Lifshitz-like black brane thermodynamics in higher dimensions

Abstract

Gravitational backgrounds in d+2 dimensions have been proposed as holographic duals to Lifshitz-like theories describing critical phenomena in d+1 dimensions with critical exponent z\geq 1. We numerically explore a dilaton-Einstein-Maxwell model admitting such backgrounds as solutions. Such backgrounds are characterized by a temperature T and chemical potential μ, and we find how to embed these solutions into AdS for a range of values of z and d. We find no thermal instability going from the (T\llμ) to the (T\ggμ) regimes, regardless of the dimension, and find that the solutions smoothly interpolate between the Lifshitz-like behaviour and the relativistic AdS-like behaviour. We exploit some conserved quantities to find a relationship between the energy density E, entropy density s, and number density n, E=\frac{d}{d+1}(Ts+nμ), as is required by the isometries of AdS_{d+2}. Finally, in the (T\llμ) regime the entropy density is found to satisfy a power law s \propto c T^{d/z} μ^{(z-1)d/z}, and we numerically explore the dependence of the constant c, a measure of the number of degrees of freedom, on d and z.

Figures (2)

• Figure 1: The plots of $\ln\left( 4G_{d+2}{s}\right)$ versus $\ln ({LT})$ for fixed $\hat{\mu}=1$. Figures (\ref{['fig:muh1d3']}), (\ref{['fig:muh1d5']}), (\ref{['fig:muh1d7']}) and (\ref{['fig:muh1d9']}) correspond to $d = 3$, $5$, $7$, and $9$, respectively. The different curves in each plot correspond to different values of $\alpha$, with $\alpha=4$ green (solid), $\alpha=2$ magenta (long-dashed), $\alpha=1$ cyan (dot-dashed), and $\alpha=0.75$ blue (dashed). $\alpha$ is related to $z$ via $\alpha=\sqrt{2d/(z-1)}$.
• Figure 2: The plot of $\ln\left( c(z,d)\right)$ as a function of $\ln (z)$ for fixed value of $\hat{\mu}=1$. The different curves correspond to different dimension with $d=3$ black (solid), $d=5$ brown (long-dash), $d=7$ red (dot-dash), and $d=9$ coral (dash). Curves for $d=2,4,6,8$ behave similarly. First of all, notice that the curves' intercept at $\ln(z)=0$ is given by the value $\ln\left({(4\pi)^d}/{(d+1)^d}\right)$ which is non-monotonic in $d$. Secondly, we find it interesting that the tails become flat out at $z\rightarrow\infty$ and that the large $z$ behaviour for various $d$ is monotonic in $d$. Given these two facts, it may be no surprise that the curves $\ln(c(z,d))$ for fixed $d$ are generically non-monotonic for small values of $z$, and that in fact they cross each other.