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

Abstract: We explore the holographic prescription for computing the refined R\'enyi entropies $\tilde S_n$ in the $n \to 0$ limit in the AdS${d+1}$/CFT${d}$ context. This can be interpreted as a high temperature limit with respect to the notion of energy provided by the modular Hamiltonian describing the state of the system reduced to a subregion. We find, to the leading order in $n$, that the system attains local equilibrium and admits a CFT description in terms of an Euclidean irrotational perfect fluid. The fluid has vortex like boundary conditions at the boundary of the region. These vortices extends into the bulk in the form of a conical singularity, in agreement with Dong's proposal for the holographic dual to the R\'enyi entropy.