  In two dimensions, intrinsic second-order topological insulators (SOTIs) are
characterized by topological corner states that emerge at the intersections of
distinct edges with reversed mass signs, enforced by spatial symmetries. Here,
we present a comprehensive investigation within the class BDI to clarify the
symmetry conditions ensuring the presence of intrinsic SOTIs in two dimensions.
We reveal that the (anti-)commutation relationship between spatial symmetries
and chiral symmetry is a reliable indicator of intrinsic corner states. Through
first-principles calculations, we identify several ideal candidates within
carbon-based polymorphic graphyne structures for realizing intrinsic SOTIs
under sublattice approximation. Furthermore, we show that the corner states in
these materials persist even in the absence of sublattice approximation. Our
findings not only deepen the understanding of higher-order topological phases
but also open new pathways for realizing topological corner states that are
readily observable.
