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An excitonic insulator (EI) is an unconventional quantum phase of matter inwhich excitons, bound pairs of electrons and holes, undergo Bose--Einsteincondensation, forming a macroscopic coherent state. While its existence wasfirst hypothesized half a century ago, the EI has eluded experimentalobservation in bulk materials for decades. In the last few years, a resurgenceof interest in the subject has been driven by the identification of severalcandidate materials suspected to support an excitonic condensate. However, oneobstacle in verifying the nature of these systems has been to find signaturesof the EI that distinguish it from a normal insulator. To address this, wefocus on a clear qualitative difference between the two phases: the existenceof Goldstone modes born by the spontaneous breaking of a $U(1)$ symmetry in theEI. Even if this mode is gapped, as occurs in the case of an approximatesymmetry, this branch of collective modes remains a fundamental feature of thelow-energy dynamics of the EI provided the symmetry-breaking is small. We studya simple model that realizes an excitonic condensate, and use mean field theorywithin the random-phase approximation to determine its collective modes. Wesubsequently develop a diagrammatic method to incorporate the effects ofdisorder perturbatively, and use it to determine the scattering rate of thecollective modes. We interpret our results within an an effective field theory.The collective modes are found to be robust against symmetry-preservingdisorder, implying an experimental fingerprint unique to the EI: the ballisticpropagation of low-lying modes over mesoscopic distances, at electronic-scalevelocities. We suggest this could affect thermal transport at low temperatures,and could be observed via spatially-resolved pump-probe spectroscopy throughthe coherent response of phonons that hybridize with the collective modes.

Effects of Disorder on the Transport of Collective Modes in an Excitonic Condensate