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 >= 1. We numerically explore one such model, characterized by a temperature T and chemical potential mu, and find how to embed these solutions into anti-de Sitter (AdS) for a range of values of z. We find no phase transition going from the T << mu to the T >> 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 epsilon, entropy density s, and number density n, epsilon = 2/3 (Ts + mu n). We show that this result is expected from general scaling arguments, and generalizes to epsilon = d/ d+1 (Ts + mu n) for a theory dual to AdS(d+2) (Poincare patch) asymptotics with a local U(1) gauge invariance.