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 St\"uckelberg fields, is
different from the one induced by elastic defects, and depending solely on the
SSB scale.Comment: v3: new section about the comparison with Hydro+, new section proving
the universal relation for the Goldstone phase relaxation; version accepted
in JHE
