Graphyne as a second-order and real Chern topological insulator in two dimensions

Higher-order topological phases and real are two emerging topics in states of matter, which have been attracting considerable research interest. However, it remains a challenge to find realistic materials that can realize these exotic phases. Here, based on first-principles calculations theoretical analysis, we identify graphyne, the representative graphyne-family carbon allotropes, as two-dimensional (2D) second-order insulator Chern insulator. We show graphyne has direct bulk band gap at three $M$ points, forming valleys. The bands feature double inversion, is characterized by nontrivial number enabled spacetime-inversion symmetry. explicitly evaluated both Wilson-loop method parity approach, dictates existence Dirac type edge corner states. Furthermore, phase transition from trivial mediated 2D Weyl semimetal phase. robustness against symmetry breaking possible experimental detection methods discussed.