Holography of Charged Dilaton Black Holes in General Dimensions

TL;DR

The paper addresses holographic descriptions of charged dilaton black holes in general dimensions by constructing exact zero- and finite-temperature solutions in Einstein-Maxwell-dilaton theory with exponential coupling $e^{2\alpha\phi}F^2$, and by analyzing transport properties. It shows that the near-horizon geometry is Lifshitz-like with $z=1/\beta$ and $\beta=\alpha^2/(\alpha^2+2d)$, and that the zero-temperature AC conductivity follows a power law ${\rm Re}\,\sigma(\omega) \propto \omega^{\delta}$ with $\delta=2\nu_0-1$, $\nu_0^2=c_0+1/4$ and $c_0$ depending on $\alpha$ and $d$ through horizon data; a Schrödinger equation with $V(z)=c_0/z^2$ underpins the result. In five dimensions, Gauss-Bonnet corrections yield $\eta/s = (1/4\pi)[1 - 12β/(2β+1)\lambda_{GB}]$, reducing to the familiar AdS value in the relativistic limit $β\to1$. Collectively, the work extends known $d=2$ results to general dimensions, elucidates transport in Lifshitz-like holographic systems, and suggests future directions for dyonic configurations and Hall response in holographic condensed matter.

Abstract

We study several aspects of charged dilaton black holes with planar symmetry in $(d+2)$-dimensional spacetime, generalizing the four-dimensional results investigated in arXiv:0911.3586 [hep-th]. We revisit the exact solutions with both zero and finite temperature and discuss the thermodynamics of the near-extremal black holes. We calculate the AC conductivity in the zero-temperature background by solving the corresponding Schrödinger equation and find that the AC conductivity behaves like $ω^δ$, where the exponent $δ$ is determined by the dilaton coupling $α$ and the spacetime dimension parameter $d$. Moreover, we also study the Gauss-Bonnet corrections to $η/s$ in a five-dimensional finite-temperature background.