Entanglement entropy for the n-sphere

TL;DR

This paper computes the entanglement entropy for a sphere and a massless scalar field across various dimensions, confirming known results in 2 and 4 dimensions and supporting the Ryu-Takayanagi holographic conjecture.

Contribution

It provides a general calculation of entanglement entropy for spheres in arbitrary dimensions using conformal transformations and heat kernel methods, extending previous specific results.

Findings

Logarithmic term coefficients match previous results in 2 and 4 dimensions.

No logarithmic contribution in odd spacetime dimensions.

Supports the Ryu-Takayanagi holographic conjecture in 4D.

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.