### 3.9 UV divergences of entanglement entropy for a scalar field

For a bosonic field described by a field operator the partition function is . The corresponding effective action on a space with a conical singularity, , is expressed in terms of the heat kernel in a standard way
The entanglement entropy is computed using the replica trick as
Using the small 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 (67). 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 [33, 152, 71, 85, 196]

##### In three dimensions
the horizon is a circle and the entropy

is linearly divergent.

##### The leading UV divergence in d dimensions
can be computed directly by using the form of the coefficient  (70) in the heat kernel expansion [33]

It is identical to expression (28) 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 the UV divergent terms in the entanglement entropy of a generic 4-dimensional black hole read [197]

We note that for a massive scalar field . Of special interest is the case of the 4d conformal scalar field. In this case and the entropy (82) takes the form
The logarithmic term in Eq. (83) is invariant under the simultaneous conformal transformations of bulk metric and the metric on the surface , . 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,

This metric has a vanishing Ricci scalar, . It has inner and out horizons, and respectively, defined by
where is the mass of the black hole and is the electric charge of the black hole. The two vectors normal to the horizon are characterized by the non-vanishing components , . The projections of the Ricci and Riemann tensors on the subspace orthogonal to are
Since for the Reissner–Nordström metric, the entanglement entropy of a massless, minimally coupled, scalar field and of a conformally-coupled scalar field coincide [197],
where and represents the UV finite term. Since is dimensionless it may depend only on the ratio 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. (87) becomes an irrelevant constant. Let us consider two cases when this happens.

##### The Schwarzschild black hole.
In this case () and so that the entropy, found by Solodukhin [196], is

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 . 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 (). The entropy of the extreme black hole is found to take the form [197]

Notice that we have omitted the irrelevant constants and in Eq. (88) and (89) respectively.

#### 3.9.2 The dilatonic charged black hole

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

with the metric functions
where is the dilaton charge, . The outer and the inner horizons are defined by
The entanglement entropy is defined for the outer horizon at . The Ricci scalar of metric (90)
vanishes at the outer horizon, . Therefore, the entanglement entropy associated with the outer horizon is the same for a minimal scalar field () and for a conformally-coupled scalar field (,
where is the area of the outer horizon.

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

so that expression (93) takes the form
In the extremal limit, , the area of the outer horizon vanishes, , and the whole black-hole entropy is determined only by the logarithmically-divergent term (using a different brick-wall method a similar conclusion was reached in [114])
In this respect the extreme dilatonic black hole is similar to a two-dimensional black hole. Notice that Eq. (95) 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, 110, 61, 104, 82, 28, 227, 48, 47, 46, 135, 137, 176, 196, 197, 117, 118, 114, 115, 116].