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