Quartic Horndeski, planar black holes, holographic aspects and universal bounds.

In this work, we consider a specific shift--invariant quartic Horndeski model where the function $G_4$ is chosen to be proportional to $\alpha \sqrt{(\partial \psi)^2/2}$, $\alpha$ denoting the coupling of the theory and $\psi$ being some massless scalar field. We take $(D-2)$--many copies of this model, and we assume a linear dependence of the scalar fields on the coordinates of the $(D-2)$--dimensional Euclidean submanifold, $\psi^{I}=p\delta^{I}_i x^{i}$. These choices allow us to construct planar black holes with a non--trivial axion profile which we explore in terms of their horizon structure and their thermodynamic properties. Since the particular scalar field profile dissipates momentum in the boundary theory, we are able to derive a sensible DC transport matrix describing the linear thermoelectric response of the holographic dual to an external electric field and a thermal gradient. We comment on the form of the conductivities and show that the heat conductivity--to--temperature ratio cannot have a universal lower bound at all scales due to the new coupling. We also verify the Kelvin formula motivated by the presence of an $\mathrm{AdS}_2\times\mathbb R^2$ horizon. Using the constants $D_{\pm}$ describing coupled diffusion at finite chemical potential, we show that these decouple in the incoherent limit of fast relaxation flowing to the usual expressions of charge/energy diffusitivities respectively, with $\alpha$ playing no qualitative role at all in this process. The bound of the refined $TD_{\pm}/v_B^2$ ratio is investigated, $v_B$ being the butterfly velocity. The coupling enters the latter only through the horizon radius, and it does not affect the $\mathcal{O}(1)$ form of the ratio in the strong dissipation regime. Finally, the viscosity--to--entropy ratio is computed by means of a (weaker) horizon formula, and the simple $(4\pi)^{-1}$--bound is found to be violated.