Tripartite information at long distances

We compute the leading term of the tripartite information at long distances for three spheres in a CFT. This falls as r-6Δ r − 6 Δ , where r r is the typical distance between the spheres, and Δ Δ 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-6Δ_f r − 6 Δ f . In particular, lattice calculations suggest a leading scaling r^(6Δ_f+1) r ( 6 Δ f + 1 ) . For free fermions in three dimensions, we show that mutual information is also non-monogamous in the long-distance regime.