  An excitonic insulator (EI) is an unconventional quantum phase of matter in
which excitons, bound pairs of electrons and holes, undergo Bose--Einstein
condensation, forming a macroscopic coherent state. While its existence was
first hypothesized half a century ago, the EI has eluded experimental
observation in bulk materials for decades. In the last few years, a resurgence
of interest in the subject has been driven by the identification of several
candidate materials suspected to support an excitonic condensate. However, one
obstacle in verifying the nature of these systems has been to find signatures
of the EI that distinguish it from a normal insulator. To address this, we
focus on a clear qualitative difference between the two phases: the existence
of Goldstone modes born by the spontaneous breaking of a $U(1)$ symmetry in the
EI. Even if this mode is gapped, as occurs in the case of an approximate
symmetry, this branch of collective modes remains a fundamental feature of the
low-energy dynamics of the EI provided the symmetry-breaking is small. We study
a simple model that realizes an excitonic condensate, and use mean field theory
within the random-phase approximation to determine its collective modes. We
subsequently develop a diagrammatic method to incorporate the effects of
disorder perturbatively, and use it to determine the scattering rate of the
collective modes. We interpret our results within an an effective field theory.
The collective modes are found to be robust against symmetry-preserving
disorder, implying an experimental fingerprint unique to the EI: the ballistic
propagation of low-lying modes over mesoscopic distances, at electronic-scale
velocities. We suggest this could affect thermal transport at low temperatures,
and could be observed via spatially-resolved pump-probe spectroscopy through
the coherent response of phonons that hybridize with the collective modes.
