Hawking radiation in large N strongly-coupled field theories

TL;DR

This work uses AdS/CFT to study Hawking radiation in strongly coupled large-$N$ field theories on black hole backgrounds, focusing on equilibrium Hartle-Hawking-like states. It identifies two holographic phases: a black funnel with a single connected bulk horizon signaling strong coupling to a deconfined plasma, and a two-horizon configuration (black droplet plus planar/hyperbolic horizon) signaling weak coupling due to finite-size plasma excitations, with a sharp transition governed by the dimensionless product $R T_H$. Explicit analytic examples in AdS$_3$ and AdS$_4$ demonstrate funnels in 1+1 dimensions and both funnels and droplets in 2+1 dimensions on backgrounds asymptotic to ${\mathbb R}\times{\bf H^2}$, illustrating the proposed phase structure and its dependence on boundary geometry and temperature. The results reveal a richer large-$N$ phase diagram for field theories on curved backgrounds and offer insights into bulk-brane world scenarios and non-equilibrium heat transport in strongly coupled plasmas.

Abstract

We consider strongly coupled field theories at large N on black hole backgrounds. At sufficiently high Hawking temperature T_H, one expects a phase where the black hole is in equilibrium with a deconfined plasma. We explore this phase in the context of the AdS/CFT correspondence, and argue that two possible behaviors may result. At a given Hawking temperature T_H, field theories on large black hole backgrounds will couple strongly to the plasma and will be dual to novel bulk spacetimes having a single connected horizon which we dub {\it black funnels}. We construct examples of black funnels in low spacetime dimensions for different classes of field theory black holes. In this case, perturbing the equilibrium state results in the field theory exchanging heat with the black hole at a rate typical of conduction through deconfined plasma. In contrast, we argue that due to the finite physical size of plasma excitations, smaller black holes will couple only weakly to the field theory excitations. This situation is dual to bulk solutions containing two disconnected horizons which remain to be constructed. Here perturbations lead to heat exchange at a level typical of confined phases, even when T_H remains far above any deconfinement transition. At least at large N and strong coupling, these two behaviors are separated by a sharp transition. Our results also suggest a richer class of brane-world black holes than hitherto anticipated.

Figures (6)

• Figure 1: A sketch of our two novel classes of solutions: (a): black funnel and (b): black droplet above a deformed planar black hole. In both cases the boundary metric is that of a black hole of temperature $T_H$ and horizon size $R$ (the top line corresponds to the boundary, the dots denote the horizon of the boundary black hole). Here we fix $T_H$ and vary $R$; (a) for sufficiently large $R$, we expect the bulk solution to have a single connected horizon in a "funnel" shape, while (b) for small $R$, we expect two disconnected horizons, a "droplet" and a planar black hole. The shaded regions are those inside the bulk horizons.
• Figure 2: The plot of the bulk horizon for the black funnel geometry whose boundary is the two dimensional black hole (\ref{['BF3formbdy']}). We plot the location of the horizon in the $\{r,z\}$ coordinates. The shaded region lies inside the black hole as in Fig. \ref{['f:fundrop']}.
• Figure 3: The functions $F(x)$ and $G(x)$ are plotted for $\mu = \mu_c$. The solid curve is the function $G(x)$ while the dashed curves correspond to $F(x)$ plotted for various values of $\lambda$. The direction of increasing $\lambda$ is indicated in the figure. The middle curve is $\lambda=0$.
• Figure 4: Real roots of $F$ (curved lines) are shown for $\lambda > -1$ with $\mu =\mu_c$ and $\kappa =1$. The values of $x_0, x_2$ (straight lines) are shown for comparison. At each $\lambda$, the smallest root $y_0$ satisfies $y_0 < x_0$. All roots lie below $x_2$, and $y_0$ is the only real root for $\lambda > 0$.
• Figure 5: Horizons (diagonal lines) are plotted in the $x,z$ plane. Note that $z$ increases downward while $x$ increases to the right. Dotted lines denote infinities at $z=0$ and $x=x_0$. As a result, each quadrant may be considered a complete spacetime unto itself. The left panel (a) corresponds to the situation when $-1 < \lambda < 0$ where $F(y)$ has three real roots and the right panel (b) corresponds to $\lambda >0$; compare with Fig. \ref{['f:fgplots1C']}. Note that in either case a conical singularity at $x = x_2$ will be visible from the point $x=x_0, z=0$ in quadrant IV.