3.9 UV divergences of entanglement entropy for a scalar field

For a bosonic field described by a field operator 𝒟 the partition function is Z(α ) = det−1∕2𝒟. The corresponding effective action W (α ) = − ln Z (α ) on a space with a conical singularity, E α, is expressed in terms of the heat kernel KE α(s) in a standard way
1 ∫ ∞ ds W (α ) = − -- ---TrKE α(s). (77 ) 2 𝜖2 s
The entanglement entropy is computed using the replica trick as
S = (α ∂α − 1)W (α)|α=1 . (78 )
Using the small s expansion one can, in principle, compute all UV divergent terms in the entropy. However, the surface terms are known only for the first few terms in the expansion (67View Equation). This allows us to derive an explicit form for the UV divergent terms in the entropy.

In two dimensions
the horizon is just a point and the entanglement entropy diverges logarithmically [33Jump To The Next Citation Point, 152, 71, 85, 196Jump To The Next Citation Point]

1 1 Sd=2 = 6 ln 𝜖 . (79 )

In three dimensions
the horizon is a circle and the entropy

-A-(Σ)- Sd=3 = 12 √π-𝜖 (80 )
is linearly divergent.

The leading UV divergence in d dimensions
can be computed directly by using the form of the coefficient aΣ1 (70View Equation) in the heat kernel expansion [33Jump To The Next Citation Point]

Sd = -------1--------A(Σ-). (81 ) 6(d − 2)(4π)d−22 𝜖d−2
It is identical to expression (28View Equation) for the entanglement entropy in flat Minkowski spacetime. This has a simple explanation. To leading order the spacetime near the black-hole horizon is approximated by the flat Rindler metric. Thus, the leading UV divergent term in the entropy is the entanglement entropy of the Rindler horizon. The curvature corrections then show up in the subleading UV divergent terms and in the UV finite terms.

The four-dimensional case
is the most interesting since in this dimension there appears a logarithmic subleading term in the entropy. For a scalar field described by a field operator − (∇2 + X ) the UV divergent terms in the entanglement entropy of a generic 4-dimensional black hole read [197Jump To The Next Citation Point]

A(Σ ) 1 ∫ ( 1 ) Sd=4 = -----2 − ----- R + 6X − -(Rii − 2Rijij) ln𝜖 . (82 ) 48 π𝜖 144π Σ 5
We note that for a massive scalar field X = − m2. Of special interest is the case of the 4d conformal scalar field. In this case X = − 1R 6 and the entropy (82View Equation) takes the form
A (Σ) 1 ∫ Sconf = -----2 + ----- (Rii − 2Rijij) ln 𝜖. (83 ) 48π 𝜖 720π Σ
The logarithmic term in Eq. (83View Equation) is invariant under the simultaneous conformal transformations of bulk metric gμν → e2σg μν and the metric on the surface Σ, γij → e2σγij. This is a general feature of the logarithmic term in the entanglement entropy of a conformally-invariant field.

Let us consider some particular examples.

3.9.1 The Reissner–Nordström black hole

A black hole of particular interest is the charged black hole described by the Reissner–Nordström metric,

ds2RN = g(r)dτ 2 + g −1(r)dr2 + r2(d𝜃2 + sin2𝜃dϕ2 ), g(r) = 1 − (r −-r+-)(r −-r−-). (84 ) r2
This metric has a vanishing Ricci scalar, ¯R = 0. It has inner and out horizons, r− and r+ respectively, defined by
∘ -------- r ± = m ± m2 − q2 , (85 )
where m is the mass of the black hole and q is the electric charge of the black hole. The two vectors normal to the horizon are characterized by the non-vanishing components nτ1 = g− 1∕2(r), ∘---- nr = g (r ) 2. The projections of the Ricci and Riemann tensors on the subspace orthogonal to Σ are
2r−- 2r+-−-4r−- Rii = − r3 , Rijij = r3 . (86 ) + +
Since R = 0 for the Reissner–Nordström metric, the entanglement entropy of a massless, minimally coupled, scalar field (X = 0) and of a conformally-coupled scalar field X = − 1R 6 coincide [197Jump To The Next Citation Point],
S = -A(Σ-) + -1-(2r+-−-3r−-)ln r+-+ s(r−) , (87 ) RN 48 π𝜖2 90 r+ 𝜖 r+
where 2 A(Σ ) = 4πr+ and r− s(r+) represents the UV finite term. Since s is dimensionless it may depend only on the ratio r−r+- of the parameters, which characterize the geometry of the black hole.

If the black hole geometry is characterized by just one dimensionful parameter, the UV finite term in Eq. (87View Equation) becomes an irrelevant constant. Let us consider two cases when this happens.

The Schwarzschild black hole.
In this case r− = 0 (q = 0) and r+ = 2m so that the entropy, found by Solodukhin [196Jump To The Next Citation Point], is

A (Σ) 1 r+ SSch = ----2-+ ---ln ---. (88 ) 48π𝜖 45 𝜖
Historically, this was the first time when the subleading logarithmic term in entanglement entropy was computed. The leading term in this entropy is the same as in the Rindler space, when the actual black-hole spacetime is approximated by flat Rindler spacetime. This approximation is sometimes argued to be valid in the limit of infinite mass M. However, we see that, even in this limit, there always exists the logarithmic subleading term in the entropy of the black hole that was absent in the case of the Rindler horizon. The reason for this difference is purely topological. The Euler number of the black-hole spacetime is non-zero while it vanishes for the Rindler spacetime; the Euler number of the black-hole horizon (a sphere) is 2, while it is zero for the Rindler horizon (a plane).

The extreme charged black hole.
The extreme geometry is obtained in the limit r → r − + (q = m). The entropy of the extreme black hole is found to take the form [197Jump To The Next Citation Point]

A (Σ ) 1 r Sext = ----2-− ---ln -+-. (89 ) 48π𝜖 90 𝜖
Notice that we have omitted the irrelevant constants s(0) and s(1) in Eq. (88View Equation) and (89View Equation) respectively.

3.9.2 The dilatonic charged black hole

The metric of a dilatonic black hole, which has mass m, electric charge q and magnetic charge P takes the form [120]:

2 2 −1 2 2 2 2 2 ds = g(r)dτ + g (r)dr + R (r)d(d𝜃 + sin 𝜃dϕ ) (90 )
with the metric functions
(r − r+)(r − r− ) g(r) = -------2--------, R2(r) = r2 − D2 , (91 ) R (r)
where D is the dilaton charge, 2 2 D = P-−2m-q. The outer and the inner horizons are defined by
∘ --2-----2----2----2 r± = m ± m + D − P − q . (92 )
The entanglement entropy is defined for the outer horizon at r = r+. The Ricci scalar of metric (90View Equation)
2(r-−-r+-)(r-−-r− ) R = − 2D (r2 − D2 )3 .
vanishes at the outer horizon, r = r+. Therefore, the entanglement entropy associated with the outer horizon is the same for a minimal scalar field (X = 0) and for a conformally-coupled scalar field (X = − 1R ) 6,
-A-Σ-- -1- 3r+-(r+-−--r− ) r+- r−- D-- Sdilaton = 48π𝜖2 + 90 ( (r2 − D2) − 1)log 𝜖 + s(r ,r ), (93 ) + + +
where AΣ = 4π(r2+ − D2 ) is the area of the outer horizon.

It is instructive to consider the black hole with only electric charge (the magnetic charge P = 0 in this case). This geometry is characterized by two parameters: m and q. In this case one finds

2 2 2 r = 2m − q---, r = q---, r2 − D2 = 4m (m − -q--) + 2m − 2m + 2m
so that expression (93View Equation) takes the form
A Σ 1 q2 r+ q Sdilaton = -----2 + ---(1 + 3(1 − ---2)) ln ---+ s( --). (94 ) 48π 𝜖 180 2m 𝜖 m
In the extremal limit, 2m2 = q2, the area of the outer horizon vanishes, A Σ = 0, and the whole black-hole entropy is determined only by the logarithmically-divergent term5 (using a different brick-wall method a similar conclusion was reached in [114Jump To The Next Citation Point])
1 r+ Sext−dil = 180-log-𝜖-. (95 )
In this respect the extreme dilatonic black hole is similar to a two-dimensional black hole. Notice that Eq. (95View Equation) is positive as it should be since the entanglement entropy is, by definition, a positive quantity.

The calculation of the entanglement entropy of a static black hole is discussed in the following papers [102, 94, 110Jump To The Next Citation Point, 61, 104Jump To The Next Citation Point, 82, 28, 227, 48, 47, 46Jump To The Next Citation Point, 135, 137, 176, 196Jump To The Next Citation Point, 197, 117, 118, 114, 115, 116].

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