# Subwavelength Chiral Spiral Acoustic Metamaterials for a Robust Topological Acoustic Insulator
## 3 Results
### Topological Edge States  
In order to verify the existence of topological edge states, a ribbon-shaped superlattice in 14a \(\times\) a is constructed, as shown in Figure 7a. The superlattice is composed of seven left-handed spiral units and seven right-handed spiral units, and an interface between them is marked by red lines. The right-handed spiral structure is located above the interface, and the left-handed spiral structure is situated below the interface. In calculation, the periodic boundary condition is imposed along the left and right boundary of the superlattice. The projected energy band of the superlattice in \(k_{x}\) direction is shown in Figure 7b. The grey and red curve is the energy diagram representing the bulk state and the edge state, respectively. The edge states only exist in the second and third band gap, which don't own the subwavelength property. Figure 7c shows the pressure and intensity field on the interface for the edge mode located in the second band gap when \(k_{x}\) = 0.6 \(\pi\)/a. We can find that the sound pressure field is mainly confined at the interface and decays exponentially towards the bulk lattice. From the direction of energy flow, it can be derived that the energy flow propagates to the right along the interface, which is in the same direction as the wave vector.  
In addition, the existence of topological edge states in the \(k_{y}\) direction is also verified by a ribbon-shaped superlattice in 14a \(\times\) a, shown in Figure 8a. The projected energy band of the superlattice in \(k_{y}\) direction is shown in Figure 8b. The grey and red curve in the energy diagram represent the bulk state and the edge state, respectively. Figure 8c shows the pressure and intensity field on the interface for the edge mode located in the second band gap when \(k_{y}\) = \(\pi\)/(\(\surd{3a}\)). We can also find that the edge states exist in the \(k_{y}\) direction.  
According to the above analysis, the interfaces composed of spiral units sharing different chirality support the edge state. In order to show the transportation of the edge state, a straight waveguide in 18a \(\times\) 20a lattice composed of left-handed and right-handed spiral units is constructed, as shown in Figure 9a. Sound waves are emitted from the left side. It can be seen from Figure 9a that the sound waves propagate to the right side along the straight interface at 3130 Hz within the second band gap and decays exponentially towards the bulk of the metamaterial. Next, to test the robustness of the edge states, defects are introduced into the interface. In Figure 9b, a sharp corner is introduced in the zigzag interface. It also shows that the edge states can propagate stably along the zigzag interface, which confirms the robustness of the edge states protected by the chirality.  
Figure 8: **(a)** The ribbon-shaped superlattice in a\(\times\)14a composed of left-handed and right-handed spiral units. **(b)** The projected band structure of the superlattice in the \(k_{y}\) direction, the edge state and bulk state are represented by the red curve and gray curve, respectively. **(c)** The pressure and intensity field on the interface for the edge states of the second band gap with \(k_{y}=\pi/(\surd{3a})\). The green arrow represents the direction of the energy flow.  
Figure 7: **(a)** The ribbon-shaped superlattice in 14a \(\times\) a composed of left-handed and right-handed spiral units. **(b)** The projected band structure of the superlattice in the \(k_{x}\) direction; the edge state and bulk state are represented by the red curve and gray curve, respectively. **(c)** The pressure and intensity field on the interface for the edge states of the second band gap with \(k_{x}=0.6\pi/a\). The purple arrow represents the direction of the energy flow.