  Excitonic insulators are insulating states formed by the coherent
condensation of electron and hole pairs into BCS-like states. Isotropic spatial
wave functions are commonly considered for excitonic condensates since the
attractive interaction among the electrons and the holes in semiconductors
usually leads to $s$-wave excitons. Here, we propose a new type of excitonic
insulator that exhibits order parameter with $p+ip$ symmetry and is
characterized by a chiral Chern number $C_\textrm{c}=1/2$. This state displays
the parity anomaly, which results in two novel topological properties:
fractionalized excitations with $\textrm{e}/2$ charge at defects and a
spontaneous in-plane magnetization. The topological insulator surface state is
a promising platform to realize the topological excitonic insulator. With the
spin-momentum locking, the interband optical pumping can renormalize the
surface electrons and drive the system towards the proposed $p+ip$ instability.
