# Subwavelength Chiral Spiral Acoustic Metamaterials for a Robust Topological Acoustic Insulator
## 1 Introduction  
Inspired by the topological insulators [1; 2] in condensed matter physics, the analogous topological insulators in acoustic realm have received significant interest. The edge state characterized by the one-way transmission without backscattering lays a new pathway for acoustic wave guiding, which enables the acoustic wave to propagate robustly along the surface or edge being immune to the defects. By emulating the quantum Hall effect (QHE) [3; 4], quantum spin Hall effect (QSHE) [5; 6], and quantum Valley Hall effect (QVHE) [7; 8] in electronic systems, similar topological transmission is also realized in classical acoustic systems [9; 10; 11; 12; 13; 14; 15]. The researchers [16; 17; 18] introduced the circulating fluid into the ring resonator, mimicking the magnetic field in an electronic system, and realized the analogue QHE. Some topological acoustic systems [19; 20; 21] realize the analogue QSHE by adjusting the filling ratio within the hexagonal honeycomb lattice to realize inversion of the acoustic energy band at a double Dirac cone. In addition, researchers [22; 23; 24; 25; 26] also broke the inversion symmetry of the structure by rotating the scatterer, making the Dirac point at the K point in the Brillouin zone split, giving rise to different Valley hall phases, thus realizing the analogue QVHE. Although these topological acoustic systems have achieved a great milestone, most of them depend on the special point group symmetry. In this way, a Dirac cone forms at the high symmetry point in the Brillouin zone, and then the reduced symmetry of the unit cell contributes to the Dirac cone split through rotation, scaling and other operations, giving rise to different topological phases. Thus, the interfaces composed of different topological phases hold the one-way transmission edge states. Furthermore, a large share of theseschemes are based on the Bragg scattering mechanism, which requires the unit cell size to be comparable with the wavelength, so it is difficult to bring the benefit to the miniaturization of a topological acoustic system. Although some systems at deep-subwavelength scale have been reported based on local resonance [27, 28], the structures of this are relatively complicated. On the other hand, the topological transmission of these systems also depends on special point group symmetry, which exerts some restrictions on the unit cell design. In acoustic metamaterials, the spiral structure [29, 30, 31, 32] featuring space-coiling creates a lengthy sound channel within a small space resulting in low-frequency resonant mode, and the whole structure possesses subwavelength characteristics. Compared with the labyrinthine space-coiling structure, the spiral structure is much simpler, and can meet the requirement of the subwavelength topological system. In addition, the topological interfaces in the valley Hall system are formed by distinct valley phases with opposite energy flow chirality. Therefore, opposite chiral structural units can be used to artificially generate topological phases with opposite energy flow chirality, and the interface composed of opposite chiral structural units will hold the edge states. In view of this, particular point group symmetry is not necessary for the generation of edge states [33, 34]. In this work, we design a topological acoustic system based on a subwavelength chiral spiral acoustic metamaterial. Due to the space folding scheme, the lengthy sound channel contributes to the subwavelength characteristics. The band gap can be reduced to the subwavelength frequency range. Furthermore, the whole system is composed of a chiral spiral structure, which does not rely on special point group symmetry. Compared with other topological systems based on QVHE, in this paper the whole system depends on the chiral symmetry. In addition, the unit cell is designed based on the Archimedean spiral, which is simpler than the space-folding structure.