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arXiv:2105.09106 · Journal of High Energy Physics

Page curve from defect extremal surface and island in higher dimensions

Jinwei Chu, Feiyu Deng, Yang Zhou

Published Oct 18, 2021• 65 citations
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Abstract

Abstract Defect extremal surface (DES) is defined by minimizing the Ryu-Takayanagi surface corrected by the quantum theory localized on the defect, which is useful when the RT surface crosses or terminates on the defect. Based on the decomposition procedure of an AdS bulk with a defect brane, proposed in [69], we derive Page curve in a time dependent set up of AdS3/BCFT2, and find that the result from island formula agrees with defect extremal surface formula precisely. We then extend the study to higher dimensions and find that the entropy computed from bulk defect extremal surface is generally less than that from island formula in boundary low energy effective theory, which implies that the UV completion of island formula gives a smaller entropy.

DOI
10.1007/jhep10(2021)149

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Abstract Defect extremal surface (DES) is defined by minimizing the Ryu-Takayanagi surface corrected by the quantum theory localized on the defect, which is useful when the RT surface crosses or terminates on the defect. Based on the decomposition procedure of an AdS bulk with a defect brane, proposed in [69], we derive Page curve in a time dependent set up of AdS3/BCFT2, and find that the result from island formula agrees with defect extremal surface formula precisely. We then extend the study to higher dimensions and find that the entropy computed from bulk defect extremal surface is generally less than that from island formula in boundary low energy effective theory, which implies that the UV completion of island formula gives a smaller entropy.
A bstract Defect extremal surface is defined by extremizing the Ryu-Takayanagi formula corrected by the quantum defect theory. This is interesting when the AdS bulk contains a defect brane (or string). We introduce a defect extremal surface formula for reflected entropy, which is a mixed state generalization of entanglement entropy measure. Based on a decomposition procedure of an AdS bulk with a brane, we demonstrate the equivalence between defect extremal surface formula and island formula for reflected entropy in AdS 3 /BCFT 2 . We also compute the evolution of reflected entropy in evaporating black hole model and find that defect extremal surface formula agrees with island formula.
We propose a doubly holographic version of the semiclassical island formula for the entanglement negativity in the framework of the defect AdS/BCFT correspondence where the anti--de Sitter (AdS) bulk contains a defect conformal matter theory. In this context, we propose a defect extremal surface (DES) formula for computing the entanglement negativity modified by the contribution from the defect matter theory on the end-of-the-world brane. The equivalence of the DES proposal and the semiclassical island formula for the entanglement negativity is demonstrated in AdS3/BCFT2{\mathrm{AdS}}_{3}/{\mathrm{BCFT}}_{2}AdS3​/BCFT2​ framework. Furthermore, in the time-dependent AdS3/BCFT2{\mathrm{AdS}}_{3}/{\mathrm{BCFT}}_{2}AdS3​/BCFT2​ scenarios involving eternal black holes in the lower dimensional effective description, we investigate the time evolution of the entanglement negativity through the DES and the island formulas and obtain the analogs of the Page curves.
A bstract The Ryu-Takayanagi formula relates entanglement entropy in a field theory to the area of extremal surfaces anchored to the boundary of a dual AdS space. It is interesting to ask if there is also an information theoretic interpretation of the areas of non-extremal surfaces that are not necessarily boundary-anchored. In general, the physics outside such surfaces is associated to observers restricted to a time-strip in the dual boundary field theory. When the latter is two-dimensional, it is known that the differential entropy associated to the strip computes the length of the dual bulk curve, and has an interpretation in terms of the information cost in Bell pairs of restoring correlations inaccessible to observers in the strip. A general realization of this formalism in higher dimensions is unknown. We first prove a no-go theorem eliminating candidate expressions for higher dimensional differential entropy based on entropic c-theorems. Then we propose a new formula in terms of an integral of shape derivatives of the entanglement entropy of ball shaped regions. Our proposal stems from the physical requirement that differential entropy must be locally finite and conformally invariant. Demanding cancelation of the well-known UV divergences of entanglement entropy in field theory guides us to our conjecture, which we test for surfaces in AdS 4 . Our results suggest a candidate c-function for field theories in arbitrary dimensions.
A bstract We investigate the mixed state entanglement structure through the reflected entropy for disjoint radiation subsystems coupled to a 2d eternal brane world black hole in a time dependent defect AdS 3 /BCFT 2 scenario. Utilizing the island prescription and the defect extremal surface (DES) formula, we demonstrate a complex mixed state entanglement structure through the reflected entropy corresponding to distinct entanglement entropy phases. In each case we verify the holographic duality of the reflected entropy with the bulk entanglement wedge cross section (EWCS) and also obtain the Page curves for both the entanglement entropy and the associated reflected entropy phases. Furthermore, we extend our analysis to adjacent radiation subsystems and obtain consistent results using both the island and the DES prescription.
Abstract The Ryu-Takayanagi formula relates entanglement entropy in a field theory to the area of extremal surfaces anchored to the boundary of a dual AdS space. It is interesting to ask if there is also an information theoretic interpretation of the areas of non-extremal surfaces that are not necessarily boundary-anchored. In general, the physics outside such surfaces is associated to observers restricted to a time-strip in the dual boundary field theory. When the latter is two-dimensional, it is known that the differential entropy associated to the strip computes the length of the dual bulk curve, and has an interpretation in terms of the information cost in Bell pairs of restoring correlations inaccessible to observers in the strip. A general realization of this formalism in higher dimensions is unknown. We first prove a no-go theorem eliminating candidate expressions for higher dimensional differential entropy based on entropic c-theorems. Then we propose a new formula in terms of an integral of shape derivatives of the entanglement entropy of ball shaped regions. Our proposal stems from the physical requirement that differential entropy must be locally finite and conformally invariant. Demanding cancelation of the well-known UV divergences of entanglement entropy in field theory guides us to our conjecture, which we test for surfaces in AdS4. Our results suggest a candidate c-function for field theories in arbitrary dimensions.
We study entanglement negativity for evaporating black hole based on the holographic model with defect brane. We introduce a defect extremal surface formula for entanglement negativity. Based on partial reduction, we show the equivalence between defect extremal surface formula and island formula for entanglement negativity in AdS _3 3 /BCFT _2 2 . Extending the study to the model of eternal black hole plus CFT bath, we find that black hole-black hole negativity decreases until vanishing, left black hole-left radiation negativity is always a constant, radiation-radiation negativity increases and then saturates at a time later than Page time. In all the time dependent cases, defect extremal surface formula agrees with island formula.