BCFT in a Black Hole Background: An Analytical Holographic Model

TL;DR

This work constructs a fully analytic holographic BCFT model for a BCFT on a two-dimensional black hole background using an AdS$_3$ black string with a Karch–Randall brane. It computes the subregion entanglement entropy for two-sided bipartitions on both the gravity side (via Hartman–Maldacena and island RT surfaces) and the field theory side (via twist operators and BCFT two-channel analysis), showing exact agreement between the two descriptions. The paper derives closed-form expressions for the HM and island entropies, and analyzes the Page time and Page angle to map out the entanglement phase structure in curved space, including asymmetric bipartitions. The results provide a tractable, analytically solvable model that sheds light on entanglement islands, Page curves, and information transfer in AdS/BCFT setups on curved backgrounds, with potential broader implications for black hole information in lower-dimensional holographic systems.

Abstract

We study the entanglement phase structure of a holographic boundary conformal field theory (BCFT) in a two-dimensional black hole background. The bulk dual is the AdS$_3$ black string geometry with a Karch-Randall brane. We compute the subregion entanglement entropy of various two-sided bipartitions to elucidate the phase space where a Page curve exists in this setup. We do fully analytical computations on both the gravity side and the field theory side and demonstrate that the results precisely match. We discuss the entanglement phase structure describing where a Page curve exists in this geometry in the context of these analytical results. This is a useful model to study entanglement entropy for quantum field theory on a curved background.

Figures (11)

• Figure 1: Two diagrams representing different time slices on the boundary ($\rho=\infty$) Penrose diagram of the eternal black hole on which the BCFT$_d$ lives. The two red vertical lines represent the asymptotic boundaries where we impose conformal boundary conditions. Each time slice is composed of a blue component on one side of the bipartition and a green component on the other side. The green intervals, determined by $u_L$ and $u_R$ are drawn as equal size here, but generically vary in our setup. The union of the two green intervals is the subsystem we considered in Geng:2021mic where we computed the entanglement entropy between the green and the blue subsystems. We label time as $t_L$ and $t_R$ on either side of the diagram, and for both sides we take time evolution to go up the diagram (note the contrast with Figure \ref{['fig:Penrose1']}). The $u$-coordinate increases along a given time slice from 0 on the red boundary to $u_H$ at the bifurcation horizon, where the black diagonal lines cross. At the jagged singularities, $u=\infty$.
• Figure 2: This diagram shows the situation for a BCFT$_2$ living on the upper half plane (UHP). We consider bulk CFT to be in the vacuum state. The boundary is specified by the red horizontal axis where we impose conformal boundary conditions. The time direction is along the horizontal axis. We take a constant time slice (dashed black vertical line) that defines the quantum state we are studying. Our goal is to compute the entanglement entropy associated with the bipartition indicated by the blue cross.
• Figure 3: a) The two Euclidean BCFTs L and R in the TFD state: time evolution is rotation with respect to the origin, hence the red circle is the time evolution of the boundary. The two black dashed lines define the zero time slice. Under the chosen time evolution, L and R evolve clockwise and counter-clockwise respectively, as indicated. We are interested in the entanglement entropy of the subsystem $\mathcal{A}=\mathcal{A}_{L}\cup\mathcal{A}_{R}$ corresponding to the solid green line segments. b) The UHP which results from the conformal mapping of the region outside of the red circle in Fig. \ref{['TFDpre1']}. The circular boundary is mapped to the real axis and infinity is mapped to $\left(0,\frac{1}{2}\right)$. The location of twist operators is mapped to the two blue crosses, separated in the horizontal direction. The branch cut is mapped (and deformed) to the dashed green line connecting the two operators.
• Figure 4: A cartoon representation of the "left-side" thermofield double is shown here in the Poincaré half-plane. The light purple shaded region is excised from the AdS bulk by the KR brane at constant angular coordinate $\rho_*$. The convenience of the $\rho$ coordinate over the $\mu$ coordinate is clear here---we have conformally compactified the braneworld's infinite extra dimension into this finite diagram, so it now runs along the angular coordinate in the Poincaré half-plane and we identify $\mu=0, \frac{\pi}{2}, \pi$ with $\rho=-\infty, 0, \infty$, respectively. Also shown are the island and radiation regions, drawn in blue, and the island surface and HM surface, drawn in green and orange, respectively. The black string horizon forms a dotted arc at the coordinate $u=u_H$. For the eternal black string, we must also consider the "right-side" thermofield double, which looks identical to the diagram above, except that we allow a bipartition for the radiation region at a different coordinate $u=u_R$, where we may have $u_L\neq u_R$.
• Figure 5: An AdS Penrose diagram shows a cartoon projection of the Hartman-Maldacena surface in blue, connecting our two bipartition points, on the left- and right-universe patches (left- and right-exterior). The Penrose diagram is shown as a projection onto a constant $\rho=\rho_\epsilon$ slice, where the purple dashed line is the singularity at $u=\infty$, the red diagonal lines are the event horizon at $u=u_H$, and the black borders on the left and right represent the defect at $u=0$. The color-gradient contours show the constant time-slices and reversed direction of the time-like Killing vector field on each side of the black hole. Notice that in this diagram, $u_L\neq u_R$, and the HM surface is not symmetric across the event horizon. Also note that the HM surface is not confined to this diagram since it generically varies in the coordinate $\rho$.