Some results on the mutual information of disjoint regions in higher dimensions

We consider the mutual Rényi information I^{(n)}(A,B)\equiv S^{(n)}_{A}+S^{(n)}_{B}-S^{(n)}_{A\cup B} 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)^{\alpha }, where α is 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 α = 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 Rényi entropies of a single spherical region for a scalar field theory with a small mass.