Dirac Hierarchy in Acoustic Topological Insulators

Dirac cones are essential features of the electronic band structure of materials like graphene and topological insulators (TIs). Lately, this avenue has found a growing interest in classical wave physics by using engineered artificial lattices. Here, we demonstrate an acoustic 3D honeycomb lattice that features a Dirac hierarchy comprising an eightfold bulk Dirac cone, a 2D fourfold surface state Dirac cone, and a 1D twofold hinge state Dirac cone. The lifting of the Dirac degeneracy in each hierarchy authorizes the 3D lattice to appear as a first-order TI with 2D topological surface states, a second-order TI exhibiting 1D hinge states, and a third-order TI of 0D midgap corner states. Analytically we discuss the topological origin of the surface, hinge, and corner states, which are all characterized by out-of-plane and in-plane winding numbers. Our study offers new routes to control sound and vibration for acoustic steering and guiding, on-chip ultrasonic energy concentration, and filtering to name a few.

Figure 1

Acoustic Dirac hierarchy. (a) 3D honeycomb lattice made of multiple stacked Kekulé monolayers. An eightfold Dirac cone (DC) appears at energy $ϵ=0$ at the $Z$ point of the Brillouin zone (BZ). (b) The eightfold degeneracy becomes a first-order TI after splitting into two fourfold DCs when the interlayer coupling $\gamma <{\gamma }^{\prime }$. (c) A second-order TI with a pair of helical hinge states (magenta and cyan lines) is unwrapped when SOC is induced via appropriate intralayer coupling, i.e., $\alpha >\beta$. (d) Last, the 3D lattice transitions into a third-order TI with a pair of midgap corner states (magenta dots) through mirror breaking by adding vertical coupling $-\delta$ to the bonds along the $y$ axis, and $+\delta /2$ to the remaining ones.

Figure 2

Bands and eigenfields of the degenerate and split DCs. (a) The eightfold degeneracy at the $Z$ point comprises overlapping symmetric ($S$) and asymmetric ($A$) DDCs. The colored lines represent the six theoretically obtained chiral bands. The circles mark the numerical results. Inset: the eigenfields of the Dirac point with dipole $\mathit{p}$ and quadrupole $\mathit{d}$ nature. (b) Splitting of the symmetric and asymmetry DDCs for interlayer coupling strengths, $\gamma <{\gamma }^{\prime }$. Insets: the eigenfields of the $S$- and $A$-type DDCs.

Figure 3

First-order TI. (a) The winding ${\mathcal{W}}_z$ vs $\gamma /{\gamma }^{\prime }$. The trajectories of $\mathit{g}(k)$ with different topologies for $\gamma /{\gamma }^{\prime }<1$ ($\gamma /{\gamma }^{\prime }>1$) display the presence (absence) of the winding around the origin. (b) Topological surface states appear in the $xy$ plane when $\gamma /{\gamma }^{\prime }<1$, (c) while they cease to exist when $\gamma /{\gamma }^{\prime }>1$.

Figure 4

Second-order TI. (a) Surface inversion diagram of the $\mathit{p}$ and $\mathit{d}$ states with a $Z_2$ topological phase transition at $\alpha =\beta$. (b) Within the gap of the projected surface states (light blue) a pair of helical hinge states (cyan and magenta, $S_{\pm }$) appears along the MZ hinges. Circles correspond to the computed results. The insets show the unidirectional propagation at 903 Hz for the clockwise $S_{+}$ and counterclockwise $S_{-}$ traversing hinge states. (c) Distributions of ${\theta }_l$ in the 2D BZ. The red dashed box marks the integral area for the MZ hinge along $q_{\parallel }$.

Figure 5

Third-order TI. (a) Computed band diagram showing the gapped surface and hinge bands (circles represent numerical simulations) and (b) the corresponding eigenstates for a finite 3D lattice at their various localizations. (c) The eigenfields of the mirror-protected corner states seen in (b).