Precisely engineered gigahertz surface acoustic wave (SAW) trapping enables diverse and controllable interconnections with various quantum systems, which are crucial to unlocking the full potential of phonons. The topological rainbow based on synthetic dimension presents a promising avenue for facile and precise localization of SAWs. In this study, we successfully developed a monolithic gigahertz SAW topological rainbow by utilizing a nanoscale translational deformation as a synthetic dimension. We observed a gapless topological boundary state, which originates from a 2π phase winding in the Zak phase during translation. These boundary states enable on-chip single-mode rainbowlike filters with an extensive range of adjustable operating frequencies. Furthermore, we construct nanoscale wedge-shaped grooves, realizing the Born-von-Karman interface. The interface generates topological rainbow resonators with high quality and small mode volume, which can trap topological phononic states of different frequencies into different positions. This study underscores the immense potential of topological acoustics in synthetic dimensions for microwave acoustics, providing a robust design framework for the precise manipulation of SAWs.
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