Universality in the geometric dependence of Renyi entropy

TL;DR

This work develops a perturbative framework for Renyi entropy in quantum field theories, tying linear geometric perturbations to stress-tensor data on replicated geometries and exposing nonperturbative constraints on the $n$-dependence of universal terms in even dimensions. In four dimensions, it derives a fundamental relation $f_c(n)=\frac{n}{n-1}\bigl(a-f_a(n)-(n-1)f'_a(n)\bigr)$, showing $f_c(n)$ is determined by $f_a(n)$, with analogous structures extended to $d=6$ and general even dimensions. The authors also provide holographic computations of Renyi entropy across non-spherical entangling surfaces via hyperbolic black holes and argue that, for a broad class of CFTs, the second-order perturbation data suggests $f_b(n)=f_c(n)$. Additional discussions address the locality of modular Hamiltonians, OPE techniques on cones, and potential deformed-hyperboloid geometries as engines for direct non-spherical Renyi calculations, with implications for the entanglement structure in strongly coupled systems.

Abstract

We derive several new results for Renyi entropy, $S_n$, across generic entangling surfaces. We establish a perturbative expansion of the Renyi entropy, valid in generic quantum field theories, in deformations of a given density matrix. When applied to even-dimensional conformal field theories, these results lead to new constraints on the $n$-dependence, independent of any perturbative expansion. In 4d CFTs, we show that the $n$-dependence of the universal part of the ground state Renyi entropy for entangling surfaces with vanishing extrinsic curvature contribution is in fact fully determined by the Renyi entropy across a sphere in flat space. Using holography, we thus provide the first computations of Renyi entropy across non-spherical entangling surfaces in strongly coupled 4d CFTs. Furthermore, we address the possibility that in a wide class of 4d CFTs, the flat space spherical Renyi entropy also fixes the $n$-dependence of the extrinsic curvature contribution, and hence that of arbitrary entangling surfaces. Our results have intriguing implications for the structure of generic modular Hamiltonians.