# A Unified Description of Translational Symmetry Breaking in Holography

## TL;DR

This paper offers a comprehensive holographic framework to understand translational symmetry breaking, exploring explicit and spontaneous cases, their interplay, and resulting collective phenomena like phonon behavior and relaxation scales.

## Contribution

It introduces a unified holographic model that smoothly interpolates between explicit and spontaneous translational symmetry breaking, analyzing associated collective modes and relaxation mechanisms.

## Key findings

- Identification of a sound-to-diffusion crossover in phonons
- Verification of the Gell-Mann-Oakes-Renner relation in the model
- Discovery of a novel relaxation scale linked to symmetry breaking scales

## Abstract

We provide a complete and unified description of translational symmetry breaking in a simple holographic model. In particular, we focus on the distinction and the interplay between explicit and spontaneous breaking. We consider a class of holographic massive gravity models which allow to range continuously from one situation to the other. We study the collective degrees of freedom, the electric AC conductivity and the shear correlator in function of the explicit and spontaneous scales. We show the possibility of having a sound-to-diffusion crossover for the transverse phonons. Within our model, we verify the validity of the Gell-Mann-Oakes-Renner relation. Despite of strong evidence for the absence of any standard dislocation induced phase relaxation mechanism, we identify a novel relaxation scale controlled by the ratio between the explicit and spontaneous breaking scales. Finally, in the pseudo-spontaneous limit, we prove analytically the relation, which has been discussed in the literature, between this novel relaxation scale, the mass of the pseudo-phonons and the Goldstone diffusivity. Our numerical data confirms this analytic result.