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arXiv:2402.15836 · Journal of High Energy Physics

Entanglement islands and cutoff branes from path-integral optimization

Ashish Chandra, Zhengjiang Li, Qiang Wen

Published Jul 9, 2024• 6 citations
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Abstract

A bstract Recently it was proposed that, the AdS/BCFT correspondence can be simulated by a holographic Weyl transformed CFT 2 , where the cut-off brane plays the role of the Karch-Randall (KR) brane [1]. In this paper, we focus on the Weyl transformation that optimizes the path integral computation of the reduced density matrix for a single interval in a holographic CFT 2 . When we take the limit that one of the endpoint of the interval goes to infinity (a half line), such a holographic Weyl transformed CFT 2 matches the AdS/BCFT configuration for a BCFT with one boundary. Without taking the limit, the induced cutoff brane becomes a circle passing through the two endpoints of the interval. We assume that the cutoff brane also plays the same role as the KR brane in AdS/BCFT, hence the path-integral-optimized purification for the interval is in the island phase. This explains the appearance of negative mutual information observed in [2]. We check that, the entanglement entropy and the balanced partial entanglement entropy (BPE) calculated via the island formulas, exactly match with the RT formula and the entanglement wedge cross-section (EWCS), which are allowed to anchor on the cutoff brane.

DOI
10.1007/jhep07(2024)069

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A bstract Recently it was proposed that, the AdS/BCFT correspondence can be simulated by a holographic Weyl transformed CFT 2 , where the cut-off brane plays the role of the Karch-Randall (KR) brane [1]. In this paper, we focus on the Weyl transformation that optimizes the path integral computation of the reduced density matrix for a single interval in a holographic CFT 2 . When we take the limit that one of the endpoint of the interval goes to infinity (a half line), such a holographic Weyl transformed CFT 2 matches the AdS/BCFT configuration for a BCFT with one boundary. Without taking the limit, the induced cutoff brane becomes a circle passing through the two endpoints of the interval. We assume that the cutoff brane also plays the same role as the KR brane in AdS/BCFT, hence the path-integral-optimized purification for the interval is in the island phase. This explains the appearance of negative mutual information observed in [2]. We check that, the entanglement entropy and the balanced partial entanglement entropy (BPE) calculated via the island formulas, exactly match with the RT formula and the entanglement wedge cross-section (EWCS), which are allowed to anchor on the cutoff brane.
A bstract We obtain the reflected entropy for bipartite mixed state configurations of two adjacent and disjoint intervals at a finite temperature in BCFT 2 s with two distinct boundaries through a replica technique in the large central charge limit. Subsequently these field theory results are reproduced from bulk computations involving the entanglement wedge cross section in the dual BTZ black hole geometry truncated by two Karch-Randall branes. Our result confirms the holographic duality between the reflected entropy and the bulk entanglement wedge cross section in the context of the AdS 3 / BCFT 2 scenario. We further investigate the critical issue of the holographic Markov gap between the reflected entropy and the mutual information for these configurations from the bulk braneworld geometry and study its variation with subsystem sizes and time.
In the context of the AdS/CFT, it was recently proposed that the boundary partial entanglement entropy structure can be represented by the so-called partial entanglement entropy (PEE) threads in the AdS bulk, which are bulk geodesics with the density determined by the boundary PEE structure [. and .]. In Poincaré anti–de Sitter (AdS) space, it was shown that the PEE threads cover the AdS space uniformly, such that the number of intersections between any bulk surface and the bulk PEE threads is always given by the area of the surface divided by 4 G . In this paper, we investigate the configurations of PEE threads when the boundary state is in the island phase. The island phase was studied in the context of the holographic Weyl transformed CFT 2 , which has been shown to capture all the main features of AdS/BCFT (Boundary Conformal Field Theory). Compared with AdS 3 / CFT 2 , in the island phase instead of modifying the distribution of the bulk PEE threads, we should replace the boundary points with the corresponding cutoff spheres. Then the two-point functions and the four-point functions of twist operators can be reproduced by identifying the bulk homologous surfaces anchored on the corresponding cutoff spheres that has the minimal number of intersections with the bulk PEE threads. This gives us a better understanding about the PEE structure in the island phase and reproduces the island formula for entanglement entropy by allowing homologous surfaces to anchor on any cutoff spheres. Furthermore, it gives a demonstration for the two basic proposals and a better understanding for the entanglement contribution that makes the foundation to compute the balanced partial entanglement entropy [.], which reproduces the entanglement wedge cross section in the island phase.
A bstract We explore constraints that arise from associating an entanglement wedge (EW) to subregions of a cutoff boundary at a finite distance in AdS/CFT, using a subcritical end-of-the-world (ETW) brane acting as a cutoff. In particular, we consider the case of two intervals in the holographic dual to a BCFT, with one interval A located at the asymptotic boundary and the second interval B located on the ETW brane. We discuss in detail subtleties that arise near the RT end-points when defining the EW for this configuration, particularly in the connected phase. Entanglement wedge nesting (EWN) requires that WE(A) {\mathcal{W}}_E(A) WE​(A) W E A ∪ WE(B) {\mathcal{W}}_E(B) WE​(B) W E B ⊆ WE(A∪B) {\mathcal{W}}_E\left(A\cup B\right) WE​(A∪B) W E A ∪ B . We demonstrate that already in the simplest example of an AdS 3 bulk geometry, EWN can be violated even if A and B are spacelike separated through the bulk and instead we must require the stronger condition that WE(A) {\mathcal{W}}_E(A) WE​(A) W E A be spacelike separated from WE(B) {\mathcal{W}}_E(B) WE​(B) W E B , which highlights the non-local nature of the cutoff theory. Our prescription to associate EWs to subregions on the ETW brane is different from the restricted maximin procedure [1] but will agree within the subset of parameter space where EWN is respected. Additionally, we study EWN in a two sided BTZ black hole geometry with an ETW brane in one of the exteriors. In the BTZ black hole example we find that our condition for EWN disallows configurations where the RT surface goes from the brane to the black hole singularity.
In this study, we calculate the m − 1 correction to the reflected entropy for two adjacent intervals on a half-infinite line within the AdS 3 / BCFT 2 framework, where m is a Renyi index for a canonical purification. We utilize the doubling trick and compute the leading terms in the large central charge expansion of correlation functions in the holographic boundary conformal field theory (BCFT). In the corresponding anti–de Sitter (AdS) space with an end of the world brane, we analyze the entanglement wedge cross section, the dual counterpart of reflected entropy. This AdS/BCFT setup allows us to explore a richer set of phases in the entanglement wedge cross section. The m − 1 correction in the holographic BCFT manifests as modifications in the entanglement wedge cross section induced by cosmic branes. For the adjacent intervals anchored to the boundary of BCFT, we show the duality between the entanglement wedge cross section with the backreaction from a cosmic brane and Renyi reflected entropy at all orders in m − 1 . Furthermore, by analyzing the entanglement wedge cross section for general adjacent intervals, we provide guidance for an ε expansion of five-point functions in the holographic conformal field theory (CFT), where ε is the rescaled conformal dimension by the central charge.
We study two-interval holographic entanglement entropy and entanglement wedge cross section in cutoff AdS. In particular, we investigate phase transitions of them. For two-interval entanglement entropy, the transition point monotonically decreases with a deformation parameter, which means that by the TT deformation the degrees of freedom in subsystems are decreasing. This implies that the effect of the TT deformation can be regarded as the rescaling of the energy scale. We also study entanglement wedge cross section in cutoff AdS, and our result implies that for the entanglement of purification in the TT deformed CFTs phase transition could occur even for fixed subsystems.
The entanglement wedge cross section in holographic picture provides a geometrical description of the entanglement for mixed states. In this paper we study the inequalities for the entanglement wedge cross section in AdS/BCFT duality. In the presence of the boundary in conformal field theory (CFT), the dual entanglement wedge cross section exhibits abundant phase structures since the extremal surface may end on the brane. We present a universal treatment which is applicable for all the possible phases such that the inequalities for the entanglement wedge cross section can be proved in an algebraic manner rather than a diagrammatic manner. We show that the entanglement wedge cross section in AdS/BCFT satisfies the same inequalities as in AdS/CFT.