Thermal behavior of charged dilatonic black branes in AdS and UV completions of Lifshitz-like geometries

TL;DR

The paper investigates finite-temperature, finite-chemical-potential physics of charged dilatonic black branes that interpolate between Lifshitz-like UV geometries and AdS IR completions in AdS$_4$. It develops a 1D effective action by reducing the four-dimensional theory, identifies conserved quantities, and analyzes horizon and asymptotic perturbations alongside numerical solutions to map the Lifshitz-to-AdS interpolation. A key result is that no phase transition occurs between the $T\ll\mu$ and $T\gg\mu$ regimes; the solutions evolve smoothly between Lifshitz-like and AdS-like behavior as the scale set by $\mu$ is varied. Thermodynamically, the energy density satisfies $\mathcal{E}=\frac{2}{3}(Ts+\mu n)$, generalizing to $\mathcal{E}=\frac{d}{d+1}(Ts+\mu n)$ for a theory with AdS$_{d+2}$ UV completion, which is argued from scaling considerations and the first law of thermodynamics.

Abstract

Several classes of gravitational backgrounds in $3+1$ dimensions have been proposed as holographic duals to Lifshitz-like theories describing critical phenomena in $2+1$ dimensions with critical exponent $z\geq 1$. We numerically explore one such model, characterized by a temperature $T$ and chemical potential $μ$, and find how to embed these solutions into AdS for a range of values of $z$. We find no phase transition going from the $T\llμ$ to the $T\gg μ$ regimes, and find that the solutions smoothly interpolate between the Lifshitz-like behavior and the relativistic AdS-like behavior. Finally, we exploit some conserved quantities to find a relationship between the energy density $\mc E$, entropy density $s$, and number density $n$, $\mc E=\frac{2}{3} \left(Ts+μn\right)$. We show that this result is expected from general scaling arguments, and generalizes to $\mc E= \frac{d}{d+1}\left(Ts+μn\right)$ for a theory dual to AdS$_{d+2}$ (Poincaré patch) asymptotics with a local $U(1)$ gauge invariance.

Figures (4)

• Figure 1: Above we have plotted $\ln(TL)$ as a function of $\ln(r_h)$ for fixed $\hat{\mu}=2$. The different graphs correspond to different values of $z$: $z=1.25 \; (\alpha=4)$ (red dashed curve), $z=2 \;(\alpha=2)$ (orange dotted curve), $z=5 \;(\alpha=1)$ (green dash-dotted curve), $z=\frac{73}{9}= 8.\bar{1} \;(\alpha=0.75)$ (blue long-dashed curve). The solid black curve is a plot for the pure AdS black brane $\ln(TL)=\ln\left(\frac{3r_h}{4\pi}\right)$, the asymptotic value of all graphs in the $\ln(r_h) \rightarrow \infty$ limit. The slopes of the graphs approach $z$ as $\ln(r_h)\rightarrow -\infty$.
• Figure 2: The above plots depict the metric functions and fields as a function of $r$, the top plots for a small value of $r_h=0.4$, and the bottom plots for a large value of $r_h=40$. The plots show $e^{A_1}$ (red dashed curve), $e^{C_1}$ (orange dotted curve), $e^{G_1}$ (green dash-dotted curve) and $e^{2\alpha \phi}$ (blue long-dashed curve). Plots (b) and (d) simply show that the asymptotics are correct with the convention that $e^{2\alpha \phi}$ is set to $1$ at the boundary.
• Figure 3: Above we have plotted the metric functions $A_1$ (red dashed curve), $C_1$ (orange dotted curve) and the fields $G_1$ (green dash-dotted curve), $2\alpha \phi$ (blue long-dashed curve) as a function of $\ln(r)$ for fixed $\hat{\mu}=2$, and horizon radius $r_h=0.2$. We see that all the functions have an approximately linear portion in the region $-1<\ln(r)<0$. The slopes of these linear portions can be shown to approximately reproduce the Lifshitz background with $z=2, (\alpha=2)$. Further, the asymptotic value of $G_1(r) \rightarrow \ln(2)$ is seen, as opposed to the asymptotic value of all other graphs $\ln(1)=0$.
• Figure 4: Above we have plotted $\ln({\mathcal{E}}_1)$ as a function of $\ln(r_h)$ for fixed $\hat{\mu}=2$. The different graphs correspond to different values of $z$: $z=1.25 \; (\alpha=4)$ (red dashed curve), $z=2 \;(\alpha=2)$ (orange dotted curve), $z=5 \;(\alpha=1)$ (green dash-dotted curve), $z=\frac{73}{9}= 8.\bar{1} \;(\alpha=0.75)$ (blue long-dashed curve). The solid black curve is a plot for the pure AdS black brane $\ln({\mathcal{E}}_1)=\ln\left(4\frac{3 \alpha r_h^3}{6\alpha}\right)$, the asymptotic value of all graphs in the $\ln(r_h) \rightarrow \infty$ limit. The slopes of the graphs approach $0$ as $\ln(r_h)\rightarrow -\infty$. This is merely an indication of a finite energy held in the $U(1)$ gauge field, and may be considered an extremal limit with "mass $=$ charge" in the right units. One can check this by graphing $\frac{Q_{\rm out}}{{{\mathcal{E}}}_1}$ and seeing that it goes to $\frac{6}{16\hat{\mu}}$ (independent of $\alpha$) for $r_h \rightarrow 0$, however we do not display such plots.