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.
Highlights
For holographic examples of these systems, calculations making use of the Ryu-Takayanagi (RT) formula [1] suggest that the entanglement entropy can undergo phase transitions related to a change in topology of the RT surface
We show that the assumption of vacuum block dominance for boundary conformal field theory (BCFT) correlators of twist operators is equivalent to a simple holographic prescription [2, 3] for the duals of BCFTs where the CFT boundary extends into the bulk as a purely gravitational end-of-the-world (ETW) brane with tension related to the boundary entropy of the BCFT.1
Starting with the vacuum state of a 1+1-dimensional CFT, the geometry of a putative bulk dual is fixed by symmetry to be AdS3, up to internal dimensions
Summary
We start with a brief review of boundary conformal field theories. Given a CFT, we can define the theory on a manifold with boundary by making a choice of boundary conditions for the fields, and possibly adding boundary degrees of freedom coupled to the bulk CFT fields. A consequence is that the kinematics (i.e. the functional form of correlators given the operator dimensions) of the BCFT in the UHP is directly related to that of a chiral CFT on the whole plane. Of bulk CFT operators Ohkhk with conformal weights (hk, hk) in the UHP are constrained to have the same functional form as chiral CFT correlators. Of bulk and boundary primary operators is constrained to have the functional form of a chiral three-point function. We discuss the general case in detail in appendix A. of scalar operators with dimension ∆ has the same functional form as a four-point function of chiral operators. The equivalence of the expressions (2.21) and (2.22) is a BCFT version of the usual crossing symmetry constraints; in this case, we have a relation between bulk OPE coefficients and boundary operator expansion coefficients.
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