Long-distance N-partite information for fermionic CFTs

Abstract

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, IN. 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 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 IN with N = 7, 9, … only vanish when the theory is free.