Zoology of solid & fluid holography - Goldstone modes and phase relaxation

We provide a comprehensive classification of isotropic solid and fluid holographic models with broken translational invariance. We describe in detail the collective modes in both the transverse and longitudinal sectors. First, we discuss holographic fluid models, i.e. systems invariant under internal volume preserving diffeomorphisms. We consider the explicit (EXB) and the spontaneous (SSB) breaking of translations and we emphasize the differences with respect to their solid counterpart. Then, we present a study of the longitudinal collective modes in simple holographic solid and fluid models exhibiting the interplay between SSB and EXB. We confirm the presence of light pseudo-phonons obeying the Gell-Mann-Oakes-Renner relation and the validity of the relation proposed in the literature between the novel phase relaxation scale, the mass of the pseudo-Golstone modes and the Goldstone diffusion. Moreover, we find very good agreement between the dispersion relation of our longitudinal sound mode and the formulae derived from the Hydro+ framework. Finally, our results suggest that the crystal diffusion mode does not acquire a simple damping term because of the novel relaxation scale proportional to the EXB. The dynamics is more complex and it involves the interplay of three modes: the crystal diffusion and two more arising from the splitting of the original sound mode. In this sense, the novel relaxation scale, which comes from the explicit breaking of the global internal shift symmetry of the Stuckelberg fields, is different from the one induced by elastic defects, and depending solely on the SSB scale.