Topological gap labeling with third Chern numbers in three-dimensional quasicrystals

We study the topological gap labeling of general three-dimensional (3D) quasicrystals and we find that every gap in the spectrum is characterized by a set of third Chern numbers. We show that a quasiperiodic structure has multiple Brillouin zones defined by redundant wave vectors, and the number of states below a gap is quantized as an integer linear combination of volumes of these Brillouin zones. The associated quantum numbers to characterize energy gaps can be expressed as third Chern numbers by considering a formal relationship between an adiabatic charge pumping under cyclic deformation of the quasiperiodic potential and a topological nonlinear electromagnetic response in 6D band insulators.