Warped Information and Entanglement Islands in AdS/WCFT

TL;DR

This work studies Hawking radiation from the eternal BTZ black hole using a doubly holographic model with holographic WCFT$_2$ matter, implemented via the swing-surface entanglement entropy in AdS$_3$/WCFT$_2$. It finds entanglement islands existing at all times and a phase transition in the entanglement surface, but the resulting entropy curve is not a unitary Page curve, instead monotonically decreases at late times due to a field-theoretic IR divergence tied to non-unitarity and non-locality of WCFT$_2$. The phase transition time depends on an infrared regulator $l_\phi$, illustrating how non-local, non-unitary matter can modify island dynamics in ways not seen in unitary AdS/CFT models. These results deepen understanding of islands beyond unitary CFT contexts and suggest a broader role for non-locality in holographic entanglement phenomena.

Abstract

We use the notion of double holography to study Hawking radiation emitted by the eternal BTZ black hole in equilibrium with a thermal bath, but in the form of warped CFT$_2$ degrees of freedom. In agreement with the literature, we find entanglement islands and a phase transition in the entanglement surface, but our results differ significantly from work in AdS/CFT in three major ways: (1) the late-time entropy decreases in time, (2) island degrees of freedom exist at all times, and not just at late times, with the phase transition changing whether or not these degrees of freedom include the black hole interior, and (3) the physics involves a field-theoretic IR divergence, emerging when the boundary interval is too big relative to the black hole's inverse temperature. This behavior in the entropy appears to be consistent with the non-unitarity of holographic warped CFT$_2$ and demonstrates that the islands are not a phenomenon restricted to black hole information in unitary setups.

Figures (8)

• Figure 1: A sketch of the renormalized entanglement entropy of Hawking radiation emitted from an eternal AdS$_d$ black hole into a bath, as a function of time $t$. The orange curve $S_{HM}^{\text{ren}}(t)$ is the entropy of the Hartman-Maldacena surface, while the blue curve represents the maximum amount of information that can be emitted. The phase transition must occur by $t = t_p$, which is larger for more entropic black holes. This Page curve is only for $t \geq 0$, but we can also extend it to negative times. By time-reversal symmetry, the Page curve is actually symmetric about $t = 0$.
• Figure 2: A simplified sketch depicting a swing surface for a boundary interval $\mathcal{A}$ in a bulk space with time coordinate $t$ and radial coordinate $z$. The dashed lines $\gamma_{(L)}$ and $\gamma_{(R)}$ are the (null) ropes, while the solid line $\gamma$ is the (spacelike) bench.
• Figure 3: The Penrose diagram for the maximally-extended, two-sided BTZ, with one of the spatial dimensions ($\phi$ in AdS-Schwarzschild \ref{['btzAS']}) suppressed. Each gray patch is an exterior region, while the purple patches comprise the interior. This geometry exhibits a $\mathbb{Z}_2$ symmetry which proves useful in our calculations. Specifically, we only need to work within a single exterior region.
• Figure 4: The exterior of the planar BTZ at $t = t_0$ with the horizon at $z = z_h$ (in red) and a brane placed at $\phi = 0$ (in blue), on which the 3-dimensional geometry induces an AdS$_2$ black hole with the same horizon $z = z_h$. The Hawking radiation (depicted by a wavy arrow) is free to propagate between the AdS$_2$ black hole and the bath. We are interested in the swing surface corresponding to the radiation region (in green, but technically including another interval on the other side of the horizon).
• Figure 5: A sketch of the renormalized entanglement entropy of $\mathcal{R}(t_0)$, $S_{\mathcal{R}(t_0)}^{\text{ren}}$, as a function of time $t_0$, with $S_{HM}^{\text{ren}}(t_0)$ and $S_E^{\text{ren}}$ being the entropies of $A_{HM}^{\text{ren}}(t_0)$ and $A_E^{\text{ren}}$, respectively. The entropy starts at some constant value, then falls after the phase transition in which the entanglement surface becomes the Hartman-Maldacena swing, reaching $0$ at $\tau$.