Investigation of the step edge states in higher-order topological insulators

The hinge and edge states of topological materials have garnered significant attention due to their extraordinary capacity to transmit current against energy loss. In particular, the step edge states (SESs) offer an intriguing avenue for robust and tunable sensing as well as information processing, greatly surpassing existing materials. However, these SESs have not been explored in classical wave systems. Here, we delve into the two-dimensional (2D) system, identifying the resulting step states (SSs) and further elucidating their counterparts as SESs in three-dimensional (3D) acoustic systems. We illustrate two zero-dimensional (0D) SSs within a 2D quadrupole topological insulator, revealing a tendency toward a second-order topology phase as the number of stepped layers increase. The tunable nature of these SSs enables us to manipulate wave transmission by adjusting the geometry of the step boundaries. Furthermore, by simply stacking the 2D models, we successfully achieve the 1D SESs in 3D second-order topological insulators. Our research presents a new pathway for generating 0D localized modes and 1D transmission channels in higher-order topological insulators. The stable SSs and SESs hold potential applications, particularly in wave sensing and trapping.