Entanglement Renyi entropies in holographic theories (1006.0047v3)

Abstract: Ryu and Takayanagi conjectured a formula for the entanglement (von Neumann) entropy of an arbitrary spatial region in an arbitrary holographic field theory. The von Neumann entropy is a special case of a more general class of entropies called Renyi entropies. Using Euclidean gravity, Fursaev computed the entanglement Renyi entropies (EREs) of an arbitrary spatial region in an arbitrary holographic field theory, and thereby derived the RT formula. We point out, however, that his EREs are incorrect, since his putative saddle points do not in fact solve the Einstein equation. We remedy this situation in the case of two-dimensional CFTs, considering regions consisting of one or two intervals. For a single interval, the EREs are known for a general CFT; we reproduce them using gravity. For two intervals, the RT formula predicts a phase transition in the entanglement entropy as a function of their separation, and that the mutual information between the intervals vanishes for separations larger than the phase transition point. By computing EREs using gravity and CFT techniques, we find evidence supporting both predictions. We also find evidence that large-$N$ symmetric-product theories have the same EREs as holographic ones.

• The paper critiques Fursaev's derivation by demonstrating that the proposed saddle points do not satisfy the Einstein equations, leading to a corrected computation for 2D CFTs.
• It verifies known results for a single interval and predicts a phase transition in mutual information for configurations with two intervals, reinforcing aspects of the Ryu-Takayanagi formula.
• Findings indicate that large-N symmetric-product theories mirror holographic ERE behavior, suggesting broader universality across different frameworks in quantum gravity.

Overview of "Entanglement Rényi Entropies in Holographic Theories"

The paper under review, authored by Matthew Headrick from Brandeis University, explores the domain of holographic theories by investigating entanglement Rényi entropies (EREs). This research builds upon Ryu and Takayanagi's conjecture regarding the entanglement entropy of spatial regions within holographic field theories. Using Euclidean gravity, this paper critiques Fursaev's earlier computation of entanglement Rényi entropies, ultimately presenting new findings for two-dimensional conformal field theories (CFTs) and verifying aspects of the Ryu-Takayanagi (RT) formula.

Summary of Key Findings

1. Critique and Reevaluation of Fursaev's Computation: The paper identifies a critical issue in Fursaev's derivation of EREs, primarily that the proposed saddle points do not satisfy the Einstein equations. This casts doubt on Fursaev’s derived formula, prompting the authors to present a rectified computation within two-dimensional CFTs for configurations involving one or two spatial intervals.
2. Entanglement Rényi Entropies in Two-Dimensional CFTs: For a single interval, the paper reproduces known EREs using gravity, aligning with existing CFT results. For two intervals, it discusses a predicted phase transition in the entanglement entropy based on the RT formula, where mutual information between the intervals vanishes beyond a critical separation.
3. Investigation into Large-$N$ Theories: Evidence suggests that large-$N$ symmetric-product theories exhibit identical EREs as holographic ones, highlighting a potentially broader universality across different theoretical frameworks.

Implications and Speculations

The findings presented have significant implications for understanding entanglement entropy in holographic theories and potentially broaden the application of the RT formula beyond its original scope. Specifically, the predicted phase transition in mutual information invites further exploration into quantum information distribution in these theories. This grounds the motivation for continued research into holographic principles, particularly concerning the effects of topology, quantum corrections, and extensions to non-holographic theories or those with large central charges.

Future Outlook

The paper hints at several open questions and directions for future research:

• Extending analyses to more complex configurations, such as multiple intervals or non-vacuum states.
• Further exploring the implications of universality in large-$c$ limits and the symmetric-product structure.
• Addressing any additional corrections or modifications to the RT formula considering quantum or higher-derivative effects.

In conclusion, this detailed paper into EREs not only solidifies some predictions of the Ryu-Takayanagi formula but also challenges researchers to ponder the broader applicability and underlying principles governing entanglement in quantum gravity and holographic theories.