# On the Mutual Information in Conformal Field Theory

## TL;DR

This paper investigates the universal behavior of mutual information between two disjoint spheres in conformal field theories, developing a framework that expresses the large-distance expansion in terms of conformal blocks and universal coefficients, with applications to free scalar and fermion theories.

## Contribution

The authors introduce a $1/n$ prescription and demonstrate the universality of leading contributions to mutual information in CFTs, applicable to various models including free scalars and fermions.

## Key findings

- Universal forms of the first few leading order contributions to mutual information.
- Good agreement between theoretical predictions and numerical lattice results for free scalars.
- Leading order results for free fermions match numerical studies.

## Abstract

In this work, we study the universal behaviors in the mutual information of two disjoint spheres in a conformal field theory(CFT). By using the operator product expansion of the spherical twist operator in terms of the conformal family, we show that the large distance expansion of the mutual information can be cast in terms of the conformal blocks. We develop the $1/n$ prescription to compute the coefficients before the conformal blocks. For a single conformal family, the leading nonvanishing contribution to the mutual information comes from the bilinear operators. We show that the coefficients of these operators take universal forms and such universal behavior persists in the bilinear operators with derivatives as well. Consequently the first few leading order contributions to the mutual information in CFT take universal forms. To illustrate our framework, we discuss the free scalars and free fermions in various dimensions. For the free scalars, we compute the mutual information to the next-to-leading order and find good agreement with the improved numerical lattice result. For the free fermion, we compute the leading order result, which is of universal form, and find the good match with the numerical study. Our formalism could be applied to any CFT potentially.