Jones index from R\'enyi entropies in the Ising conformal field theory

Abstract

We study the relation between the Jones index and the R\'enyi entropies of two disjoint intervals on the line and of the ground state for a generic value of the R\'enyi index in the two conformal field theory models given by the Ising model and a free Majorana fermion, where Haag duality is satisfied. The analytic expressions of the crossing asymmetry for all the submodels displaying a violation of the Haag duality that are closed under the fusion rules are obtained. In the limiting regime where the two intervals become adjacent, the leading term of the expansion of the crossing asymmetry provides the Jones global index, for any finite value of the R\'enyi index.