Figure 5: Topologically protected transmission from a resonator.

(a) Strong backscattering of a topologically trivial edge state from a coupled resonator. Complete reflection can be seen in both the displacement fields (top panel) and the transmission spectrum between the input and the output (bottom panel), when the input is tuned to the resonance (grey arrow) of a side-coupled cavity (red area in the middle panel). The edge state (dispersion relation in Supplementary Figs 5 and 6) travels along a line defect formed by half-filled air holes (middle panel). Red areas indicate the shape of the defects, which are also filled with the phononic metamaterial (the same as grey area). The bulk of the crystal is similar to that given in Fig. 1. Arrows and crosses illustrate the direction of the on-resonance power flow. (b) Lack of backscattering for a phononic helical edge state, evident in the displacement fields (top panel) and the transmission spectrum (bottom panel), even at the resonance frequency of a side-coupled cavity (blue L-shaped regions in the middle panel). Both the edge state and the resonant mode are guided by domain walls between two crystals with opposite effective mass (grey and blue areas in the middle panel). The direction of the on-resonance power flow is shown in arrows. The presence of the resonator only manifests in the phase spectrum (bottom panel).