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

Abstract

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. 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 Stückelberg fields, is different from the one induced by elastic defects, and depending solely on the SSB scale.

A unified description of translational symmetry breaking in holography

“…The crystal diffusion mode in the SSB limit is not damped. This has been shown by some of the authors of this work in two preprints [75,76] appeared at the same time of our work.…”

Longitudinal sound and diffusion in holographic massive gravity

“…As expected, the quasi-normal modes calculated in this limit show the presence of a diffusive mode with dispersion relation 27) which is shown in figure 11. Given the simplicity of the decoupled regime, we are able to obtain the Green's function of the scalar operator analytically, using perturbative techniques.…”

“…In particular the longitudinal sound mode and the diffusive mode have been identified numerically but without any systematic study or theoretical background. Related results have recently appeared in [26] in the presence of electromagnetic interactions and in [27] for holographic fluid models. Other recent works have analysed the origin of this extra diffusive mode in similar holographic models with a global U (1) symmetry [28,29].…”

Scale invariant solids

“…The latter disagreement for b < 5/2 has been explicitly checked for in [49]. The validity of formula (2.15) has been ascertained directly by a direct comparison with the QNMs spectrum for transverse [37,37] and longitudinal [35,66] waves. Two important results follow:…”