Synchronized in-gap edge states and robust copropagation in topological insulators without magnetic flux.

Copropagation of antichiral edge states in the metallic phase requires the bulk states as counterpropagating modes. Without the band gap protection, the copropagation along the boundaries is easily scattered into the bulk and the counterpropagation in the bulk is not robust against disorder and defects. Here, we propose a novel time-reversal symmetric topological insulator holding the synchronized in-gap edge states. To prevent the participation of bulk states, we introduce the time-reversal symmetry to detach the edge states from the bulk band, then the robust copropagation is realized through widening the band gap across a metal-insulator transition with anisotropic next-nearest-neighbor couplings. The time-reversal symmetry ensures the in-gap edge states with opposite momenta as the counterpropagating modes. The inversion symmetry protects the quantized polarization as a topological invariant. The synchronized in-gap edge states not only enrich the family of topological edge states, but also provide additional flexibility in the design of reconfigurable topological optical devices. Our findings open a new avenue for the tailored robust wave transport using the in-gap edge states for future acousto-optic topological metamaterials.