Abstract
We consider entanglement entropy of a cap-like region for a conformal field theory living on a sphere times a circle in d space-time dimensions. Assuming that the finite size of the system introduces a unique ground state with a nonzero mass gap, we calculate the leading correction to the entanglement entropy in a low temperature expansion. The correction has a universal form for any conformal field theory that depends only on the size of the mass gap, its degeneracy, and the angular size of the cap. We confirm our result by calculating the entanglement entropy of a conformally coupled scalar numerically. We argue that an apparent discrepancy for the scalar can be explained away through a careful treatment of boundary terms. In an appendix, to confirm the accuracy of the numerics, we study the mutual information of two cap-like regions at zero temperature.
Highlights
Aand in the high temperature limit, becomes dominated by it
Where ∆ is the smallest scaling dimension among the set of operators including the stress tensor and all primaries not equal to the identity and g is their degeneracy. (See [9] for the specific case of the stress tensor.) In order for this result to hold, the CFT needs to have a unique ground state separated from the first excited state by a nonzero mass gap
We will assume that the Sd−1 gaps the spectrum and leads to a unique ground state. (Maximally supersymmetric Yang-Mills in 3+1 dimensions on an S3 would be an example.) We write down the density matrix as a Boltzmann sum, keeping only the ground state |0 and the first excited states |ψi with i = 1, . . . , g:
Summary
We are interested in a d dimensional CFT on Sd−1 at finite temperature. We will assume that the Sd−1 gaps the spectrum and leads to a unique ground state. (Maximally supersymmetric Yang-Mills in 3+1 dimensions on an S3 would be an example.) We write down the density matrix as a Boltzmann sum, keeping only the ground state |0 and the first excited states |ψi with i = 1, . . . , g:. (We will argue in section 4 that at least for a conformally coupled scalar, this integral may differ from the true modular Hamiltonian by boundary terms.) The covariance of a CFT under Weyl rescaling allows us to rewrite HM in terms of Ttt on R × Sd−1. The states |ψi must transform under some representation of SO(d) because of the rotational symmetry of the sphere while the operator Ttt(p) will transform as a scalar under rotations. It follows that IA must transform as a scalar. Note that since sin θ is invariant under θ → π − θ, the odd part is insensitive to potential ambiguities in the sind−2 θ0 term
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