Entanglement, defects, and $T\bar{T}$ on a black hole background

TL;DR

This work tests the proposed duality between the island prescription and the defect extremal surface (DES) prescription in a KR braneworld setup with a radiation bath on an AdS2 black hole background, using an AdS3 black string as the bulk. It computes the fine-grained entanglement entropy for radiation subsystems via both island and DES formalisms and demonstrates their agreement for both island and no-island phases in the undeformed case; it then extends the analysis to TTbar-deformed CFTs, obtaining first-order corrections that maintain the island–DES agreement. The paper derives explicit expressions for island and DES entropies in both the boundary and bulk channels, including the defect contributions on the brane, and analyzes the Page curves, showing deformation-dependent shifts in the Page time and island entropy. The results bolster the robustness of the island framework and its bulk-boundary DES dual in nontrivial backgrounds and clarify how TTbar deformation modifies entanglement structure and Page-time behavior. The findings have implications for understanding information recovery in holographic models with defects and finite-cutoff boundaries, and they point to avenues for exploring higher-dimensional generalizations and other deformations.

Abstract

In this article, we investigate the proposed duality between the island and the defect extremal surface (DES) prescriptions using the fine-grained entanglement entropy in Karch-Randall (KR) brane-world models with gravitating radiation baths. We consider the AdS$_3$ black string geometry and compute the entanglement entropy for radiation subsystems on an AdS$_2$ eternal black hole background using both the island and the DES prescriptions. We find an agreement between the two proposals for the island and the no-island phases, thus verifying the validity of the proposed duality. We further extend to a $T\bar{T}$ deformed AdS$_3$ black string geometry with a cut-off and find consistent results for both phases. We finally plot and compare the Page curves for the undeformed and deformed scenarios, and discuss the modifications due to $T\bar{T}$ deformation.

Figures (8)

• Figure 1: Holographic dual of a BCFT$_2$ defined on a half plane $(x > 0)$. The EOW brane is considered to be at a constant hyperbolic angle $\rho_0$. (Figure modified from Deng:2020ent.)
• Figure 2: The Island phase of the entanglement entropy. The radiation subsystem $A$ is denoted by the yellow subregions on the asymptotic boundary, while the corresponding entanglement entropy island $I_S(A)$ are illustrated by the red regions on the EOW brane. The green curves represent the RT surfaces.
• Figure 3: The No-Island phase of the entanglement entropy. The radiation subsystem is once again denoted by the yellow subregions on the asymptotic boundary. The RT surface in this scenario is a Hartman-Maldacena type surface, and is represented by the green curve.
• Figure 4: The Island phase of the entanglement entropy in the presence of $T\bar{T}$ deformation. The radiation subsystem $A$ is denoted by the yellow subregions on a cut-off boundary at $\rho_c$, while the corresponding entanglement entropy island $I_S(A)$ are illustrated by the red regions on the EOW brane. The green curves represent the RT surfaces.
• Figure 5: The No-Island phase of the entanglement entropy in the presence of $T\bar{T}$ deformation. The radiation subsystem is once again denoted by the yellow subregions on a cut-off boundary at $\rho_c$. The RT surface in this scenario is a Hartman-Maldacena type surface, and is represented by the green curve.