  Quadrupole phase, as a novel high-order topological phase, exhibits
nontrivial gapless states at the boundaries whose dimension is lower than bulk
by two. However, this phase has not been observed experimentally in
two-dimensional (2D) materials up to now. In this work, using first-principles
calculations and tight-binding (TB) model, we propose that the experimentally
synthesized C2N is a 2D quadrupole topological insulator with one-dimensional
gapped edge states and zero-dimensional gapless corner states. C2N is found to
have a large bulk gap of 2.45 eV and an edge gap of 0.32 eV, making it an
excellent candidate to evidently present the nontrivial corner states in
experiments. The robustness of the corner states against the edge disorders has
been explicitly identified. Moreover, another three C2N-like materials are also
found to host the nontrivial quadrupole phase including an experimentally
synthesized material aza-fused microporous polymers (CMP). The four 2D
quadrupole topological phases proposed in our present work provide excellent
candidates for studying the novel high-order topological properties in future
experiments.
