A simple holographic model for spontaneous breaking of translational symmetry

It has been shown that holographic massive gravities can effectively realize spontaneous breaking of translational symmetry in homogenous manners. In this work, we consider a toy model of such category by adding a special gauge-axion coupling to the bulk action. Firstly, we identify the existence of spontaneous breaking of translations by the analysis on the UV expansion. In the absence of explicit breaking, the black hole solution is simply the same as the Reissner-Nodström(RN) black holes, regardless of the non-trivial profile of the bulk axions. The associated Goldstone modes exist only when the charge density is non-zero. Then, we investigate the optical conductivity both analytically as well as numercially. Our result perfectly agrees with that for a clean system, while the incoherent conductivity gets modified due to the symmetry breaking. The transverse Goldstone modes are dispersionless since the solution is dual to a liquid state. Finally, the effect of momentum relaxation to the transverse modes is considered. In this case, the would-be massless modes are pinned at certain frequency, which is another key difference from the unbroken states.