On the dynamics of fluctuations in time crystals

We study the evolution of quantum fluctuations in systems known as time crystals, hypothetical systems for which the lowest energy state performs a periodic motion. We first discuss some general properties shared by time crystals, and deduce the effective field theory parametrizing the evolution of their fluctuations. We show that these fluctuations fall into categories analogous to acoustic and optical phonons, encountered in conventional crystals. The acoustic phonons correspond to gapless Goldstone boson modes parametrizing the broken time translation invariance of the crystal, whereas the optical phonons are identified with modes perpendicular to the broken symmetry of the system, which generically remain gapped. We study how these two modes decay and interact together, and discuss some observable features that could be tested in experimental realizations of time crystals.