Topological origin of antichiral edge states induced by a nonchiral phonon

In contrast to the chiral edge modes, the antichiral edge modes propagate in the same direction along two parallel boundaries, representing a new state that provides more possibility for the dissipationless transportation of information. In a recent work by Medina Due\~nas et al. [Phys. Rev. Lett. 128, 066801 (2022)], a chiral phonon was shown to induce antichiral edge states on the zigzag edges of a honeycomb lattice through electron-phonon interaction (EPI). In this work we demonstrate that the EPI can promise antichiral edge modes even without the chirality of the phonon at K/K^'. To illustrate this, we consider a model of EPI with a nonchiral phonon and show that antichiral edge states appear independent of the phonon chirality. The antichiral edge modes carry quantized Zak phases if the model further respects the combination of time reversal (T) and inversion (I), i.e., TI. Our work successfully extends the EPI-induced antichiral edge modes to the nonchiral phonon case, significantly increasing the experimental feasibility on realizing antichiral edge states in realistic materials.