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 μ, 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≪μ to the T≫μ 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 E, entropy density s, and number density n are related via E=(d)/(d+1)(Ts+μn), as is required by the isometries of AdS_d+2. Finally, in the T≪μ regime, the entropy density is found to satisfy a power law s∝cT^d/zμ^(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.