order 3D TI. Our Letter not only expands the fundamental research scope of Dirac physics, but also opens up a new route for multidimensional robust wave manipulation.

Figure 1

Dirac hierarchy and topological phase hierarchy. The Dirac hierarchy comprises (a) an eightfold-degenerate bulk DC to (b) a fourfold-degenerate 2D surface DC in a first-order 3D TI, followed by (c) a twofold-degenerate 1D hinge DC in a second-order 3D TI and finally (d) a pair of 0D corner states in a third-order 3D TI. At each hierarchy, the degenerate DC in the higher dimension is broken by appropriate symmetry breaking, resulting in a complete band gap that hosts a DC dispersion in a lower dimension. The topological phase hierarchy comprises (e) a gapless phase, (f) a first-order TI, (g) a second-order TI, and (h) a third-order TI.

Figure 2

Design and band diagram of the first-order 3D acoustic TI. (a) Photograph of the sample of the first-order 3D acoustic TI. The inset shows the top view of the sample. (b) Design of the unit cell of the acoustic crystal. The detailed geometrical parameters are provided in the Supplemental Material [55]. (c) Bulk band structure corresponding to equal out-of-plane and in-plane coupling strengths ($r_s=r_t$, $r_w=r_v$), hosting an eightfold-degenerate bulk DC at high-symmetry point $Z$. (d) Bulk band structure for unequal out-of-plane coupling strengths and equal in-plane coupling strengths ($r_s<r_t$, $r_w=r_v$), in which case the eightfold-degenerate bulk DC is split into two fourfold-degenerate surface DCs with a complete 3D band gap indicated by gray region.

Figure 3

Experimental observation of the fourfold surface DC in the first-order 3D acoustic TI. (a) Measured transmission spectra for the bulk (red region) and surface (blue region) states, respectively. (b) Measured and simulated acoustic field distributions of the topological surface states excited by a point source placed at the center of the top surface. (c) Measured (background color) and simulated (cyan hollow circles and transparent gray dots) band diagrams of the topological surface states. The inset shows the projective Brillouin zone on the top surface. (d) Measured (background color) and simulated (white circles) isofrequency contours of the topological surface states at different frequencies. White dashed lines are guides to the eyes that indicate the shape of the cones intersecting at the surface Dirac point, and the double white solid circles in each isofrequency contour indicate the fourfold degeneracy of the surface DC. The color scale measures the acoustic energy density.

Figure 4

Experimental observation of the twofold hinge DC in the second-order 3D acoustic TI. (a) Photograph of the sample of the second-order 3D acoustic TI. Left inset illustrates the in-plane Kekulé distortion with unequal intracell and intercell couplings. Right inset shows the molecule-zigzag hinge. (b) Measured (background color) and simulated (cyan hollow circles) band diagram along the molecule-zigzag hinge direction ($x$ axis), which clearly exhibits a twofold-degenerate gapless hinge Dirac point. (c) Measured transmission spectra of the surface (blue region) and hinge (green region) states. (d) Measured and simulated acoustic pressure distribution of the hinge states excited by a point source (cyan star) placed at the middle of the hinge.

Figure 5

Experimental observation of 0D corner states in the third-order 3D acoustic TI. (a) Sample of the third-order acoustic TI. Inset illustrates the $M_y$ mirror symmetry breaking. (b) Measured transmission spectra for the surface (blue region), hinge (green region), and corner (yellow region) states, respectively. (c) Calculated eigenfrequency spectra of the fabricated acoustic crystal. (d) Measured and simulated acoustic pressure distributions at the corner-mode frequency (i.e., 4180 Hz) under a point source excitation placed near the sample corner.