Abstract
In this paper, we study the fine structure of entanglement in holographic two-dimensional boundary conformal field theories (BCFT) in terms of the spatially resolved quasilocal extension of entanglement entropy — entanglement contour. We find that the boundary induces discontinuities in the contour revealing hidden localization-delocalization patterns of the entanglement degrees of freedom. Moreover, we observe the formation of “islands” where the entanglement contour vanishes identically implying that these regions do not contribute to the entanglement at all. We argue that these phenomena are the manifestation of the entanglement islands recently discussed in the literature. We apply the entanglement contour proposal to the recently discussed BCFT black hole models reproducing the Page curve — moving mirror model and the pair of BCFT in the thermofield double state. From the viewpoint of entanglement contour, the Page curve also carries the imprint of strong delocalization caused by dynamical entanglement islands.
Highlights
JHEP03(2022)033 we would like to investigate how the entanglement contour resolves the Page curve in certain examples of two different black hole boundary conformal field theories (BCFT) models proposed recently [32, 34–36]
In this paper, we study the fine structure of entanglement in holographic twodimensional boundary conformal field theories (BCFT) in terms of the spatially resolved quasilocal extension of entanglement entropy — entanglement contour
The second model we study is the pair of BCFT in the thermofield double (TFD BCFT model for short) argued to have the properies of a black hole which is in equilibrium with the Hawking radiation [32, 34]
Summary
The entanglement contour of a subsystem A is a function fA(x) defined as. In [23] the proposal for a contour function in terms of the partial entanglement entropy has been given. For the entanglement entropy of a single interval (x1, x2) given by some function S(x1, x2) in 1+1 dimensional theory with the spatial direction x after taking the size limit A2 → 0 it is straightforward to obtain the entanglement contour [25] of this interval in the form fA(x) = 1 ∂S (x1, x) − ∂S (x, x2). S(x1, x2) = c log sinh(πT (x2 − x1)) , πT ε πcT fA(x) = 6 (coth (πT (x − x1)) + coth (πT (x2 − x))) These contour functions diverge near the endpoints and take its minimum in the center of the entangling interval. The divergent term comes from the entanglement between infinite number of degrees of freedom across the junction of interval and its complement
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