Long-distance N-partite information for fermionic CFTs

The mutual information, I₂, of general spacetime regions is expected to capture the full data of any conformal field theory (CFT). For spherical regions, this data can be accessed from long-distance expansions of the mutual information of pairs of regions as well as of suitably chosen linear combinations of mutual informations involving more than two regions and their unions — namely, the N-partite information, I_N. In particular, the leading term in the I₂ long-distance expansion is fully determined by the spin and conformal dimension of the lowest-dimensional primary of the theory. When the operator is a scalar, an analogous formula for the tripartite information I₃ contains information about the OPE coefficient controlling the fusion of such operator into its conformal family. When it is a fermionic field, the coefficient of the leading term in I₃ vanishes instead. In this paper we present an explicit general formula for the long-distance four-partite information I₄ of general CFTs whose lowest-dimensional operator is a fermion ψ. The result involves a combination of four-point and two-point functions of ψ and $\overline{\psi }$ \( \overline{\psi} \) evaluated at the locations of the regions. We perform explicit checks of the formula for a (2 + 1)-dimensional free fermion in the lattice finding perfect agreement. The generalization of our result to the N-partite information (for arbitrary N) is also discussed. Similarly to I₃, we argue that I₅ vanishes identically at leading order for general fermionic theories, while the I_N with N = 7, 9, … only vanish when the theory is free.