  The influence of phonons on the formation of the excitonic insulator has
hardly been analyzed so far. Recent experiments on $\rm Ta_2NiSe_5$, 1$T$-$\rm
TiSe_2$, and $\rm TmSe_{0.45}Te_{0.55}$, being candidates for realizing the
excitonic-insulator state, suggest, however, that the underlying lattice plays
a significant role. Employing the Kadanoff-Baym approach we address this issue
theoretically. We show that owing to the electron-phonon coupling a static
lattice distortion may arise at the excitonic instability. Most importantly
such a distortion will destroy the acoustic phase mode being present if the
electron-hole pairing and condensation is exclusively driven by the Coulomb
interaction. The absence of off-diagonal long-range order, when lattice degrees
of freedom are involved, challenges that excitons in these materials form a
superfluid condensate of Bose particles or Cooper pairs composed of electrons
and holes.
