Second-order and real Chern topological insulator in twisted bilayer <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> -graphyne

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