The Diagonal Approximation for Holographic R\'{e}nyi Entropies

TL;DR

This paper derives and validates a diagonal approximation method for computing holographic Rényi entropies, revealing modifications to the cosmic brane prescription depending on the Rényi index and removing certain symmetry assumptions.

Contribution

It provides a derivation of the diagonal approximation for Rényi entropies and clarifies its implications for the cosmic brane prescription in holography.

Findings

Diagonal approximation accurately computes Rényi entropies up to logarithmic corrections.

For α<1, it modifies the cosmic brane prescription at leading order.

For α>1, it reproduces the original cosmic brane prescription without assuming replica symmetry.

Abstract

Recently Dong, Rath and Kudler-Flam proposed a modified cosmic brane prescription for computing the R\'{e}nyi entropy $S_\alpha$ of a holographic system in the presence of multiple extremal surfaces. This prescription was found by assuming a diagonal approximation, where the R\'{e}nyi entropy is computed after first measuring the areas of all extremal surfaces. We derive this diagonal approximation and show that it accurately computes R\'{e}nyi entropies up to $O(\log G)$ corrections. For $\alpha<1$, this allows us to derive the modified cosmic brane prescription, which differs from the original cosmic brane prescription at leading order in $G$. For $\alpha>1$, it leads to the original cosmic brane prescription without needing to assume that replica symmetry is unbroken in the bulk.