Holographic entanglement entropy: near horizon geometry and disconnected regions

TL;DR

This work analyzes the finite term of holographic entanglement entropy for one or two parallel strips in black hole backgrounds, focusing on how near-horizon geometry governs the large-volume (large-L) behavior and how Lifshitz-like scaling modifies the entropy's scaling with width. It develops a near-horizon expansion framework, derives general area-functionals for a broad class of metrics, and applies it to charged AdS black holes, warped black holes, and perturbed Lifshitz backgrounds; notably, it provides an analytic all-orders result for a Lifshitz black hole in four dimensions. For two strips, the paper studies the holographic mutual information transition, obtaining analytic AdS results and characterizing the curved transition in charged BHs that signals a finite boundary correlation length at finite temperature. Overall, the results illuminate how horizon and dynamical exponents shape entanglement structure in strongly coupled boundary theories at large central charge.

Abstract

We study the finite term of the holographic entanglement entropy for the charged black hole in AdS(d+2) and other examples of black holes when the spatial region in the boundary theory is given by one or two parallel strips. For one large strip it scales like the width of the strip. The divergent term of its expansion as the turning point of the minimal surface approaches the horizon is determined by the near horizon geometry. Examples involving a Lifshitz scaling are also considered. For two equal strips in the boundary we study the transition of the mutual information given by the holographic prescription. In the case of the charged black hole, when the width of the strips becomes large this transition provides a characteristic finite distance depending on the temperature.

Figures (7)

• Figure 1: Charged black hole, extremal case and $z_0=1$. Plot of the finite term $\mathcal{A}_2(z_{\textrm{max}},0)$ as function of $z_{\textrm{max}}$ (red curve). Close to the boundary (i.e. when $z_{\textrm{max}} \rightarrow 0$) it coincides with the curve corresponding to $AdS_4$ (blue curve), which can be read from (\ref{['A expansion empty']}).
• Figure 2: Extremal charged black hole in $AdS_4$ with $z_0=1$ (left) and $z_0=1.5$ (right). Plot of the finite term $\mathcal{A}_2(z_{\textrm{max}},0)$ as a function of $L$ (red line). For small $L$ it recovers the corresponding quantity for $AdS_4$ (blue curve) obtained from (\ref{['Atilde free']}). The black line provides the large $L$ behavior given by (\ref{['expansion nh rn T0']}).
• Figure 3: Holographic mutual information in $AdS_4$ with $L_2=L_1$. On the left we show $M_2(L_1,L_1;L_0)$ for $L_0=0.87$ (red), $L_0=1.91$ (blue) and $L_0=3.93$ (black). On the right, in the parameter space $(L_1,L_0)$, we plot the position of the transition point at which the mutual information starts to be non zero.
• Figure 4: Two equal and parallel strips. Angular coefficient of the line characterizing the transition of the holographic mutual information in $AdS_{d+2}$ in terms of $d$. The red point corresponds to $AdS_3$, which is not described by the equation (\ref{['AdS transition eq']}). In this case the transition occurs at the value $x=1/2$ of the conformal ratio Headrick:2010zt.
• Figure 5: Extremal charged black hole in $AdS_4$ with $z_0=1$. On the left the position of the transition point of the holographic mutual information in the parameter space $(L_1,L_0)$. On the right, a zoom of the same plot: the asymptotic line is provided by the equation (\ref{['trans eq asympt L0']}) and the green line corresponds to the transition point of $AdS_4$ (figure \ref{['2intM0free']}, plot on the right).