# Subwavelength Chiral Spiral Acoustic Metamaterials for a Robust Topological Acoustic Insulator
## 3 Results
### Band Structure
Figure 4 shows the energy band of the spiral structure with a different number of turns \(n\), where the \(n=1\), 1.5, and 2, respectively. It can be derived that the frequency of the first band gap decreases with the increase of the number of turns, which is owing to the length extension of the spiral structure. In this way, the frequency of the first resonant mode decreases. Moreover, with the increase of \(n\), the number of local resonant modes increases, which generates new energy bands and band gaps. When \(n=2\), there are two dispersion curves and two local resonant band gaps below 0.5 the normalized frequency. In the band diagram, there always exists tiny gaps around the reduced frequency 0.55 at K point, whatever the value of n. It reveals that the tiny gap is owed to the Bragg Scattering mechanism, which doesn't show the subwavelength property. Figure 3: The first, second and third band gap varies with geometric parameters marked by grey, blue, and orange areas, respectively. (**a**) Rotation angle \(\theta\) ranging from \(0^{\circ}\)to \(120^{\circ}\). (**b**) Width of spiral \(w\) ranging from 0.05 cm to 0.5 cm. Figure 2: (**a**) The band structure of the left-handed spiral structure with \(n=2\). (**b**) Eigenmodes at K point in Brillouin zone corresponding to four energy bands. For the spiral structure with \(n=2\), we explore the topological characteristics of the first, second and third band gaps by calculating the Berry curvature of the first, second, third and fourth energy bands in the first Brillouin zone. The topological properties of the band gap can be characterized by the topological properties of the upper and lower band of the band gap. We calculate the Berry curvature \(\Omega=\int\Gamma_{k}\times u(k)|\
abla_{k}|u(k)\mathrm{d}k^{2}\) through the numerical method [35]. The calculated Berry curvature of the first, second, third and fourth energy bands is shown in Figure 5. From the Berry curvature distribution, the first and second energy bands are topological trivial since the Berry curvature across the whole Brillouin zone is 0. The third and fourth energy bands are topological non-trivial since there exist extrema near the K, K\({}^{\prime}\) point in the Berry curvature distribution, which is protected by the valley topology. Therefore, the first band gap between the first and second energy bands is the topological trivial band gap, indicating no topological edge state. The second band gap between the second and third energy bands is a topological non-trivial local resonant band gap, which promises the existence of a topological edge state. The third band gap between the third and fourth energy bands is a topological non-trivial Bragg scattering band gap, which also holds the topological edge state. Figure 4: The energy band of the spiral structure with the number of turns \(n=1\), \(1.5\), and \(2\). Figure 5: Berry curvature over the first Brillouin zone for each energy band of the spiral structure. (**a**) The first band (**b**) The second band (**c**) The third band (**d**) The fourth band. To construct the topological edge state with different chiral spiral units, we have calculated the Valley Chern numbers for different chirality. The Valley Chern numbers of the third band are calculated by integrating the Berry curvature over a finite square around the K point. The side length of the finite square is \(\frac{\pi}{a},k_{x}\) ranges from \(\frac{5\pi}{60}\) to \(\frac{11\pi}{6a}\) and \(k_{y}\) ranges from \(-\frac{\pi}{2a}\) to \(\frac{\pi}{2a}\). Figure 6 shows the Valley Chern numbers of the left-handed and right-handed spiral, respectively. The Valley Chern number of a left-handed spiral is 0.2, shown in Figure 6a, and the right-handed spiral is \(-0.2\), shown in Figure 6b.