Topological Phononic Fiber of Second Spin-Chern Number.

The discovery of quantum spin Hall effect characterized by the first spin-Chern numbers in 2D systems has significantly advanced topological materials. To explore its 4D counterpart is of fundamental importance, but so far remains elusive in experiments. Here, we realize a topological phononic fiber protected by the second spin-Chern number in a 4D manifold, using a 3D geometric structure combined with a 1D rotational parameter space. We experimentally observe spin-momentum-locked core states traveling along a vortex line where the Dirac mass varies continuously. A novel higher-order face-centered bound state is further demonstrated. These findings underscore the interplay between higher-dimensional topological physics and defects, opening up a topological path for fibers.