# DC Conductivities with Momentum Dissipation in Horndeski Theories

## TL;DR

This paper investigates holographic DC conductivities in four-dimensional Horndeski gravity theories with scalar fields causing momentum dissipation, analyzing black hole solutions and their physical properties in dual boundary theories.

## Contribution

It constructs new black hole solutions in Horndeski theories with momentum dissipation and analytically computes the holographic DC conductivity, exploring effects of magnetic charge and critical points.

## Key findings

- Conductivity behaves like a conductor below the critical point.
- At the critical point, conductivity becomes semiconductor-like.
- Beyond the critical point, ghost-like fields lead to unphysical conductivities.

## Abstract

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.