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It has been shown that holographic massive gravities can effectively realizethe states with spontaneous breaking of translational symmetry in a homogenousmanner. In this work, we consider a toy model of such category by adding aspecial gauge-axion coupling to the bulk action. Firstly, we identify theexistence of spontaneous breaking of translations by the UV analysis. In theabsence of explicit breaking, the black hole solution is simply the same as theReissner-Nodstrom(RN) black holes, regardless of the non-trivial profile of thebulk axions. The associated Goldstone modes exist only when the charge densityis non-zero. Then, we investigate the optical conductivity both analytically aswell as numercially. Our numerical result perfectly agrees with that for aclean system, while the incoherent part gets modified due to the symmetrybreaking. The transverse Goldstone modes are dispersionless, which reflects thefact that the solution is dual to a liquid state. Finally, the effect ofmomentum relaxation to the transverse modes is also considered. In this case,the would-be massless modes are pinned at certain frequency, which is anotherkey difference from unbroken phases.

A simple holographic model for spontaneous breaking of translational symmetry