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

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 $\mu$, and find how to embed these solutions into AdS for a range of values of $z$. We find no phase transition going from the $T\ll\mu$ to the $T\gg \mu$ 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 $n$, $\mc E=\frac{2}{3} \left(Ts+\mu n\right)$. We show that this result is expected from general scaling arguments, and generalizes to $\mc E= \frac{d}{d+1}\left(Ts+\mu n\right)$ for a theory dual to AdS$_{d+2}$ (Poincar\'e patch) asymptotics with a local $U(1)$ gauge invariance.