  Quantum spin-hall insulator (QSHI) processes nontrivial topology. We notice
that the electronic structures of some particular QSHIs are favorable for
realization of excitonic insulators (EIs). Using first-principles many-body
perturbation theory ($GW$+BSE) and $k \cdot p$ model, we show that
high-temperature ($T$) topological EIs with unlike spin can exist in such QSHIs
with non-vanishing band gaps, e.g. 2D AsO and
$\text{Mo}_2\text{Ti}\text{C}_2\text{O}_2$. Spin-triplet type EI phase induced
by strong electron-hole interaction preserves time-reversal symmetry and the
topological characteristics. A novel optical selection rule exists, upon going
through the phase transition from the normal QSHIs to the topological EIs,
absorption spectroscopy shows pronounced $T$-dependent changes, providing
guidance for future experimental detections. The demonstrated coupling between
EIs and topology also means that rich physics exists in such materials which
retain such interdisciplinary features.
