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Exciton condensation indicating the spontaneous formation of electron-holepair can cause the phase transition from a semimetal to an excitonic insulatorby gap opening at the Fermi surface. While the idea of this excitonic insulatorhas been proposed for decades, current theoretical approaches can only providequalitative descriptions, and a quantitative predicting tool is still missing.To shed insight on this problem, we developed an $ab~initio$ method based onthe finite-temperature density functional theory and many-body perturbationtheory to compute the exciton condensation critical behavior. Applying ourapproach to the monolayer $\rm TiSe_2$, we find a lattice distortionaccompanied by the formation of the excitonic gap via electron-phonon couplingwithout phonon softening, proving that the exciton condensation is the originof the charge-density-wave state observed in this compound. Overall, themethodology introduced in this work is general and paves the way to searchingfor candidate excitonic insulators in natural material systems.

Development of $ab ~initio$ method for exciton condensation and its application to $\bf TiSe_2$