Higher-order topological sound transport in synthetic spaces

We design a four-dimensional system with straight waveguides composed of coordinate-dependent coupled waveguides. By controlling the phase in synthetic space and the excitation of the corresponding sound source mode, we show that the pumping incorporating adiabatic modulation of discretely varied coupling waveguides along a given orbit in the phason space occurs only in specific space directions. In addition, robust corner-to-corner topological pumping is achieved in finite-sized two-dimensional modulated acoustic structures. Our results provide a general strategy to achieve topological pumping through discrete property modulations and pave new paths for studying higher-dimensional topological phases in discrete systems.