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 not equal 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 and argue that it should not be present at T not equal 0.