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 μ, 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≪μ to the T≫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 E, entropy density s, and number density n, E=(2)/(3)(Ts+μn). We show that this result is expected from general scaling arguments, and generalizes to E=(d)/(d+1)(Ts+μn) for a theory dual to AdS_d+2 (Poincaré patch) asymptotics with a local U(1) gauge invariance.