[Paper Review] The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT

Thomas Faulkner|arXiv (Cornell University)|Mar 28, 2013

Black Holes and Theoretical PhysicsPhysics and Astronomy42 references145 citations

TL;DR

This paper computes entanglement Renyi entropies (EREs) for disjoint intervals in 1+1D conformal field theories with classical gravity duals using the AdS/CFT correspondence. By applying the replica trick and constructing smooth, replica-symmetric handlebody solutions in 3D gravity, the authors derive a simple prescription for the regularized classical action, which yields the EREs at leading order in Newton's constant. The method reproduces the Ryu-Takayanagi formula in the limit n → 1.

ABSTRACT

We study entanglement Renyi entropies (EREs) of 1+1 dimensional CFTs with classical gravity duals. Using the replica trick the EREs can be related to a partition function of n copies of the CFT glued together in a particular way along the intervals. In the case of two intervals this procedure defines a genus n-1 surface and our goal is to find smooth three dimensional gravitational solutions with this surface living at the boundary. We find two families of handlebody solutions labelled by the replica index n. These particular bulk solutions are distinguished by the fact that they do not spontaneously break the replica symmetries of the boundary surface. We show that the regularized classical action of these solutions is given in terms of a simple numerical prescription. If we assume that they give the dominant contribution to the gravity partition function we can relate this classical action to the EREs at leading order in G_N. We argue that the prescription can be formulated for non-integer n. Upon taking the limit n -> 1 the classical action reproduces the predictions of the Ryu-Takayanagi formula for the entanglement entropy.

Motivation & Objective

• To compute entanglement Renyi entropies (EREs) for disjoint intervals in 1+1D CFTs with classical gravity duals.
• To identify smooth, replica-symmetric three-dimensional gravitational solutions that correspond to the boundary replica construction for two disjoint intervals.
• To derive a general prescription for the regularized classical action of these solutions, valid for non-integer replica index n.
• To show that the classical action of these solutions reproduces the Ryu-Takayanagi formula in the limit n → 1.
• To establish a gravity-based computation of EREs that generalizes the minimal surface prescription to non-integer n and multiple intervals.

Proposed method

• The replica trick is applied to the boundary CFT, mapping the computation of EREs to a partition function on a genus n−1 Riemann surface formed by gluing n copies of the CFT along the disjoint intervals.
• A family of smooth, handlebody-type gravitational solutions in 3D AdS space is constructed, with the genus n−1 surface as the boundary, preserving the replica symmetry.
• The regularized classical action of these solutions is computed and shown to follow a simple numerical prescription independent of the specific geometry.
• The prescription is extended to non-integer values of the replica index n via analytic continuation.
• The leading-order contribution to the ERE is obtained by relating the classical action to the partition function in the gravity path integral.
• The n → 1 limit of the action is shown to reproduce the Ryu-Takayanagi formula for entanglement entropy.

Experimental results

Research questions

• RQ1 How can entanglement Renyi entropies for two disjoint intervals be computed in 1+1D CFTs with classical gravity duals using the AdS/CFT correspondence?
• RQ2 What are the properties of smooth 3D gravitational solutions that realize the replica construction for two disjoint intervals on the boundary?
• RQ3 Can a simple, universal prescription be derived for the regularized classical action of these replica-symmetric bulk solutions?
• RQ4 How does the classical action of these solutions relate to the entanglement Renyi entropy at leading order in G_N?
• RQ5 Does the proposed prescription reduce to the Ryu-Takayanagi formula in the limit n → 1?

Key findings

• The authors identify two distinct families of smooth, replica-symmetric handlebody solutions in 3D gravity that correspond to the boundary replica construction for two disjoint intervals.
• The regularized classical action of these solutions is given by a simple numerical prescription that depends only on the topology and not on the specific embedding of the intervals.
• The prescription is valid for non-integer values of the replica index n, enabling analytic continuation to the entanglement entropy limit.
• In the limit n → 1, the classical action reproduces the Ryu-Takayanagi formula for the entanglement entropy.
• The dominant contribution to the entanglement Renyi entropy at leading order in G_N is captured by these replica-symmetric solutions, supporting their physical relevance.