Antichiral edge states in a square lattice

Recent advances in topological phases with broken time-reversal symmetry unveil a novel gapless topological phase, i.e., antichiral edge state, featuring co-propagating along the parallel edges. However, to date, such antichiral edge states are only realized in the modified Haldane model, which are based on the honeycomb lattice. Here, we realize the antichiral edge states in a square-lattice tight-binding model with complex nearest-neighbor coupling and both positive and negative next-nearest-neighbor couplings. In contrast to previous proposals, the complex nearest-neighbor coupling breaks the time-reversal symmetry, and the negative next nearest-neighbor coupling shift two Dirac points in energy. We also propose a possible scheme to realize our model with the assistance of acoustic metamaterials. The existence of antichiral edge states is revealed through full-wave simulation of the band structure and acoustic fields excited by a point source.