  We discuss the possible phases dual to the AdS hairy black holes in Horndeski
theory. In the probe limit breaking the translational invariance, we study the
conductivity and we find a non-trivial structure indicating a collective
excitation of the charge carriers as a competing effect of momentum dissipation
and the coupling of the scalar field to Einstein tensor. Going beyond the probe
limit, we investigate the spontaneous breaking of translational invariance near
the critical temperature and discuss the stability of the theory. We consider
the backreaction of the charged scalar field to the metric and we construct
numerically the hairy black hole solution. To determine the dual phases of a
hairy black hole, we compute the conductivity. When the wave number of the
scalar field is zero, the DC conductivity is divergent due to the conservation
of translational invariance. For nonzero wave parameter with finite DC
conductivity, we find two phases in the dual theory. For low temperatures and
for positive couplings, as the temperature is lower, the DC conductivity
increases therefore the dual theory is in metal phase, while if the coupling is
negative we have the opposite behavior and it is dual to an insulating phase.
We argue that this behavior of the coupling of the scalar field to Einstein
tensor can be attributed to its role as an impurity parameter in the dual
theory.
