Charged Lifshitz Black Holes

TL;DR

This work addresses charged Lifshitz black holes in $n+1$ dimensions with arbitrary dynamical exponent $z$ and horizon topology, clarifying how Maxwell charge modifies neutral Lifshitz solutions and their holographic observables. The authors derive the Einstein–Maxwell–Lifshitz field equations, obtain exact solutions in special cases, and perform extensive numerical analysis to map out near-horizon, large-$r$, and thermodynamic behavior across dimensions and topologies. A key result is the generalized energy–entropy–chemical potential relation $\mathcal{E}=\frac{n-1}{n+z-1}\,[TS+\mathcal{Q}\Phi]$ for $z<n-1$, alongside the identification of a finite extremal charge $Q_c$ that governs extremality and the charge dependence of Wilson loops. These findings advance holographic modeling of quantum critical systems with anisotropic scaling by detailing how charge affects the geometry, thermodynamics, and boundary observables in higher-dimensional Lifshitz spacetimes.

Abstract

We investigate modifications of the Lifshitz black hole solutions due to the presence of Maxwell charge in higher dimensions for arbitrary $z$ and any topology. We find that the behaviour of large black holes is insensitive to the topology of the solutions, whereas for small black holes significant differences emerge. We generalize a relation previously obtained for neutral Lifshitz black branes, and study more generally the thermodynamic relationship between energy, entropy, and chemical potential. We also consider the effect of Maxwell charge on the effective potential between objects in the dual theory.

Figures (14)

• Figure 1: Metric functions $f(r)$ in 4-dimensions with: Left) $Q=0.5$ for $k=-1$ (green), $k=0$ (red) and $k=1$ (blue). Right) $Q=30$ for all $k$'s.
• Figure 2: Metric functions $f(r)$ for zero modes in 4, 5 and 7-dimensions from bottom to top respectively. Left) $r_{0}=20$ for all $k$. Right) $r_{0}=0.6$ for $k=0$.
• Figure 3: Metric functions $f(r)$ for zero modes with $r_{0}=0.6$ in 4 (red), 5 (blue) and 7 (green)-dimensions for: Left) $k=1$; Right) $k=-1$.
• Figure 4: Metric functions $f(r)$ for $k=-1$ in 7-dimensions with $z=2$: Left) $r_{0}=0.92$. Right) $r_{0}=20$ while the massive gauge field strength $h_{0}$ increases and temperature decreases from top to bottom in both cases.
• Figure 5: Metric functions $f(r)$ in 4, 5 and 7-dimensions with $z=$4, 5 and 7 from bottom to top, respectively. Left) $r_{0}=20$ and all values of $k$'s. For a given dimension the different values of $k$ lie almost exactly on the same curve. Right) $r_{0}=0.6$ and $k=0$ .