Fractal topological band-gap structure induced by singularities in the one-dimensional Thue–Morse system

The physical origin of the fractal topological band-gap structure in the one-dimensional Thue–Morse system has been revealed,which is characterized by the evolutions of two types of topological singularities with zero-scattering properties and the paths of phase vortex points,which are the mirrored paths of the first-type singularities.The field distribution of the upper and lower gap-edge states will interchange when the traditional gaps are closed and reopened.The topologically protected edge-states are found at both traditional gaps and fractal gaps.Our work broadens the topological properties of quasicrystals or aperiodic systems and provides potential applications in new optoelectronic devices.