BCFT entanglement entropy at large central charge and the black hole interior

TL;DR

The paper shows that entanglement and Rényi entropies in BCFTs and CFTs prepared by Euclidean BCFT path integrals reproduce holographic RT phase transitions under large central charge and vacuum-block dominance. By computing twist-operator correlators in the replicated BCFT and analyzing bulk versus boundary exchanges, the authors derive the connected/disconnected entropy phases and link them to whether the bulk entanglement wedge includes the black hole interior. They establish explicit spectral and OPE constraints on holographic BCFTs to justify vacuum-block dominance, extend the analysis to multiple intervals, and relate the BCFT results to gravity-side replica wormholes. The work strengthens the AdS/BCFT dictionary, clarifies microscopic criteria for interior reconstruction and Page-like behavior, and connects black hole physics to BCFT bootstrap data and monodromy techniques.

Abstract

In this note, we consider entanglement and Renyi entropies for spatial subsystems of a boundary conformal field theory (BCFT) or of a CFT in a state constructed using a Euclidean BCFT path integral. Holographic calculations suggest that these entropies undergo phase transitions as a function of time or parameters describing the subsystem; these arise from a change in topology of the RT surface. In recent applications to black hole physics, such transitions have been seen to govern whether or not the bulk entanglement wedge of a (B)CFT region includes a portion of the black hole interior and have played a crucial role in understanding the semiclassical origin of the Page curve for evaporating black holes. In this paper, we reproduce these holographic results via direct (B)CFT calculations. Using the replica method, the entropies are related to correlation functions of twist operators in a Euclidean BCFT. These correlations functions can be expanded in various channels involving intermediate bulk or boundary operators. Under certain sparseness conditions on the spectrum and OPE coefficients of bulk and boundary operators, we show that the twist correlators are dominated by the vacuum block in a single channel, with the relevant channel depending on the position of the twists. These transitions between channels lead to the holographically observed phase transitions in entropies.

Figures (11)

• Figure 1: Top: transition in the RT surface from connected to disconnected topology; in black hole applications, the latter is associated with an entanglement wedge that includes the black hole interior. Bottom: BCFT interpretation in terms of the two-point function of twist operators used to compute entanglement entropy via the replica method. Phase transition comes from a switch of dominance between the identity block in a bulk channel to the identity block in a boundary channel.
• Figure 2: Holographic calculation of entanglement entropy for an interval $A$ containing the boundary. The RT surface $\tilde{A}$ sits at a fixed location on the AdS$_2$ fibers of the dual geometry. Here $A$ is homologous to $\tilde{A}$ since the ETW brane represents a smooth part of full microscopic geometry.
• Figure 3: Holographic calculation of entanglement entropy for an interval $A$ away from the boundary. The RT surface has two possible topologies, a connected (solid curve) and disconnected (dashed curves).
• Figure 4: The dual geometry for $|b,\tau_0\rangle$ for sufficiently small $\tau_0$ is a portion of the maximally-extended AdS-Schwarzchild geometry, cut off by a spherically symmetric ETW brane. The pictures on the right show the spatial slice at $t=0$ and the connected and disconnected topologies for the RT surface corresponding to a large interval on the boundary circle.
• Figure 5: Dual geometry to the thermofield double state of two BCFTs, showing two possible topologies for the RT surface for the region corresponding to the union of points in either CFT at a distance larger than $x_0$ from the boundary.