Second-order and real Chern topological insulator in twisted bilayer α -graphyne

Abstract

The study of higher-order and real topological states as well as the material realization have become a research forefront of topological condensed matter physics in recent years. Twisted bilayer graphene (tbG) is proved to have higher-order and real topology. However, whether this conclusion can be extended to other two-dimensional twisted bilayer carbon materials and the mechanism behind it lack explorations. In this paper, we identify the twisted bilayer $\ensuremath{\alpha}$-graphyne (tbGPY) at large twisting angle as a real Chern insulator (also known as Stiefel-Whitney insulator) and a second-order topological insulator. Our first-principles calculations suggest that the tbGPY at $21.{78}^{\ensuremath{\circ}}$ is stable at 100 K with a larger bulk gap than the tbG. The nontrivial topological indicators, including the real Chern number and a fractional charge, and the localized in-gap corner states are demonstrated from first-principles and tight-binding calculations. Moreover, with ${\mathcal{C}}_{6z}$ symmetry, we prove the equivalence between the two indicators, and explain the existence of the corner states. To decipher the real and higher-order topology inherited from the moir\'e heterostructure, we construct an effective four-band tight-binding model capturing the topology and dispersion of the tbGPY at large twisting angle. A topological phase transition to a trivial insulator is demonstrated by breaking the ${\mathcal{C}}_{2y}$ symmetry of the effective model, which gives insights on the trivialization of the tbGPY as reducing the twisting angle to $9.{43}^{\ensuremath{\circ}}$ suggested by our first-principles calculations.