Revealing the Boundary Weyl Physics of the Four-Dimensional Hall Effect via Phason Engineering in Metamaterials

Quantum Hall physics has been theoretically predicted in four dimensions and higher. In hypothetical 2n dimensions, the topological characters of both the bulk and the boundary are manifested as quantized nonlinear transport coefficients that respectively connect to the Chern number of the bulk gap projection and to the winding number of the Weyl spectral singularities on the ()-dimensional boundaries. Here, we introduce the concept of phason engineering in metamaterials and use it as a vehicle to access and apply the quantum Hall physics in arbitrary dimensions. Using these specialized design principles, we fabricate a reconfigurable two-dimensional aperiodic acoustic crystal with a phason living on a 2-torus, giving us access to the four-dimensional quantum Hall physics. Also, we supply a direct experimental confirmation that the topological boundary spectrum assembles in a Weyl singularity when mapped as a function of the quasimomenta. We also demonstrate topological wave steering enabled by the Weyl physics of the three-dimensional boundaries.