6 Logarithmic Term in the Entropy of Generic Conformal Field Theory

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 d 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 δ = 2π(1 − α ) at the horizon surface Σ, compute the effective action W (E α) on a manifold E α with a singular surface and then apply the replica formula S = (α∂ − 1)| W (E α) α α=1 and obtain from it the entanglement entropy. In d dimensions the effective action has the general structure

α ad a1 an (α) WCFT (E ) = -d − -d−2 − .. − -d−2n − ..− ad∕2ln𝜖 + w (g ). (269 ) 𝜖 𝜖 𝜖
The logarithmic term in this expansion appears only if dimension d is even. Thus, only even d will be considered in this section. The terms ad, ad−2,…representing the power UV divergences, are not universal, while the term ad∕2 is universal and is determined by the integrated conformal anomaly. The term w(g(α)) is the UV finite part of the effective action. Under a global rescaling of the metric on E α, (α) 2 (α) g → λ g, one has
w(λ2g (α)) = w (g (α)) + a ln λ . (270 ) d∕2

An important property of the expansion (269View Equation) for a quantum field theory, which classically is conformally invariant, is that the logarithmic term ad∕2 is conformally invariant (see [219] and references therein),

ad∕2[e−2ωg] = ad∕2[g].
On a manifold with a conical singularity at the surface Σ, the coefficients ad− 2n have a bulk part and a surface part. To first order in (1 − α ) one finds that
α bulk Σ 2 ad−2n(E ) = αa d−2n(E ) + (1 − α )a d− 2n + O (1 − α) (271 )
For n = d∕2 the coefficients abulk(E ) d∕2 and a (Σ) d∕2 are the integrated bulk and the surface conformal anomalies respectively. The bulk and surface terms are independently invariant under conformal transformation g → e−2ωg,
abulk(e−2ωg) = abulk(g) and aΣ (e− 2ωg ) = aΣ (g) . (272 ) d∕2 d∕2 d∕2 d∕2
Applying the replica formula one obtains the entanglement entropy
S = sd−-2+ ..+ sd−2n-+ ..+ s Σln 𝜖 + s(g) , 𝜖d− 2 𝜖d−2n 0 s0 = aΣ , sd−2n = aΣ , n = 1,..,d∕2 − 1 d∕2 n s(λ2g) = s(g) − aΣd∕2ln λ . (273 )
For a regular manifold each term ad− 2n is the integral of a polynomial of degree n in the Riemann curvature. Respectively, the surface terms Σ ad−2n are the integral over the singular surface Σ of polynomial of degree n − 1 in the Riemann curvature and its projections onto the subspace orthogonal to surface Σ. The concrete structure of the polynomials depends on the dimension d.

 6.1 Logarithmic terms in 4-dimensional conformal field theory
 6.2 Logarithmic terms in 6-dimensional conformal field theory
 6.3 Why might logarithmic terms in the entropy be interesting?

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