Replica Wormholes and Entanglement Islands in the Karch-Randall Braneworld

TL;DR

The paper investigates how entanglement islands and the Page curve arise in the Karch-Randall braneworld by deriving the Ryu-Takayanagi prescription from a boundary replica trick within AdS/BCFT. It demonstrates that replica wormholes on the Karch-Randall brane correspond to the island phase, with the entanglement entropy computed as the minimum of connected and island-ending RT surfaces, reproducing the Page curve in (2+1)-dimensional ambient space and connecting to Einstein-matter dynamics on the brane in higher dimensions. The results provide a first-principles bridge between island prescriptions and gravitational path integrals in braneworld holography, and discuss how these mechanisms extend beyond AdS_3 to higher dimensions and more general brane actions. Overall, the work solidifies the replica-wormhole origin of entanglement islands on the KR brane and clarifies the role of brane geometry and conformal matter in the gravitational computation of Hawking radiation entropy.

Abstract

The Karch-Randall braneworld provides a natural set-up to study the Hawking radiation from a black hole using holographic tools. Such a black hole lives on a brane and is highly quantum yet has a holographic dual as a higher dimensional classical theory that lives in the ambient space. Moreover, such a black hole is coupled to a nongravitational bath which is absorbing its Hawking radiation. This allows us to compute the entropy of the Hawking radiation by studying the bath using the quantum extremal surface prescription. The quantum extremal surface geometrizes into a Ryu-Takayanagi surface in the ambient space. The topological phase transition of the Ryu-Takayanagi surface in time from connecting different portions of the bath to the one connecting the bath and the brane gives the Page curve of the Hawking radiation that is consistent with unitarity. Nevertheless, there doesn't exit a derivation of the quantum extremal surface prescription and its geometrization in the Karch-Randall braneworld. In this paper, we fill this gap. We mainly focus on the case that the ambient space is (2+1)-dimensional for which explicit computations can be done in each description of the set-up. We show that the topological phase transition of the Ryu-Takayanagi surface corresponds to the formation of the replica wormhole on the Karch-Randall brane as the dominate contribution to the replica path integral. For higher dimensional situations, we show that the geometry of the brane satisfies Einstein's equation coupled with conformal matter. We comment on possible implications to the general rule of gravitational path integral from this equation.

Figures (19)

• Figure 1: a) The Penrose diagram of an eternal black hole in AdS$_{d}$ coupled to $d$-dimensional baths. We specify the two red vertical lines as the conformal boundary of the AdS$_{d}$ black hole. The orange arrows are the radiation coming in and out of the black hole. We choose time evolution as indicated in the diagram. We also specify two Cauchy slices of this time evolution as the two blue curves and on each of them we denote the subsystem $R=R_{I}\cup R_{II}$ in red. We emphasize that the Cauchy slices of this time evolution all go through the bifurcation horizon so they don't touch the black interior. b) We draw a putative configuration with entanglement island as the purple interval in the black hole spacetime. Its causal diamond overlaps the black hole interior.
• Figure 2: The demonstration of the Karch-Randall realization of the set-up in Fig.\ref{['pic:penroseoriginal']} on a constant time slice. For simplicity we only draw the configuration on the left hand side on the bulk eternal black hole. The dashed line is the horizon. The brane is the blue curve. The orange and the green curves are the two candidate entangling surfaces, that calculate $S_{\text{vN}}(R)=S_{\text{vN}}(R_{I}\cup R_{II})$, among which one goes through the bulk black hole interior connecting the boundaries of $R_{I}$ and $R_{II}$, and the other has two disconnected components one on each side of the bulk black hole and they end on the brane.
• Figure 3: The brane configuration in an empty AdS$_{3}$. The configuration is static so we just draw a constant time slice. The brane subtend an angle $\theta$ with the asymptotic boundary $z=0$. The gray region behind the brane (for which $x<0$) is cut off by the brane.
• Figure 4: a) Intermediate description: Gravitational AdS$_{2}$ coupled with a nongravitational (1+1)-dimensional half-Minkowski bath. b) Boundary description: a BCFT$_{2}$ in the ground state.
• Figure 5: The brane configuration in an eternal BTZ black hole background. The configuration is static so we just draw a constant time slice and for simplicity we only draw one exterior. The dashed line is the black hole horizon $z=z_{h}$. The gray region behind the brane is cut off by the brane.