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

We study the topological gap labeling of general three-dimensional quasicrystals and we find that every gap in the spectrum is characterized by a set of third Chern numbers.

Quasicrystals are non-periodic but long-range ordered crystals found in a wide variety of physical systems including metallic alloys, photonic systems and twisted two-dimensional materials. Despite the increasing importance of quasicrystalline systems, the theoretical description of their physical properties is limited by the lack of Bloch theorem. In periodic crystals, each energy gap is characterized by an integer, which is the number of Bloch bands below the gap. In contrast, it is supposed that quasicrystals do not have any quantum units to count the number of states below the gap.

In this work, 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. This result is analogous to the gap labeling of 1D quasicrystals using first Chern numbers and 2D quasicrystals using second Chern numbers .