Lifshitz-like black brane thermodynamics in higher dimensions

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 >= 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 mu, and we find how to embed these solutions into anti-de Sitter (AdS) for a range of values of z and d. We find no thermal instability going from the T << mu to the T >> mu regimes, regardless of the dimension, and find that the solutions smoothly interpolate between Lifshitz-like behavior and relativistic AdS- like behavior. We show, using some conserved, that the energy density epsilon, entropy density s, and number density n are related via epsilon = d/d+1(Ts + mu n), as is required by the isometries of AdS(d+2). Finally, in the T << mu regime, the entropy density is found to satisfy a power law s proportional to cT(d/z) mu((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.