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Lifshitz-like black brane thermodynamics in higher dimensions

2011; American Physical Society; Volume: 83; Issue: 12 Linguagem: Inglês

10.1103/physrevd.83.126006

1550-7998

Gaetano Bertoldi, Benjamin A. Burrington, Amanda W. Peet, Ida G. Zadeh,

Galaxies: Formation, Evolution, Phenomena

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 $z\ensuremath{\ge}1$. We numerically explore a dilaton Einstein-Maxwell model, admitting such backgrounds as solutions. Such backgrounds are characterized by a temperature $T$ and chemical potential $\ensuremath{\mu}$, 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 $T\ensuremath{\ll}\ensuremath{\mu}$ to the $T\ensuremath{\gg}\ensuremath{\mu}$ 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 $\mathcal{E}$, entropy density $s$, and number density $n$ are related via $\mathcal{E}=\frac{d}{d+1}(Ts+\ensuremath{\mu}n)$, as is required by the isometries of ${\mathrm{AdS}}_{d+2}$. Finally, in the $T\ensuremath{\ll}\ensuremath{\mu}$ regime, the entropy density is found to satisfy a power law $s\ensuremath{\propto}c{T}^{d/z}{\ensuremath{\mu}}^{(z\ensuremath{-}1)d/z}$, and we numerically explore the dependence of the constant $c$, a measure of the number of degrees of freedom, on $d$ and $z$.