Tripartite information at long distances

Abstract

We compute the leading term of the tripartite information at long distances for three spheres in a CFT. This falls as r −6∆, where r is the typical distance between the spheres, and ∆ the lowest primary field dimension. The coefficient turns out to be a combination of terms coming from the two- and three-point functions and depends on the OPE coefficient of the field. We check the result with three-dimensional free scalars in the lattice finding excellent agreement. When the lowest-dimensional field is a scalar, we find that the mutual information can be monogamous only for quite large OPE coefficients, far away from a perturbative regime. When the lowest-dimensional primary is a fermion, we argue that the scaling must always be faster than r −6∆f . In particular, lattice calculations suggest a leading scaling r −(6∆f +1) . For free fermions in three dimensions, we show that mutual information is also non-monogamous in the long-distance regime.