Spontaneous breaking of a discrete symmetry and holography

We present an exactly solvable model of a scalar field in an AdS$_{d+1}$ like background interpolating between a $Z_2$ preserving and a $Z_2$ breaking minima of the potential. We define its holographic dual through the AdS/CFT dictionary and argue that at zero temperature the $d-$dimensional strongly coupled system on the boundary of AdS$_{d+1}$ exhibits a phase with a spontaneously broken discrete symmetry. In the presence of a black hole in the bulk ($T\neq 0$) we find that, although the metastable phase is present, the discrete symmetry gets restored. We compute exactly the lowest order boundary correlation functions in the spontaneously broken phase at T=0, finding out a pole of the propagator for zero momenta that signals the presence of a massless mode.