  In this paper, we consider two four-dimensional Horndeski-type gravity
theories with scalar fields that give rise to solutions with momentum
dissipation in the dual boundary theories. Firstly, we study Einstein-Maxwell
theory with a Horndeski axion term and two additional free axions which are
responsible for momentum dissipation. We construct static electrically charged
AdS planar black hole solutions in this theory and calculate analytically the
holographic DC conductivity of the dual field theory. We then generalize the
results to include magnetic charge in the black hole solution. Secondly, we
analyze Einstein-Maxwell theory with two Horndeski axions which are used for
momentum dissipation. We obtain AdS planar black hole solutions in the theory
and we calculate the holographic DC conductivity of the dual field theory. The
theory has a critical point $\alpha + \gamma \Lambda = 0$, beyond which the
kinetic terms of the Horndeski axions become ghost-like. The conductivity as a
function of temperature behaves qualitatively like that of a conductor below
the critical point, becoming semiconductor-like at the critical point. Beyond
the critical point, the ghost-like nature of the Horndeski fields is associated
with the onset of unphysical singular or negative conductivities. Some further
generalisations of the above theories are considered also.

High Energy Physics - Theory