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

TL;DR

This work shows that in holographic CFTs the refined Rényi entropy in the $n\to0$ limit is governed by a Euclidean irrotational perfect fluid with vortex boundary conditions at the entangling surface. By constructing a bulk ansatz tied to the boundary fluid data and validating it against Einstein equations, the authors derive a universal relation where $\tilde S_{n\to0}$ reduces to a boundary entropy flux controlled by the local inverse temperature $\beta(x)$ and velocity $u^\mu$. They provide explicit 2D results using Banados geometries, obtain closed-form expressions for multiple intervals, and connect these to known free-fermion structures, while discussing implications for the density of states and the role of diagonal saddles in the small-$n$ regime. The results highlight a universal holographic picture linking high-modular-temperature behavior to fluid dynamics and brane geometries, with potential cross-theory applicability and numerical avenues for solving the boundary fluid equations in nontrivial regions.

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.

Figures (1)

• Figure 1: Brane geometry representations for different values of $n$. On the left figure, we illustrate the back-reacted geometry of a brane $\gamma_1$ in the standard $n>1$ case. Here, $\gamma_1$ and $\gamma_2$ are the two topologically distinct branes, homologous to the entangling region $A$, with associated areas $A_1$ and $A_2$, respectively. The holographic refined Rényi entropy is given by the area of $\gamma_1$ which we assumed to be minimal: $A_1<A_2$. The centered figure illustrates an example of a geometry that contains both back-reacted branes $\gamma_1$ and $\gamma_2$, with $A_1=A_2$. This is the relevant geometry for sufficiently small $n$ ---below a certain model dependent $n_0<1$--- as it was recently argued in Dong:2023bfy. On the right figure, we schematically draw the topology of our ansatz in which a single brane-like object carries the full tension. This geometry can be thought of as the limiting $n\to0$ case of the figure in the center.