Quantum Extremal Islands Made Easy, Part I: Entanglement on the Brane

TL;DR

The paper presents a higher-dimensional holographic framework in which quantum extremal islands arise from standard Ryu–Takayanagi surfaces crossing a Randall–Sundrum brane, thereby generalizing the island paradigm beyond black holes. It analyzes the brane-induced gravity on AdS_d, including DGP corrections and a special treatment of two dimensions, and develops three complementary viewpoints (bulk/brane/boundary) that consistently reproduce island physics. By computing holographic entanglement entropy for regions on a boundary CFT with a defect, the authors show how island contributions emerge as brane portions of RT surfaces and match Wald-Dong entropy on the brane. The results indicate that islands are a universal feature of effective gravitational theories and remain robust across dimensions, provided suitable brane couplings are chosen; the work also elucidates the relationship between gravitational entropy on the brane and holographic entanglement entropy in the bulk/boundary pictures.

Abstract

Recent progress in our understanding of the black hole information paradox has lead to a new prescription for calculating entanglement entropies, which involves special subsystems in regions where gravity is dynamical, called \textit{quantum extremal islands}. We present a simple holographic framework where the emergence of quantum extremal islands can be understood in terms of the standard Ryu-Takayanagi prescription, used for calculating entanglement entropies in the boundary theory. Our setup describes a $d$-dimensional boundary CFT coupled to a ($d$-1)-dimensional defect, which are dual to global AdS${}_{d+1}$ containing a codimension-one brane. Through the Randall-Sundrum mechanism, graviton modes become localized at the brane, and in a certain parameter regime, an effective description of the brane is given by Einstein gravity on an AdS${}_d$ background coupled to two copies of the boundary CFT. Within this effective description, the standard RT formula implies the existence of quantum extremal islands in the gravitating region, whenever the RT surface crosses the brane. This indicates that islands are a universal feature of effective theories of gravity and need not be tied to the presence of black holes.

Figures (17)

• Figure 1: A sketch of our holographic setup illustrating the various elements appearing in eq. \ref{['eq:sad0']}, which manifests the island rule in our analysis.
• Figure 2: Panel (a): Our Randall-Sundrum construction involves foliating with AdS$_d$ slices. Then identical portions of two such AdS$_{d+1}$ geometries are glued together along an common AdS$_d$ slice. Panel (b): The jump in the extrinsic curvature across the interface between the two geometries is supported by a(n infinitely) thin brane. The brane is represented by a green line in the figures and the bulk AdS$_{d+1}$ spacetime is blue with a $d$-dimensional CFT at the asymptotic boundary.
• Figure 3: This figure shows the relation between a time-slice in our construction and the holographic setup of Almheiri:2019hni. The top row illustrates three perspectives with which the system discussed here can be described, while the bottom row displays the analogous descriptions for the model in Almheiri:2019hni. The comparison can be made more precise by performing a $\mathbb Z_2$ orbifold quotient across the bulk brane/conformal defect in the top row. a. Bulk gravity perspective, with an asymptotically AdS$_{d+1}$ space (shaded blue) which contains a co-dimension one Randall-Sundrum brane (shaded grey). b. Brane perspective, with dual CFT$_d$ on the asymptotic boundary geometry (blue) and also extending on the AdS$_d$ region (shaded green) where gravity is dynamical. c. Boundary perspective, with the holographic CFT$_d$ on $S^{d-1}$ (blue) coupled to a codimension-one conformal defect (green). d. AdS$_3$ formulation with two boundary components: the flat asymptotic boundary (straight black line) and a "Planck brane" (curved black line) with an AdS$_2$ geometry. e. The holographic CFT extends over a region with a fixed metric (blue) and an AdS$_2$ region with JT gravity (green). f. The microscopic description as a two-dimensional BCFT (blue) coupled to a quantum mechanical system at its boundary (green).
• Figure 4: This figure illustrates the spatial profile of the first few normalized graviton modes in the presence of a large tension brane, and a $\mathbb Z_2$ orbifolding across the brane. We use the spatial coordinate $\mu$, related to $\rho$ in eq. (\ref{['metric']}) by $\cot \mu = \sinh\rho/L$. The tension is adjusted such that the location of the brane is at $\mu = \mu_\textrm{\tiny B}$ with $\mu_\textrm{\tiny B}\lesssim\pi$. As discussed in the main text, the presence of the brane creates new bulk modes (orange), which are highly localized at the brane, and which play the role of a (nearly massless) graviton on the brane. The remaining bulk modes appear as KK modes in the brane theory.
• Figure 5: A timeslice of our CFT setup. A conformal defect running along the equator separates the two halves of R and its corresponding engangling surface $\Sigma_\textrm{\tiny CFT}$.