Unitarity of Entanglement and Islands in Two-Sided Janus Black Holes

TL;DR

This work analyzes entanglement evolution in two-sided Janus black holes that embed in string theory, showing that the Page curve is recovered unitarily via bulk geodesics and two complementary pictures: an ICFT perspective and a 2d gravity theory on branes (shadow region). It identifies island regions in the radiation entanglement wedge and demonstrates consistency between holographic entanglement entropy and quantum-extremization approaches, including a transitional Page curve before $t_P$ driven by emergent brane matter. A large deformation limit reveals an effective separation into Grav$_+$ and Grav$_-$ sectors, while ICFT computations capture interface-induced entanglement corrections and boundary operator dynamics. The results provide a concrete realization of double holography with higher-dimensional gravity, clarifying when islands arise naturally and how new matter on branes reshapes the Page curve in a controlled, string-theory–consistent setting.

Abstract

We explore the entanglement evolution of boundary intervals in eternal Janus black holes that can be embedded consistently into string theory in the low-energy limit. By studying the geodesics we show that there is a transition in the entanglement characteristic around the Page time, which manifests the unitarity of the evolution. We reproduce and reinterpret these bulk results from two different lower-dimensional perspectives: first as an interface CFT in the usual AdS/CFT correspondence and second as an effective gravity theory in one lower dimension coupled to a radiation background. In the limit where the number of interface degrees of freedom becomes large, we obtain an effective theory on appropriate branes that replace the deep interior region in the bulk, coined the shadow region. In this effective theory, we also identify the island of the radiation entanglement wedge and verify the newly proposed quantum extremization method. Our model clarifies that double holography with gravity in two higher dimensions can be realized in a concrete and consistent way and that the occurrence of islands is natural in one higher dimension. Furthermore, our model reveals that there can be a transitional behavior of the Page curve before the Page time, which is related to the emergence of new matter degrees of freedom on the branes.

Figures (13)

• Figure 1: We draw the constant $t$ section of the BTZ spacetime where we show $(\mu, w)$ together with $(r,x)$ coordinates. The middle line with $w=1$ and $r=L$ represents the horizon. The red lines are representing constant $\mu$ surfaces whereas the blue lines are constant $w$ surfaces. The top/ bottom line represents the spatial direction of the R/L boundary respectively.
• Figure 2: We draw the constant $t$ section of the two-sided Janus black hole where we show $(\mu, w)$ together with $(r,x)$ coordinates. The middle line with $w=1$ represents the horizon. The red lines are representing constant $\mu$ surfaces whereas the blue lines are constant $w$ surfaces. The $\mu$ coordinate is ranged over $[-\mu_0,\mu_0]$ with $\mu_0 > \pi/2$. This leads to the $x=0$ angled-joints of the R-L boundaries.
• Figure 3: We draw ICFT$\, \times \,$ICFT living on the L and R boundaries of our two-sided Janus black hole. Each ICFT consists of three components of CFT$_- \times {\rm QM}_0 \times {\rm CFT}_+$, which preserves 1d conformal symmetries of $SO(1,2)$.
• Figure 4: We draw the shadow region specified by $\mu = constant$ slices ranged over $\mu \in [-\mu_I, \mu_I]$. One may integrate out the bulk degrees of freedom in this shadow region and view the resulting 2d gravity theories as living on the two slices at $\mu=\pm \mu_I$, respectively.
• Figure 5: We have depicted the RL geodesic at $\tau=0$ and at a later time $\tau >0$ with a shadow region. Though we have depicted a constant $\tau$ slice, the curve $\mu=\pm\mu_{I}$ takes nearly the same form as in Figure \ref{['fig2']}.