Subwavelength topological valley-spin states in the space-coiling acoustic metamaterials

Phononic crystals possess Dirac linear dispersion bands. In the vicinity of Dirac cones, phononic crystals exhibit topological properties which have good application prospects in control of acoustic waves. Up to now, the topological edge states of phononic crystals, based on the band structures arising from the Bragg scattering, cannot realize low-frequency sound waves by the topologically protected one-way edge transmission. In this paper, by introducing the space-coiling structure, a space-coiling phononic metamaterial with C3v symmetry is designed. At the K (K') points of the Brillouin zone, the bands linearly cross to a subwavelength Dirac degenerated cones. With a rotation of the acoustic metamaterials, the mirror symmetry will be broken and the Dirac degenerated cones will be reopened, leading to subwavelength topological phase transition and subwavelength topological valley-spin states. Lastly, along the topological interface between acoustic metamaterials with different topological valley-spin states, we successfully observe the phononic topologically valley-spin transmission. The subwavelength Dirac conical dispersion and the subwavelength topological valley-spin state breakthrough the limitation of the geometric dimension of the phononic topological insulator, and provide a theoretical basis for the application of the phononic topologically robust transmission in a subwavelength scale.