Holographic Rényi $n\to 0$ entropy and Euclidean fluids

Abstract

We explore the holographic prescription for computing the refined Rényi entropies ${\tilde{S}}_n$ in the $n\to 0$ limit within the AdS${}_{d+1}$/CFT${}_d$ framework. This limit can be interpreted as a high-temperature regime with respect to the energy defined by the modular Hamiltonian of the state reduced to a subregion. To leading order in $n$, we find that the system attains local equilibrium and admits a CFT description in terms of a Euclidean, irrotational perfect fluid. This fluid exhibits vortex-like boundary conditions at the entangling surface. Guided by this physical picture, we construct an ansatz for the dual geometry in terms of the boundary fluid variables. We show that our anzats solves Einstein's equations coupled to a cosmic brane at leading order in $n$, in agreement with Dong's proposal for the holographic dual to the refined Rényi entropy. The resulting conical singularity, signaling the brane's location, can be understood from this perspective as the bulk extension of the boundary fluid vortices.

27 pages and 1 figure. v2 contains a few improvements to match the version accepted for publication at JHEP. It contains appendices B and C. App B discusses our two-dimensional results from the Schottky uniformization perspective, and App C analyzes the relevance of the region behind the zero-determinant wall