Note on non-vacuum conformal family contributions to Rényi entropy in two-dimensional CFT**Supported by ERC Starting Grant 637844-HBQFTNCER

Abstract

We calculate the contributions of a general non-vacuum conformal family to Rényi entropy in two-dimensional conformal field theory (CFT). The primary operator of the conformal family can be either non-chiral or chiral, and we denote its scaling dimension by Δ. For the case of two short intervals on a complex plane, we expand the Rényi mutual information by the cross ratio x to order x2Δ+2. For the case of one interval on a torus with low temperature, we expand the Rényi entropy by q=exp(−2πβ/L), with β being the inverse temperature and L being the spatial period, to order qΔ+2. To make the result meaningful, we require that the scaling dimension Δ cannot be too small. For two intervals on a complex plane we need Δ > 1, and for one interval on a torus we need Δ > 2. We work in the small Newton constant limit on the gravity side and so a large central charge limit on the CFT side, and find matches of gravity and CFT results.