Thermodynamics of black branes in asymptotically Lifshitz spacetimes

TL;DR

The paper addresses thermodynamics of black branes in asymptotically Lifshitz spacetimes (dual to 2+1D quantum critical theories with dynamical exponent $z$) and develops a purely analytic framework based on a Noether charge that links horizon data to boundary data. By performing near-horizon and asymptotic perturbations and tracking gauge-invariant quantities, it derives a conserved quantity $D_0$ and uses it to show the energy density $\mathcal{E}$, temperature $T$, and entropy density $s$ satisfy $\mathcal{E}=\frac{2}{2+z}Ts$, in agreement with Lifshitz scaling. The work also validates the $z=1$ AdS$_4$ limit, derives the equation of state $2P=z\mathcal{E}$, and generalizes the thermodynamic relations to arbitrary spatial dimension with $\mathcal{E}=Ts\frac{d}{d+z}$, providing a concrete holographic thermodynamic framework for Lifshitz backgrounds. These results bolster the interpretation of Lifshitz black branes as duals to quantum critical systems and offer a robust method to extract thermodynamics from horizon and boundary data.

Abstract

Recently, a class of gravitational backgrounds in 3+1 dimensions have been proposed as holographic duals to a Lifshitz theory describing critical phenomena in 2+1 dimensions with critical exponent $z\geq 1$. We continue our earlier work \cite{Bertoldi:2009vn}, exploring the thermodynamic properties of the "black brane" solutions with horizon topology $\mathbb{R}^2$. We find that the black branes satisfy the relation $\mathcal{E}=\frac{2}{2+z}Ts$ where $\mathcal{E}$ is the energy density, $T$ is the temperature, and $s$ is the entropy density. This matches the expected behavior for a 2+1 dimensional theory with a scaling symmetry $(x_1,x_2)\to λ(x_1,x_2)$, $t\to λ^z t$.