The Gravity Dual of Renyi Entropy

A remarkable yet mysterious property of black holes is that their entropy is proportional to the horizon area. This area law inspired the holographic principle, which was later realized concretely in gauge/gravity duality. In this context, entanglement entropy is given by the area of a minimal surface in a dual spacetime. However, discussions of area laws have been constrained to entanglement entropy, whereas a full understanding of a quantum state requires Renyi entropies. Here we show that all Renyi entropies satisfy a similar area law in holographic theories and are given by the areas of dual cosmic branes. This geometric prescription is a one-parameter generalization of the minimal surface prescription for entanglement entropy. Applying this we provide the first holographic calculation of mutual Renyi information between two disks of arbitrary dimension. Our results provide a framework for efficiently studying Renyi entropies and understanding entanglement structures in strongly coupled systems and quantum gravity.Comment: 9 pages, 1 figure; v2: typos fixed, references and minor clarifications added; v3: references added, title and format changed to match version to be published in Nature Communication

Holographic Entanglement Entropy for General Higher Derivative Gravity

We propose a general formula for calculating the entanglement entropy in theories dual to higher derivative gravity where the Lagrangian is a contraction of Riemann tensors. Our formula consists of Wald's formula for the black hole entropy, as well as corrections involving the extrinsic curvature. We derive these corrections by noting that they arise from naively higher order contributions to the action which are enhanced due to would-be logarithmic divergences. Our formula reproduces the Jacobson-Myers entropy in the context of Lovelock gravity, and agrees with existing results for general four-derivative gravity. We emphasize that the formula should be evaluated on a particular bulk surface whose location can in principle be determined by solving the equations of motion with conical boundary conditions. This may be difficult in practice, and an alternative method is desirable. A natural prescription is simply minimizing our formula, analogous to the Ryu-Takayanagi prescription for Einstein gravity. We show that this is correct in several examples including Lovelock and general four-derivative gravity.Comment: 1+35 pages, 2 figures; v2: typos fixed, references added, and other improvements; v3: corrected a previous omission in counting, other clarifications; v4: minor clarifications, references added, published versio

Holographic Entropy Cone with Time Dependence in Two Dimensions

In holographic duality, if a boundary state has a geometric description that realizes the Ryu-Takayanagi proposal then its entanglement entropies must obey certain inequalities that together define the so-called holographic entropy cone. A large family of such inequalities have been proven under the assumption that the bulk geometry is static, using a method involving contraction maps. By using kinematic space techniques, we show that in two boundary (three bulk) dimensions, all entropy inequalities that can be proven in the static case by contraction maps must also hold in holographic states with time dependence.Comment: 37 pages, 10 figure

Constraints on RG Flows from Holographic Entanglement Entropy

We examine the RG flow of a candidate c-function, extracted from the holographic entanglement entropy of a strip-shaped region, for theories with broken Lorentz invariance. We clarify the conditions on the geometry that lead to a break-down of monotonic RG flows as is expected for generic Lorentz-violating field theories. Nevertheless we identify a set of simple criteria on the UV behavior of the geometry which guarantee a monotonic c-function. Our analysis can thus be used as a guiding principle for the construction of monotonic RG trajectories, and can also prove useful for excluding possible IR behaviors of the theory.Comment: 5 pages, no figure

Entropy, Extremality, Euclidean Variations, and the Equations of Motion

We study the Euclidean gravitational path integral computing the Renyi entropy and analyze its behavior under small variations. We argue that, in Einstein gravity, the extremality condition can be understood from the variational principle at the level of the action, without having to solve explicitly the equations of motion. This set-up is then generalized to arbitrary theories of gravity, where we show that the respective entanglement entropy functional needs to be extremized. We also extend this result to all orders in Newton's constant $G_N$, providing a derivation of quantum extremality. Understanding quantum extremality for mixtures of states provides a generalization of the dual of the boundary modular Hamiltonian which is given by the bulk modular Hamiltonian plus the area operator, evaluated on the so-called modular extremal surface. This gives a bulk prescription for computing the relative entropies to all orders in $G_N$. We also comment on how these ideas can be used to derive an integrated version of the equations of motion, linearized around arbitrary states.Comment: 37 pages; v2: typos fixed and new references added; v3: new references and minor clarifications adde

Generalized Gravitational Entropy from Total Derivative Action

We investigate the generalized gravitational entropy from total derivative terms in the gravitational action. Following the method of Lewkowycz and Maldacena, we find that the generalized gravitational entropy from total derivatives vanishes. We compare our results with the work of Astaneh, Patrushev, and Solodukhin. We find that if total derivatives produced nonzero entropy, the holographic and the field-theoretic universal terms of entanglement entropy would not match. Furthermore, the second law of thermodynamics could be violated if the entropy of total derivatives did not vanish.Comment: 24 pages; v2: added references, Sec. 5.2 for corner entanglement, a toy model in Sec. 5.3, and minor corrections; v3: added one reference, published versio