Acoustic Realization of a Four-Dimensional Higher-Order Chern Insulator and Boundary-Modes Engineering

We present a theoretical study and experimental realization of a system that is simultaneously a four-dimensional (4D) Chern insulator and a higher-order topological insulator. The system sustains the coexistence of (4-1)-dimensional chiral topological hypersurface modes (THMs) and (4-2)-dimensional chiral topological surface modes (TSMs). Our study reveals that the THMs are protected by second Chern numbers, and the TSMs are protected by a topological invariant composed of two first Chern numbers, each belonging to a Chern insulator existing in subdimensions. With the synthetic coordinates fixed, the THMs and TSMs, respectively, manifest as topological edge modes and topological corner modes (TCMs) in the real space, which are experimentally observed in a 2D acoustic lattice. These TCMs are not related to quantized polarizations, making them fundamentally distinctive from existing examples. We further show that our 4D topological system offers an effective way for the manipulation of the frequency, location, and number of TCMs, which is highly desirable for applications.