Entanglement entropy for the n-sphere

TL;DR

This work analytically computes the universal logarithmic contribution to the entanglement entropy of a massless scalar across a spherical boundary in arbitrary dimensions by mapping the reduced density matrix to a thermal state on hyperbolic space and evaluating the partition function with heat-kernel techniques. It demonstrates that odd spacetime dimensions have no logarithmic term, while even dimensions yield a dimension-dependent coefficient $g_0$ with explicit values (e.g., $g_0=-\tfrac{1}{3}$ for $d+1=2$ and $g_0=\tfrac{1}{90}$ for $d+1=4$), in agreement with Solodukhin’s conformal-anomaly framework and the Ryu-Takayanagi holographic conjecture. The results extend to Renyi entropies through corresponding $g_0^{(n)}$ and are cross-validated by independent methods, reinforcing the holographic perspective on universal entanglement contributions in conformal field theories.

Abstract

We calculate the entanglement entropy for a sphere and a massless scalar field in any dimensions. The reduced density matrix is expressed in terms of the infinitesimal generator of conformal transformations keeping the sphere fixed. The problem is mapped to the one of a thermal gas in a hyperbolic space and solved by the heat kernel approach. The coefficient of the logarithmic term in the entropy for 2 and 4 spacetime dimensions are in accordance with previous numerical and analytical results. In particular, the four dimensional result, together with the one reported by Solodukhin, gives support to the Ryu-Takayanagi holographic anzats. We also find there is no logarithmic contribution to the entropy for odd space time dimensions.