Prediction of at topological $p+ip$ excitonic insulator with parity anomaly

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.