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We address the question of the origin of the recently discovered chiralproperty of the charge-density-wave phase in 1$T$-TiSe$_2$ which so far lacks amicroscopic understanding. We argue that the lattice degrees of freedom seemsto be crucial for this novel phenomenon. We motivate a theoretical model thattakes into account one valence and three conduction bands, a strongly screenedCoulomb interaction between the electrons, as well as the coupling of theelectrons to a transverse optical phonon mode. The Falicov-Kimball modelextended in this way possesses a charge-density-wave state at low temperatures,which is accompanied by a periodic lattice distortion. The charge ordering isdriven by a lattice deformation and electron-hole pairing (excitonic)instability in combination. We show that both electron-phonon interaction andphonon-phonon interaction must be taken into account at least up to quarticorder in the lattice displacement to achieve a stable chiral charge order. Thechiral property is exhibited in the ionic displacements. Furthermore, weprovide the ground-state phase diagram of the model and give an estimate of theelectron-electron and electron-phonon interaction constants for 1$T$-TiSe$_2$.

Chiral charge order in 1$T$-TiSe$_2$: importance of lattice degrees of freedom