Entanglement Phase Structure of a Holographic BCFT in a Black Hole Background

TL;DR

The paper investigates how entanglement entropy for subregions of a BCFT on a nongravitating black-hole background evolves in a doubly holographic setting. By modeling the bulk as a planar AdS$_{d+1}$ black string with a Karch-Randall brane at angle $\theta_b$ and comparing Hartman-Maldacena and island surfaces, the authors map a rich phase structure controlled by the anchor $\Gamma$ and brane angle $\theta_b$, including a constant-entropy belt and an atoll confinement below the critical angle. Contrary to zero-temperature results, islands persist at finite temperature even below the critical angle, though their support is confined to a finite brane region (the atoll); above the critical angle the atoll expands to cover the brane. The findings illuminate how bulk geometry and brane dynamics govern unitarity-maintaining island formation and have implications for entanglement-wedge reconstruction in higher-dimensional braneworlds and BCFT holography.

Abstract

We compute holographic entanglement entropy for subregions of a BCFT thermal state living on a nongravitating black hole background. The system we consider is doubly holographic and dual to an eternal black string with an embedded Karch-Randall brane that is parameterized by its angle. Entanglement islands are conventionally expected to emerge at late times to preserve unitarity at finite temperature, but recent calculations at zero temperature have shown such islands do not exist when the brane lies below a critical angle. When working at finite temperature in the context of a black string, we find that islands exist even when the brane lies below the critical angle. We note that although these islands exist when they are needed to preserve unitarity, they are restricted to a finite connected region on the brane which we call the atoll. Depending on two parameters -- the size of the subregion and the brane angle -- the entanglement entropy either remains constant in time or follows a Page curve. We discuss this rich phase structure in the context of bulk reconstruction.

Figures (17)

• Figure 1:
• Figure 2: The bulk gravitational system we study in this paper is dual to a BCFT thermal state on a nongravitating black-hole background. We compute the entropy of the subregion $\mathcal{R}$, a connected interval that runs through the bifurcation surface (the horizon at $t = 0$ separating the past and future horizons, represented by the point in the middle of the Penrose diagram above), using the holographic RT prescription. This bipartition is similar to that in a recent work Geng:2021wcq studying entanglement islands in de Sitter space.
• Figure 3: This cartoon illustrates the relevant RT surfaces for different brane angles $\theta_{1,2}$ below the critical angle. Island surfaces end on the brane at right angles, and HM surfaces cross the black-string horizon and traverse the Einstein-Rosen bridge. Islands always attach to the brane inside the atoll, which begins at a point called the critical anchor and extends to the black-hole horizon. Decreasing the brane angle reduces the size of the atoll, which shrinks to the horizon as $\theta \rightarrow 0$. Increasing the size of $\mathcal{R}$, bounded by $\Gamma$, pushes the anchors (in purple) toward the black-hole horizon; increasing the brane angle pushes the anchors toward the defect. The critical anchor, which is the point on the brane that determines the size of the largest island on the brane (which always corresponds to $\Gamma=0$ ), coincides with the defect above the critical angle. See Figure \ref{['fig:surfaces']} for numerical results.
• Figure 4: These figures were included in our previous work Geng:2020fxl, but we repeat them here for emphasis. The critical angle is a monotonically increasing function of the number of spatial dimensions $d$ and takes the same value for any geometry which is asymptotically AdS$_{d+1}$. This is distinct from the behavior of the critical anchor defining the beginning of the atoll for the black string. The critical anchor instead decreases monotonically with brane angle and coincides with the defect at the critical angle (about 0.98687 for $d = 4$). It can be shown that the critical anchor also increases monotonically with the number of dimensions for the space.
• Figure 5: Here we see the setup together with numerical data in the planar black string. The KR brane is green, the HM surfaces are red, the island surfaces are blue, and the black-string horizon is the semicircle in black. The area difference between the island surface and the HM surface for various anchor points $\Gamma$ is displayed in the set of second plots below. Sending $\Gamma \rightarrow 0$ recovers the finite area differences from our previous paper Geng:2020fxl for angles below the critical angle. When the brane lies above the critical angle, the area difference diverges to negative infinity. These divergences are explained physically in Section \ref{['secdiv']}.