Some Results on Mutual Information of Disjoint Regions in Higher Dimensions

TL;DR

This paper investigates the mutual Renyi information between disjoint regions in higher-dimensional conformal field theories, deriving universal formulas and explicit results for specific geometries, and exploring corrections for massive scalar fields.

Contribution

It provides a general formula for mutual Renyi information in higher-dimensional CFTs and explicit calculations for free scalar fields, including geometric and mass-related corrections.

Findings

Mutual information scales as (R_A R_B / r^2)^a with universal constants.

Explicit formulas for mutual information for spherical and ellipsoidal regions.

Universal logarithmic correction to the entanglement entropy area law for massive scalars.

Abstract

We consider the mutual Renyi information I^n(A,B)=S^n_A+S^n_B-S^n_{AUB} of disjoint compact spatial regions A and B in the ground state of a d+1-dimensional conformal field theory (CFT), in the limit when the separation r between A and B is much greater than their sizes R_{A,B}. We show that in general I^n(A,B)\sim C^n_AC^n_B(R_AR_B/r^2)^a, where a the smallest sum of the scaling dimensions of operators whose product has the quantum numbers of the vacuum, and the constants C^n_{A,B} depend only on the shape of the regions and universal data of the CFT. For a free massless scalar field, where 2x=d-1, we show that C^2_AR_A^{d-1} is proportional to the capacitance of a thin conducting slab in the shape of A in d+1-dimensional electrostatics, and give explicit formulae for this when A is the interior of a sphere S^{d-1} or an ellipsoid. For spherical regions in d=2 and 3 we obtain explicit results for C^n for all n and hence for the leading term in the mutual information by taking n->1. We also compute a universal logarithmic correction to the area law for the Renyi entropies of a single spherical region for a scalar field theory with a small mass.