As we have already seen, in even dimensions, there typically appears a logarithmic term in the entanglement entropy. This term is universal in the sense that it does not depend on the scheme, which is used to regularize the UV divergences. In conformal field theories the logarithmic terms in the entropy are closely related to the conformal anomaly. In this section we discuss in detail this aspect and formulate precisely the relation between entanglement entropy and conformal anomalies.
Consider a conformal field theory in spacetime dimensions. As we have discussed throughout this
review the most efficient way to calculate the corresponding entanglement entropy for a non-extremal black
hole is to introduce a small angle deficit
at the horizon surface
, compute the effective
action
on a manifold
with a singular surface and then apply the replica formula
and obtain from it the entanglement entropy. In
dimensions the effective
action has the general structure
An important property of the expansion (269) for a quantum field theory, which classically is
conformally invariant, is that the logarithmic term
is conformally invariant (see [219] and references
therein),
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Living Rev. Relativity 14, (2011), 8
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