Holographic entanglement plateaux

Abstract

We consider the entanglement entropy for holographic field theories in finite volume. We show that the Araki-Lieb inequality is saturated for large enough subregions, implying that the thermal entropy can be recovered from the knowledge of the region and its complement. We observe that this actually is forced upon us in holographic settings due to non-trivial features of the causal wedges associated with a given boundary region. In the process, we present an infinite set of extremal surfaces in Schwarzschild-AdS geometry anchored on a given entangling surface. We also offer some speculations regarding the homology constraint required for computing holographic entanglement entropy.

Figures (12)

• Figure 1: Minimal surfaces (geodesics) in BTZ geometry with black hole size $r_+=1$ (left) and $r_+=0.2$ (right). In each panel, the thick black circle represents the AdS boundary, and the thick red circle the black hole horizon. The minimal surfaces are depicted by the thin curves, color-coded by $\theta_\infty$, with $\theta_\infty$ varying between $0$ (red) and $\pi$ (purple) in increments of $0.05\pi$. For ease of visualization we use a compactified radial coordinate $\rho = \tan^{-1} r$. The boundary region ${\cal A}$ is centered at $\theta = 0$ which is plotted on top.
• Figure 2: Comparison of the two families ${\mathfrak M}_{1}$ and ${\mathfrak M}_{2}$ of the minimal surfaces (geodesics) in BTZ, with black hole sizes $r_+=1$ (left) and $r_+=0.2$ (right) as in Fig. \ref{['f:BTZgeods']}, plotted at the critical value of $\theta_\infty$ where they exchange dominance, i.e. at $\theta_\infty = \theta_\infty^{\cal X}$. The connected family ${\mathfrak M}_{1}$ is represented by the solid curve (color-coded by $\theta_\infty$ as in Fig. \ref{['f:BTZgeods']}), while the disconnected family ${\mathfrak M}_{2}$ is given by the two dashed curves (color-coded by $\pi-\theta_\infty$). The thick orange arc on the boundary represents the region ${\cal A}$.
• Figure 3: Plot of the curves $\delta S_{\cal A}/S_{\rho_\Sigma}$ for a Dirac fermion in $1 +1$ dimensions in the canonical ($T \neq 0$) Azeyanagi:2007bj and grand canonical ($T, \mu \neq 0$) Ogawa:2011bz. We examine the behaviour for a range of temperatures and chemical potential. The solid curves from bottom are $\beta = 4$ (red), $\beta = 2$ (blue), $\beta = 1$ (magenta) and $\beta = 0.1$ (black). The situation with the chemical potential turned on is indicated with the same colour coding with $\mu = 0.1$ represented by circles and $\mu = 0.5$ by triangles. The symmetry $\mu \leftrightarrow 1-\mu$ is used to restrict $\mu \in [0,1/2]$ and we see that for $\mu = 0.5$ the normalized $\delta S_{\cal A}$ is essentially the same at any temperature. This behaviour should be contrasted against the large $c$ holographic result displayed in Fig. \ref{['f:dSnorm']} for $d\geq 2$ thermal CFTs.
• Figure 4: Minimal surfaces for $r_+=1$ (left) and $r_+=0.2$ (right) black holes in Schwarzschild-AdS$_{}$ in $3+1$ (top) and $4+1$ (bottom) dimensions, analogous to the plots of Fig. \ref{['f:BTZgeods']}. In each panel, the thick black circle represents the AdS boundary, and the thin red circle the black hole horizon. The boundary region ${\cal A}$ is centered at $\theta = 0$ (North Pole) which is plotted on top. Further plots with other black hole sizes and higher dimensions can be found as http://arxiv.org/src/1306.4004/anc with the submission.
• Figure 5: Entanglement entropy plateaux for Schwarzschild-AdS$_{}$ black holes dual to thermal field theories. (a). Behaviour in Schwarzschild-AdS$_{}$$_5$ for varying $r_+$. From right to left we have $r_+ = \frac{1}{4}$ (blue), $r_+ = \frac{1}{2}$ (purple), $r_+ = 1$ (yellow), $r_+ = 2$ (green) and $r_+ =4$ (blue). (b). Dimension dependence of the plateaux, at the Hawking-Page transition point $R \, T=\frac{d-1}{2\pi}$ for $d=2,3,4,5,6$ (red, yellow, green, blue, purple respectively). Note that the horizon size is related to the temperature via $r_+=\frac{2\pi}{d} \left(R\,T+\sqrt{(RT)^2-\frac{d(d-2)}{4\pi^2}}\right)$.

Theorems & Definitions (1)

• proof