Black holes in asymptotically Lifshitz spacetime

TL;DR

This paper develops a gravity dual for Lifshitz-type critical phenomena in 2+1 dimensions by studying a 3+1D gravity model with negative $\Lambda$ coupled to abelian gauge fields. It constructs a global, static setup that admits Lifshitz asymptotics with $z=2$ (fixed by a topological coupling $c$), and analyzes both charged Lifshitz black holes and non-singular Lifshitz-star solutions, showing that a one-parameter family of black holes arises when a marginal zero mode is suppressed. The authors compute finite-temperature thermodynamics, finding thermodynamic stability for all black-hole sizes and a linear scaling of entropy with temperature for large black holes, along with extremality at small sizes; they also evaluate Wilson loops to probe the boundary theory, revealing finite-temperature screening with a characteristic length $\ell_c$ that scales as $T^{-1/2}$ for $z=2$. Together, these results support a holographic framework for 2+1D Lifshitz critical systems and motivate further work to extract finite-temperature correlators and to explore connections with string theory embeddings and Horava-type nonrelativistic theories.

Abstract

A model of 3+1 dimensional gravity with negative cosmological constant coupled to abelian gauge fields has been proposed as a gravity dual for Lifshitz like critical phenomena in 2+1 dimensions. The finite temperature behavior is described by black holes that are asymptotic to the Lifshitz fixed point geometry. There is a one-parameter family of charged black holes, where the magnitude of the charge is uniquely determined by the black hole area. These black holes are thermodynamically stable and become extremal in the limit of vanishing size. The theory also has a discrete spectrum of localized objects described by non-singular spacetime geometries. The finite temperature behavior of Wilson loops is reminiscent of strongly coupled gauge theories in 3+1 dimensions, including screening at large distances.

Figures (10)

• Figure 1: Large $r$ behavior of numerical black hole solutions. The figure plots the amplitude of the zero mode $\gamma_0$ at $r=10^6$ as a function of $h(r_0)$, the electric field at the horizon, for a black hole with $r_0=10$. The curve has a single zero at which the zero mode amplitude vanishes, and this uniquely determines a value of $h(r_0)$ for which the black hole is asymptotic to the Lifshitz fixed point geometry.
• Figure 2: The charge of an asymptotically Lifshitz black hole is uniquely determined by the horizon area. The figure plots $h(r_0)$, the electric field strength at the horizon, as a function of $r_0$, the area coordinate at the horizon. The dashed curve is the upper bound on $h_0$ in equation (\ref{['upperbound']}). The bound is saturated in the small black hole limit.
• Figure 3: A large black hole with $r_0=20$. The figure on the left shows the metric functions $f(r)$ and $g(r)$, while the figure on the right shows the electric field $f(r)$ and the charge density $j(r)$.
• Figure 4: A small black hole with $r_0=0.1$. The figure on the left shows the metric functions $f(r)$ and $g(r)$, while the figure on the right shows the electric field $f(r)$ and the charge density $j(r)$.
• Figure 5: Large $r$ behavior of the non-singular solutions discussed in Section \ref{['lstars']}. The figure indicates the amplitude of the zero mode $\gamma_0$ at $r=10^4$ as a function of $j_0$, the charge density at the origin. The zeroes correspond to Lifshitz stars which are asymptotic to the Lifshitz fixed point geometry. We plot the product $j_0\gamma_0$ rather than $\gamma_0$ itself. This does not affect the location of the zeroes but makes the plot more readable.