Recently, the study of topological phase transitions and edge states for acoustic wave systems has become a research hotspot. However, most current studies on topological edge states are based on Bragg scattering, which is not practical to apply in situations involving low-frequency sound because of the large structural dimensions. Therefore, we construct in this study a graphene-like structure based on a sub-wavelength resonant unit Helmholtz resonator and adjust the acoustic capacitance diameter of adjacent units to change the local resonance frequency, and thereby impose the degeneracy of the Dirac cone and topological spin states, which is characterized by valley Chern numbers of opposite sign. We also check topological valley edge states at zigzag and armchair interfaces and find that gapless topological valley edge states only appear at zigzag interfaces, whereas armchair interfaces host gap edge states. Moreover, the results show that the transmission properties of edge states in a zigzag rectangular waveguide are immune to backscattering and defects.
