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.
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 algeb…
The relation between kinematic space metric and entanglement entropy provides us with a differential equation for entanglement entropy. For BCFT on upper half plane we solve this equation to obtain an expression for entanglement entropy consistent with known results in the literature. We also discuss how this relation can be used to recast the RG flow, under relevant deformations of a CFT, as a flow in the space of kinematic space metrics.
We study entanglement entropy inequalities in boundary conformal field theory (BCFT) by holographic correspondence. By carefully classifying all the configurations for different phases, we prove the strong subadditiviy and the monogamy of mutual information for holographic entanglement entropy in BCFT at both zero and finite temperatures
We study the entanglement phase structure of a holographic boundary conformal field theory (BCFT) in a two-dimensional black hole background. The bulk dual is the AdS$_3$ black string geometry with a Karch-Randall brane. We compute the subregion entanglement entropy of various two-sided bipartitions to elucidate the phase space where a Page curve exists in this setup. We do fully analytical computations on both the gravity side and the field theory side and demonstrate that the results precisely match. We discuss the entanglement phase structure describing where a Page curve exists in this geo…
We compute holographic entanglement entropy for subregions of a BCFT thermal state living on a nongravitating black hole background. The system we consider is doubly holographic and dual to an eternal black string with an embedded Karch-Randall brane that is parameterized by its angle. Entanglement islands are conventionally expected to emerge at late times to preserve unitarity at finite temperature, but recent calculations at zero temperature have shown such islands do not exist when the brane lies below a critical angle. When working at finite temperature in the context of a black string, w…
In this paper we review the AdS/BCFT proposal of T. Takayanagi for holographic description of systems with boundaries, in particular, boundary conformal field theories (BCFTs). Motivated by better understanding of the proposed duality we employ entanglement entropy as a probe of familiar properties of impurities and defects. Using the dual gravity description, we check that in two spacetime dimensions the impurity entropy does not depend on a particular state of the theory, which is a well-known CFT result. In three dimensions different, and not necessarily equivalent, definitions of the defec…
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 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 …
We propose a new holographic dual of conformal field theory defined on a manifold with boundaries, i.e. BCFT. Our proposal can apply to general boundaries and agrees with arXiv:1105.5165 for the special case of a disk and half plane. Using the new proposal of AdS/BCFT, we successfully obtain the expected boundary Weyl anomaly and the obtained boundary central charges satisfy naturally a c-like theorem holographically. We also investigate the holographic entanglement entropy of BCFT and find that the minimal surface must be normal to the bulk spacetime boundaries when they intersect. Interestin…
We study the AdS/BCFT duality between two-dimensional conformal field theories with two boundaries and three-dimensional anti-de Sitter space with two Karch-Randall branes. We compute the entanglement entropy of a bipartition of the BCFT, on both the gravity side and the field theory side. At finite temperature this entanglement entropy characterizes the communication between two braneworld black holes, coupled to each other through a common bath. We find a Page curve consistent with unitarity. The gravitational result, computed using double-holographically realized quantum extremal surfaces, …
We study the partial entanglement entropy (PEE) aspects of the holographic BCFT setup with an entanglement island, inspired by the holographic triality of the AdS/BCFT setup developed in the recent study on the black hole information problem, and the "PEE=CFF (component flow flux)" prescription, which is proposed recently to investigate the holographic PEE in the framework of bit thread formulation. Our study provides a bit thread description of the AdS/BCFT setup, which characterizes the specific entanglement details between the different parts of the system with an entanglement island, and m…
In this paper we provide a Chern-Simons gravity dual of a two dimensional conformal field theory on a manifold with boundaries, so called boundary conformal field theory (BCFT). We determine the correct boundary action on the end of the world brane in the Chern-Simons gauge theory. This reproduces known results of the AdS/BCFT for the Einstein gravity. We also give a prescription of calculating holographic entanglement entropy by employing Wilson lines which extend from the AdS boundary to the end of the world brane. We also discuss a higher spin extension of our formulation.
We study the conductivities and entanglement structures of two different holographic gapped systems at zero density in the presence of boundaries within AdS/BCFT. The first gapped system is described by the Einstein-scalar gravity and the second one is the dual of AdS soliton geometry. We show that in both these two systems the bulk and boundary conductivities along the spatial direction of the boundary of BCFT are trivial. For the first system, when we increase the size of the subsystem the renormalized entanglement entropy is always non-negative and monotonically decreasing with discontinuou…
We study the holographic entanglement entropy of spatial regions with corners in the AdS4/BCFT3 correspondence by considering three dimensional boundary conformal field theories whose boundary is a timelike plane. We compute analytically the corner function corresponding to an infinite wedge having one edge on the boundary. A relation between this corner function and the holographic one point function of the stress tensor is observed. An analytic expression for the corner function of an infinite wedge having only its tip on the boundary is also provided. This formula requires to find the globa…
This paper is an extended version of our short letter on a new proposal for holographic boundary conformal field, i.e., BCFT. By using the Penrose-Brown-Henneaux (PBH) transformation, we successfully obtain the expected boundary Weyl anomaly. The obtained boundary central charges satisfy naturally a c-like theorem holographically. We then develop an approach of holographic renormalization for BCFT, and reproduce the correct boundary Weyl anomaly. This provides a non-trivial check of our proposal. We also investigate the holographic entanglement entropy of BCFT and find that our proposal gives …
In this note, we study the first law of entanglement in a boundary conformal field theory (BCFT) dual to warped AdS cut off by a brane. Exploiting the symmetry of boundary-centered half-balls in the BCFT, and using Wald's covariant phase space formalism in the presence of boundaries, we derive constraints from the first law for a broad range of covariant bulk Lagrangians. We explicitly evaluate these constraints for Einstein gravity, and find a local equation on the brane which is precisely the Neumann condition of Takayanagi [arXiv:1105.5165] at linear order in metric perturbations. This is a…
By combining the AdS/BCFT correspondence and the brane world holography, we expect an equivalence relation between a boundary conformal field theory (BCFT) and a gravitational system coupled to a CFT. However, it still remains unclear how the boundary condition of the BCFT is translated in the gravitational system. We examine this duality relation in a two-dimensional setup by looking at the computation of entanglement entropy and energy flux conservation. We also identify the two-dimensional gravity which is dual to the boundary dynamics of a BCFT. Moreover, we show that by considering a grav…
We investigate the time evolution of reflected entropy and entanglement negativity for mixed state configurations involving two adjacent and disjoint intervals in the radiation flux of moving mirrors by utilizing the $AdS/BCFT$ duality. These measures are computed for the required mixed state configurations by using the respective replica techniques in the large central charge limit of the $BCFT_{1+1}$. We demonstrate that the results obtained exactly agree with the corresponding holographic computations in the dual bulk $AdS_3$ geometry with an end of the world brane. In this context, the ana…
We study the holographic entanglement entropy of spatial regions having arbitrary shapes in the AdS4/BCFT3 correspondence with static gravitational backgrounds, focusing on the subleading term with respect to the area law term in its expansion as the UV cutoff vanishes. An analytic expression depending on the unit vector normal to the minimal area surface anchored to the entangling curve is obtained. When the bulk spacetime is a part of AdS4, this formula becomes the Willmore functional with a proper boundary term evaluated on the minimal surface viewed as a submanifold of a three dimensional …
In this paper we construct time-dependent solutions of three-dimensional gravity in AdS space dual to systems with boundaries (BCFTs), following the AdS/BCFT prescription. Such solutions can be discussed in the context of the dynamics of first order phase transitions, or more generally, in the description of quantum quenches. As an example, we apply the holographic model to calculate the dynamics of the entanglement entropy of a local quench corresponding to a nucleation of a Euclidean bubble. As in the known 1+1 CFT examples of local cut and glue quenches, the holographic entanglement entropy…
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 $AdS_3/BCFT_2$ sc…
Boundary flow and geometric realization in holographic $T\bar T$-deformed BCFT
Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）
Boundary conformal field theory (BCFT) and interface conformal field theory (ICFT) attract attention in the context of the information paradox problem. On this background, we develop the idea of the reflected entropy in BCFT/ICFT. We first introduce the left-right reflected entropy (LRRE) in BCFT and show that its holographic dual is given by the area of the entanglement wedge cross section (EWCS) through AdS/BCFT. We also present how to evaluate the reflected entropy in ICFT. By using this technique, we can show the universal behavior of the Markov gap in some special cases. Furthermore, we c…
CFT derivation of entanglement phase transition in pseudo entropy
Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University（京都大学 引力物理与量子信息中心、汤川理论物理研究所）；Inamori Research Institute for Science（稻盛理科研究所）；Society of Fellows, Harvard University（哈佛大学 院士社）
In this paper we calculate the entanglement entropy for topological interfaces in rational conformal field theories for the case where the interface lies at the boundary of the entangling interval and for the case where it is located in the center of the entangling interval. We compare the results to each other and also to the recently calculated left/right entropy of a related BCFT. We also comment of the entanglement entropies for topological interfaces for a free compactified boson and Liouville theory.
We consider entanglement entropies of finite spatial intervals in Minkowski radiation baths coupled to the eternal black hole in JT gravity, and the related problem involving free fermion BCFT in the thermofield double state. We show that the non-monotonic entropy evolution in the black hole problem precisely matches that of the free fermion theory in a high temperature limit, and the results have the form expected for CFTs with quasiparticle description. Both exhibit rich behaviour that involves at intermediate times, an entropy saddle with an island in the former case, and in the latter a sp…
In this note, following [arXiv:1905.08255, arXiv:1905.08762, arXiv:1908.10996], we introduce and study various holographic systems which can describe evaporating black holes. The systems we consider are boundary conformal field theories for which the number of local degrees of freedom on the boundary ($c_{bdy}$) is large compared to the number of local degrees of freedom in the bulk CFT ($c_{bulk}$). We consider states where the boundary degrees of freedom on their own would describe an equilibrium black hole, but the coupling to the bulk CFT degrees of freedom allows this black hole to evapor…
In this paper, we investigate AdS/BCFT for curvature-squared gravity. To warm up, we start with Gauss-Bonnet gravity. We derive the one point function of stress tensor and show that the central charge related to the norm of displacement operator is positive for the couplings obeying causality constraints. Furthermore, by imposing the null energy condition on the end-of-the-world brane, we prove the holographic g-theorem for Gauss-Bonnet gravity. This corrects a wrong point of view in the literature, which claims that the holographic g-theorem is violated for Gauss-Bonnet gravity. As a by-produ…
We investigate the time evolution of entanglement negativity following a global quench for mixed state configurations of two disjoint and adjacent intervals in a ($1+1$)-dimensional conformal field theory ($CFT_{1+1}$) dual to the eternal black hole sliced in half by an end of the world brane, through the $AdS_3/CFT_2$ correspondence. To this end we obtain the time evolution of the holographic entanglement negativity for such mixed states from a dual bulk eternal black hole geometry and elucidate the relevant geodesic structures. The holographic entanglement negativity for such mixed states, f…
