A bstract We provide a derivation of the Ryu-Takayanagi (RT) formula in 3D gravity for generic boundary subregions — including RT surface phase transitions — directly from the dual two-dimensional conformal field theory (CFT). Our approach relies on the universal statistics of the algebraic conformal data and the large- c behavior of conformal blocks with Cardy boundaries involved. We observe the emergence of 3D multi-boundary black holes with Karch-Randall branes from entangled states of any number of CFTs with and without Cardy boundaries. The RT formula is obtained directly from the CFT in the high-temperature regime. Two direct applications are: 1) A simple derivation of the multi-interval entanglement entropy for the vacuum state of a single CFT; 2) A CFT-based detection of the emergence of replica wormholes in the context of entanglement islands and black hole microstate counting. Our framework yields the first holographic random tensor network that faithfully captures the entanglement structure of holographic CFTs. These results imply that bulk spacetime geometries indeed emerge from the eigenstate thermalization hypothesis (ETH) in the dual field theory in the large- c limit — a paradigm we refer to as It from ETH .
A bstract We provide a derivation of the Ryu-Takayanagi (RT) formula in 3D gravity for generic boundary subregions — including RT surface phase transitions — directly from the dual two-dimensional conformal field theory (CFT). Our approach relies on the universal statistics of the algebraic conformal data and the large- c behavior of conformal blocks with Cardy boundaries involved. We observe the emergence of 3D multi-boundary black holes with Karch-Randall branes from entangled states of any number of CFTs with and without Cardy boundaries. The RT formula is obtained directly from the CFT in the high-temperature regime. Two direct applications are: 1) A simple derivation of the multi-interval entanglement entropy for the vacuum state of a single CFT; 2) A CFT-based detection of the emergence of replica wormholes in the context of entanglement islands and black hole microstate counting. Our framework yields the first holographic random tensor network that faithfully captures the entanglement structure of holographic CFTs. These results imply that bulk spacetime geometries indeed emerge from the eigenstate thermalization hypothesis (ETH) in the dual field theory in the large- c limit — a paradigm we refer to as It from ETH .
We define a generalized entanglement measure in the context of AdS3/CFT2. Compared to the ordinary entanglement entropy between spatial degrees of freedom dual to the area of the Ryu-Takayanagi surface, we take into account both entanglement between spatial degrees of freedom as well as between different fields of the boundary CFT. We then calculate this generalized entanglement measure in a thermal state dual to the BTZ black hole in the setting of the D1/D5 system at and close to the orbifold point. We find that the entanglement entropy defined in this way is dual to the length of a geodesic with non-zero winding number. Such geodesics probe the entire bulk geometry, including regions known as entanglement shadows which are not reached by any Ryu-Takayanagi surface. This allows us to describe regions close to the black hole horizon in the one-sided black hole and the wormhole growth in the case of a two-sided black hole from entanglement data in the boundary field theory, leading us to propose that entanglement is in fact enough to reconstruct the full BTZ geometry from boundary data.
Tensor networks provide a natural framework for exploring holographic duality because they obey entanglement area laws. They have been used to construct explicit toy models realizing many of the interesting structural features of the AdS/CFT correspondence, including the non-uniqueness of bulk operator reconstruction in the boundary theory. In this article, we explore the holographic properties of networks of random tensors. We find that our models naturally incorporate many features that are analogous to those of the AdS/CFT correspondence. When the bond dimension of the tensors is large, we show that the entanglement entropy of all boundary regions, whether connected or not, obey the Ryu-Takayanagi entropy formula, a fact closely related to known properties of the multipartite entanglement of assistance. We also discuss the behavior of Rényi entropies in our models and contrast it with AdS/CFT. Moreover, we find that each boundary region faithfully encodes the physics of the entire bulk entanglement wedge, i.e., the bulk region enclosed by the boundary region and the minimal surface. Our method is to interpret the average over random tensors as the partition function of a classical ferromagnetic Ising model, so that the minimal surfaces of Ryu-Takayanagi appear as domain walls. Upon including the analog of a bulk field, we find that our model reproduces the expected corrections to the Ryu-Takayanagi formula: the bulk minimal surface is displaced and the entropy is augmented by the entanglement of the bulk field. Increasing the entanglement of the bulk field ultimately changes the minimal surface behavior topologically, in a way similar to the effect of creating a black hole. Extrapolating bulk correlation functions to the boundary permits the calculation of the scaling dimensions of boundary operators, which exhibit a large gap between a small number of low-dimension operators and the rest. While we are primarily motivated by the AdS/CFT duality, the main results of the article define a more general form of bulk-boundary correspondence which could be useful for extending holography to other spacetimes.
Entanglement entropy in holographic conformal field theories with boundaries of generic spacetime signature cannot be described entirely within the standard holographic duality to a real bulk spacetime. We show, both in static and time-dependent setups, that an exact match with the conformal field theory predictions in two boundary dimensions entails an extension of the dual spacetime to complex coordinates. As a consequence, relevant bulk objects such as Ryu-Takayanagi surfaces and branes are in general complex, yet anchored on a real asymptotic boundary preserving the physical meaning of field-theoretical quantities. We also provide a third numerical check of the holographic prediction with a two-dimensional system of free Gaussian fermions exhibiting the same asymptotic regimes across different entanglement phases.
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.
A central insight of holography is that spacetime geometry emerges from the quantum entanglement of a boundary theory. This work proposes and validates a precise, local manifestation of this principle: in a 1+1D boundary conformal field theory (CFT) dual to an asymptotically bulk, the Ricci scalar curvature is proportional to the negative second derivative of the entanglement entropy density. This hypothesis, motivated by the Ryu-Takayanagi formula and the geodesic deviation equation, posits that local fluctuations in boundary entanglement directly encode local bulk curvature. The relation is tested numerically using a tensor network toy model inspired by the Multi-scale Entanglement Renormalization Ansatz (MERA). Localized suppressions of boundary entanglement, analogous to removing disentanglers in MERA, are shown to induce localized positive curvature spikes in the emergent bulk geometry. The results provide direct computational evidence for the entanglement-curvature correspondence, reinforcing the paradigm that spacetime geometry is built from quantum entanglement.
We show that in holographic 2d CFTs, the entanglement entropies across several choices of global state and subregion can be written in a way that at once has a microscopic interpretation and matches the leading large−c organization of the Ryu-Takayanagi formula. This representation is obtained by applying crossing symmetry to the replica manifolds. From the boundary point of view, each rewritten entropy looks like an algebraic entanglement entropy for the Virasoro algebra restricted to the region, with center labels obtained by coarse-graining heavy primaries of the BCFT on the regulated region into bins labeled by Liouville momenta. At large c, a resulting sum over bins is dominated by a saddle, and the O(c) part of the entropy comes from the Cardy density of heavy primaries in the dominant bin. We identify this O(c) part of the entropy with the Ryu-Takayanagi area. Physically, this suggests a concrete statistical origin for the Ryu-Takayanagi area as coming from coarse-grained Virasoro intertwiners across the entangling cut. The result also provides quantitative criteria for the amount of coarse-graining allowed for the consistency of the interpretation.