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Thermal behavior of charged dilatonic black branes in AdS and UV completions of Lifshitz-like geometries

2010; American Physical Society; Volume: 82; Issue: 10 Linguagem: Inglês

10.1103/physrevd.82.106013

1550-7998

Gaetano Bertoldi, Benjamin A. Burrington, Amanda W. Peet,

Quantum Chromodynamics and Particle Interactions

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 $z\ensuremath{\ge}1$. We numerically explore one such model, characterized by a temperature $T$ and chemical potential $\ensuremath{\mu}$, 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 $T\ensuremath{\ll}\ensuremath{\mu}$ to the $T\ensuremath{\gg}\ensuremath{\mu}$ 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 $\mathcal{E}$, entropy density $s$, and number density $n$, $\mathcal{E}=\frac{2}{3}(Ts+\ensuremath{\mu}n)$. We show that this result is expected from general scaling arguments, and generalizes to $\mathcal{E}=\frac{d}{d+1}(Ts+\ensuremath{\mu}n)$ for a theory dual to ${\mathrm{AdS}}_{d+2}$ (Poincar\'e patch) asymptotics with a local $U(1)$ gauge invariance.