  We employ mean-field approximation to investigate the interplay between the
nontrivial band topology and the formation of excitonic insulator (EI) in a
one-dimensional chain of atomic $s-p$ orbitals in the presence of repulsive
inter-orbital Coulomb interaction. We find that our model, in a non-interacting
regime, admits topological and trivial insulator phases, whereas, in strong
Coulomb interaction limit, the chiral symmetry is broken and the system
undergoes a topological-excitonic insulator phase transition. The latter phase
transition stems from an orbital pseudomagnetization and band inversion around
$k=0$. Our findings show that contrary to the topological insulator phase,
electron-hole bound states do not form exciton condensate in the trivial band
insulator phase due to lack of band inversion. Interestingly, the EI phase in
low $s-p$ hybridization limit hosts a Bardeen-Cooper-Schrieffer
(BCS)/Bose-Einstein condensation (BEC) crossover. Irradiated by a pump pulse,
our findings reveal that the oscillations of exciton states strongly depend on
the frequency of the laser pulse. We further explore the signatures of dynamics
of the exciton condensate in optical measurements.
