Kitaev formula for periodic, quasicrystal, and fractal Floquet topological insulators

Topological insulators are attracting researchers to reveal related fundamental physics, whose fractal versions are expanding topological physics into the fractal regime. The fractal topological systems, however, show a challenge to the Kitaev formula, because it leads to a noninteger Chern number. The Chern number within the band gap oscillates more and more actively as the system size increases. We uncover that such a challenge should be attributed to the coupling, brought about by the lattice fractalization, between the internal and external edge states. To confirm this, examples of periodic, quasicrystal, and fractal Floquet topological insulators are compared with each other by analyzing the topological edge states and local Chern markers. The sum of the local Chern markers on all the edges of the fractal lattice presents a fractional power relation with the system size, reflecting the fractalization of the topological edge states. We reveal that once such a coupling is shut down, the Kitaev formula would once again result in integers.