  Spontaneous condensation of excitons is a long sought phenomenon analogous to
the condensation of Cooper pairs in a superconductor. It is expected to occur
in a semiconductor at thermodynamic equilibrium if the binding energy of the
excitons---electron (e) and hole (h) pairs interacting by Coulomb
force---overcomes the band gap, giving rise to a new phase: the 'excitonic
insulator' (EI). Transition metal dichalcogenides are excellent candidates for
the EI realization because of reduced Coulomb screening, and indeed a
structural phase transition was observed in few-layer systems. However,
previous work could not disentangle to which extent the origin of the
transition was in the formation of bound excitons or in the softening of a
phonon. Here we focus on bulk MoS2 and demonstrate theoretically that at high
pressure it is prone to the condensation of genuine excitons of finite
momentum, whereas the phonon dispersion remains regular. Starting from
first-principles many-body perturbation theory, we also predict that the
self-consistent electronic charge density of the EI sustains an out-of-plane
permanent electric dipole moment with an antiferroelectric texture in the layer
plane: At the onset of the EI phase, those optical phonons that share the
exciton momentum provide a unique Raman fingerprint for the EI formation.
Finally, we identify such fingerprint in a Raman feature that was previously
observed experimentally, thus providing direct spectroscopic confirmation of an
ideal excitonic insulator phase in bulk MoS2 above 30 GPa.
