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\geq 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 AdS for a range of values of z and d. We find no thermal instability going from the (T\llμ) to the (T\ggμ) regimes, regardless of the dimension, and find that the solutions smoothly interpolate between the Lifshitz-like behaviour and the relativistic AdS-like behaviour. We exploit some conserved quantities to find a relationship between the energy density E, entropy density s, and number density n, E=\frac{d}{d+1}(Ts+nμ), as is required by the isometries of AdS_{d+2}. Finally, in the (T\llμ) regime the entropy density is found to satisfy a power law s \propto c T^{d/z} μ^{(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.

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