Diagonal Approximation for Holographic Rényi Entropies

Recently, Dong et al., [A modified cosmic brane proposal for holographic Renyi entropy, J. High Energy Phys. 06 (2024) 120JHEPFG1029-847910.1007/JHEP06(2024)120] proposed a modified cosmic brane prescription for computing the Rényi entropy S_{α} of a holographic system in the presence of multiple extremal surfaces. This prescription was found by assuming a diagonal approximation, where the Rényi entropy is computed after first measuring the areas of all extremal surfaces. We derive this diagonal approximation for the case of two extremal surfaces and show that it accurately computes Rényi entropies up to O(logG) corrections. For α<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 α>1, it leads to the original cosmic brane prescription without needing to assume that replica symmetry is unbroken in the bulk.