The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT

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.

• The paper introduces the replica trick to translate the calculation of Renyi entropies into a geometric problem on a Riemann surface.
• It identifies two families of handlebody solutions that preserve replica symmetry and link the gravitational action to entropy measures.
• Numerical methods for computing the classical gravitational action are detailed, providing consistency with the Ryu-Takayanagi formula in the n→1 limit.

The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT

The paper "The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT" by Thomas Faulkner investigates the computation of entanglement Renyi entropies (EREs) in $(1+1)$-dimensional Conformal Field Theories (CFTs) with Anti-de Sitter/Conformal Field Theory (AdS/CFT) gravity duals. The study uses the replica trick, involving multiple copied CFTs, to analyze the EREs for systems with disjoint intervals. The work extends the scope of existing insights from the Ryu-Takayanagi (RT) prescription, aiming to provide a comprehensive understanding of quantum entanglement in systems described by AdS/CFT correspondences.

Key Contributions

1. Replica Trick Application: The paper uses the replica trick to relate EREs to the partition function of $n$ copies of a CFT arranged in a specific configuration. Through this correspondence, the work translates the ERE problem into a geometric one involving a Riemann surface of genus $n-1$.
2. Handlebody Solutions: Two families of handlebody solutions, parameterized by the replica index $n$, are identified and explored. These solutions are crucial because they preserve the replica symmetry at the boundary, ensuring that the results adhere to the requisite symmetry principles of the problem.
3. Numerical Prescription for Classical Action: The paper offers a numerical method to compute the regularized classical action of the gravitational configurations. By linking this to the gravity partition function, the study derives a connection between the classical action and the EREs, valid at the leading order in Newton's constant, $G_N$.
4. Generalization to Non-Integer $n$: The study argues that the approach can be extended to non-integer replica indices. By taking the limit as $n \rightarrow 1$, the results reproduce the predictions of the RT formula for entanglement entropy, aligning with known theoretical results in holography.

Implications of the Research

The paper provides a framework that captures the entanglement properties of quantum states in systems governed by CFTs with gravity duals. By confirming the consistency of these methods with known results like the RT formula, this research aids the validation and potential expansion of holographic principles in theoretical physics.

Future Directions

The study leaves several intriguing directions open for future exploration:

• Exploration of Non-Handlebody Saddles: The investigation hints at the existence of other potential solutions that could contribute to the gravity partition function. These solutions, not captured within the particular symmetries assumed, could require extending the replica symmetry framework or finding more general numerical techniques.
• Quantum and Higher Derivative Corrections: The study focuses on classical actions, leaving room for exploration of quantum corrections and higher-derivative terms. These corrections are especially relevant in understanding the full quantum mechanical nature of the AdS/CFT correspondence.
• Applications to Higher Dimensions: Extending the methods to higher-dimensional AdS/CFT correspondences represents a substantial yet potentially rewarding challenge, given the increased complexity of higher-dimensional geometries.

In summary, this paper significantly contributes to the intersection of quantum entropy and holographic principles through a detailed understanding of EREs in CFTs with gravity duals. The methods and results provide a robust platform for further theoretical investigations in quantum gravity and holography.