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

We study the topological gap labeling of general 3D quasicrystals and we find that every gap in the spectrum is characterized by a set of the third Chern numbers. We show that a quasi-periodic structure has multiple Brillouin zones defined by redundant wavevectors, 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 quasi-periodic potential and a topological nonlinear electromagnetic response in 6D insulators.