  Exciton condensation indicating the spontaneous formation of electron-hole
pair can cause the phase transition from a semimetal to an excitonic insulator
by gap opening at the Fermi surface. While the idea of this excitonic insulator
has been proposed for decades, current theoretical approaches can only provide
qualitative descriptions, and a quantitative predicting tool is still missing.
To shed insight on this problem, we developed an $ab~initio$ method based on
the finite-temperature density functional theory and many-body perturbation
theory to compute the exciton condensation critical behavior. Applying our
approach to the monolayer $\rm TiSe_2$, we find a lattice distortion
accompanied by the formation of the excitonic gap via electron-phonon coupling
without phonon softening, proving that the exciton condensation is the origin
of the charge-density-wave state observed in this compound. Overall, the
methodology introduced in this work is general and paves the way to searching
for candidate excitonic insulators in natural material systems.
