# Topological Zero-Energy Modes in Gapless Commensurate Aubry-André-Harper Models

The Aubry-André or Harper (AAH) model has been the subject of extensive theoretical research in the context of quantum localization. Recently, it was shown that one-dimensional quasicrystals described by the incommensurate AAH model has a nontrivial topology. In this Letter, we show that the commensurate off-diagonal AAH model is topologically nontrivial in the gapless regime and supports zero-energy edge modes. Unlike the incommensurate case, the nontrivial topology in the off-diagonal AAH model is attributed to the topological properties of the one-dimensional Majorana chain. We discuss the feasibility of experimental observability of our predicted topological phase in the commensurate AAH model.

1. Ganeshan, S., Sun, K. & Sarma, Das, S. Topological Zero-Energy Modes in Gapless Commensurate Aubry-André-Harper Models. *Phys. Rev. Lett.* **110,** 180403 (2013).