#### Logarithmic black hole entropy corrections and holographic Rényi entropy

Eur. Phys. J. C
Logarithmic black hole entropy corrections and holographic Rényi entropy
Subhash Mahapatra 0 1
0 Department of Physics, KU Leuven Campus Kortrijk - KULAK , Etienne Sabbelaan 53 bus 7800, 8500 Kortrijk , Belgium
1 The Institute of Mathematical Sciences , Chennai 600113 , India
The entanglement and Rényi entropies for spherical entangling surfaces in CFTs with gravity duals can be explicitly calculated by mapping these entropies first to the thermal entropy on hyperbolic space and then, using the AdS/CFT correspondence, to the Wald entropy of topological black holes. Here we extend this idea by taking into account corrections to the Wald entropy. Using the method based on horizon symmetries and the asymptotic Cardy formula, we calculate corrections to the Wald entropy and find that these corrections are proportional to the logarithm of the area of the horizon. With the corrected expression for the entropy of the black hole, we then find corrections to the Rényi entropies. We calculate these corrections for both Einstein and Gauss-Bonnet gravity duals. Corrections with logarithmic dependence on the area of the entangling surface naturally occur at the order G0D. The entropic c-function and the inequalities of the Rényi entropy are also satisfied even with the correction terms.
1 Introduction
Quantum entanglement is one of the most remarkable
properties of quantum systems; it is essential to most
quantuminformation applications. The quantification and
characterization of entanglement is an important problem in
quantuminformation science and a number of measures have been
suggested for its definition in the literature [
1
]. Two such
measures are entanglement and Rényi entropies. Formally,
given a subsystem v and its complement v¯, the entanglement
entropy of the subsystem v is defined by the von Neumann
entropy of its reduced density matrix ρv,
SEE = −T r [ρv ln ρv],
(1)
where ρv is obtained by tracing out the degrees of freedom of
the subsystem v¯ from the full density matrix. Similarly, the
Rényi entropy is defined by a one-parameter generalization
of the von Neumann entropy,
(2)
1
Sq = 1 − q ln T r [ρvq ],
where q is a positive real number. Given the above
definition, one finds that limq→1 Sq = SEE, i.e., the entanglement
entropy is recovered from the Rényi entropy by taking the
limit q → 1.
The properties of entanglement and Rényi entropies have
been the subject of intense investigations in the last two
decades and form the basis for many applications ranging
from condensed matter physics to quantum gravity. The
standard way of calculating these entropies in quantum field
theories is the replica method [
2
]. In this method, one maps the
trace of the qth power of the density matrix ρv to the
partition function on a singular q-folded Riemann surface.
Geometrically, this q-fold space is a flat cone with angle deficit
2π(1 − q) at the entangling surface, and the Euclidean path
integral over fields defined on this q-fold space can be
explicitly computed for a few simple cases. However, for generic
quantum field theories the computation of the partition
function on the singular surfaces is rather difficult, which limits
the usefulness of the replica method.
On the other hand, recent developments in the AdS/CFT
correspondence [
3
] have suggested an elegant and geometric
way of computing entanglement entropy in conformal field
theories (CFTs) which have gravity duals. In the seminal
work of Ryu and Takayanagi (RT) [
4,5
], the entanglement
entropy of a d-dimensional boundary CFT was conjectured
to be given by the area of a minimal surface in the bulk AdS
space as
SEE =
Area(γv) ,
4G D
(3)
where γv is the (d − 1)-dimensional minimal-area
hypersurface, which extends into the bulk spacetime and shares
the same boundary ∂v of the subsystem v. This proposal has
been extensively tested for a variety of systems and by now
there is a good amount of evidence to support this
conjecture [
6
]. Indeed, further indications for the correctness of this
conjecture was provided in [
7
], where the generalized replica
method was applied in the bulk AdS space to prove the RT
conjecture.
An alternative approach to calculating the holographic
entanglement entropy, which works only for spherical
entangling surfaces, was proposed in [
8
]. In [
8
], it was observed
that the reduced density matrix for a spherical entangling
region in flat space can be conformally mapped to the
thermal density matrix on the hyperbolic space; or equivalently
the original entanglement entropy can be mapped to
thermal entropy on the hyperbolic space [
9
]. Then, using the
AdS/CFT correspondence, the latter thermal entropy can
be again mapped on the dual gravity side to black hole
entropy of certain topological black holes with hyperbolic
horizons. This observation therefore also provided an
alternative derivation of the RT conjecture for a spherical
entangling surface in d-dimensional CFT. Moreover, in [
10
], it
was observed that the idea of [
8
] can be further extended
to the study of the Rényi entropy for a spherical entangling
region. For the record, let us also mention that a holographic
variant for the Rényi entropy, which works for any
entangling surface, has recently been put forth in [
11
]. See also
[
12,13
], for a discussion of the holographic Rényi entropy in
two-dimensional boundary CFTs.
The procedure of mapping entanglement and Rényi
entropies of the boundary CFT to the entropy of the black
hole have many advantages. In particular, this mapping can
easily be generalized to calculate Rényi entropy of
higherdimensional CFTs. Moreover, this mapping can also be
generalized to calculate Rényi entropy of boundary theories
which are dual to higher derivative gravity theories.
An essential point which comes in the derivation of [
8,10
]
is the notion of black hole entropy, which can be computed
using the standard Wald formula. Using the Wald entropy,
the above procedure indeed reproduces independent known
results of Rényi entropy. However, an important point, which
we want to stress here and that will play a significant role in
this paper, is that the Wald entropy only gives the leading
order contribution to the entropy of the black hole. There
are additional subleading quantum corrections to the usual
Wald entropy, which are generally proportional to the
logarithm of the area of the horizon. Indeed, recent activities in
most quantum gravity models, including string theory and
loop quantum gravity (LQG) or methods based on
diffeomorphism symmetry arguments, have predicted logarithmic
corrections to the entropy of the black hole, with a general
expression like
The coefficient C has been calculated for a variety of cases.
For example, for the Schwarzschild black hole the
coefficient C was found to be equal to − 3/2 in LQG [14]. There,
the horizon was treated as a boundary spacetime which was
quantized within the “quantum geometry” program.
Similarly in string theory, using the Euclidean gravity approach,
logarithmic corrections to the entropy of extremal and
nonextremal asymptotically flat black holes have been computed
in [
15–18
], also see [
19–22
]. There, the logarithmic
correction arises at the one-loop level from the massless fields of
the theory and the coefficient C was found to be dependent
on the total number of spacetime dimensions as well as the
number of massless fields. A simple comparison showed
disagreement on the value of C in these two quantum gravity
models.1
The black hole entropy and its logarithmic correction can
also be calculated using an approach based on the
argument of diffeomorphism symmetry on the horizon. This
approach, using the work of [
28
], was initiated in [
29
] and
later emphasized in [
30–32
] to calculate black hole entropy.
In this approach one first identifies a set of vector fields
(based on some physical consideration) and then construct
an algebra for the Fourier modes of the charges
(corresponding to an appropriate diffeomorphism symmetry) from these
vector fields. Using this algebra, which usually has a
twodimensional Virasoro algebra like structure, one then extracts
the central charge and the zero mode from it and then finally
uses the Cardy formula [
34
] for the asymptotic density of
states to compute the entropy of the black hole. It has been
shown in [
30–32
] that this approach does indeed correctly
reproduce the expression for the black hole entropy.
Moreover, in [
35
], it was shown that this symmetry-based approach
can be further used to calculate the logarithmic correction to
the BTZ black hole and it was found that C = − 3/2.2
Importantly, the above procedure can also be generalized
to compute logarithmic corrections to the entropy of
hyperbolic black holes which are relevant for our purpose here,
especially since it is unclear (at-least to the author) how to
calculate logarithmic corrections to entropy of these black holes
in LQG or in string theory context. In this paper, using the
methodology of [
36
], we calculate these corrections for two
gravity theories, namely Einstein and Gauss–Bonnet
gravity. We find C = −3/2 for both cases. However, we want to
stress from the outset that the whole purpose for this exercise
is to show at-least one method by which corrections to the
entropy of the hyperbolic black holes can be explicitly
cal1 Black hole entropy can also acquires logarithmic corrections due to
thermal fluctuations in the black hole extensive parameters. More details
can be found in [
23–27
].
2 This result was based on certain assumptions. We will discuss these
assumptions in more detail in Sect. 4.
culated and that it is non-zero. Although important in its own
way, it is not our main objective to settle down the
controversial nature of C here. Instead, we want to probe the effects
of this C on the Rényi entropy. For this reason, we will work
with arbitrary C in this paper for most of the time.
Using the corrected expression for the entropy of the black
hole (Eq. (4)) into the prescription of [
8,10
], we find that there
are corrections to the standard results of Rényi entropies.
These corrections are both logarithmic and non-logarithmic
in nature. For a two-dimensional CFT, we find that the
correction terms in the Rényi entropy are a function of the
logarithmic of the central charge as well as index q. However, the
dependence on the size of the system is doubly logarithmic.3
For higher dimension CFTs, Rényi entropy is a complicated
function of q. For the entanglement entropy our results
simplify tremendously. We find that correction term in the
entanglement entropy is proportional to C times the logarithm of its
standard expression. Interestingly, at the order G0D, the
correction term in the entanglement entropy depends
logarithmically on the area of the spherical entangling surface. This
is important since quantum corrections to the holographic
entanglement entropy are expected to be of order G0D.
Moreover, using the renormalized entanglement entropy
prescription of [37], we find that the corrected entanglement entropy
expression can again be properly renormalized and that the
universal part of the entanglement entropy remains the same
even with the correction terms. Therefore, the entropic
ctheorem is again satisfied. Our analysis further suggests a
similar kind of corrections in the Rényi entropy too. With a
Gauss–Bonnet black hole as a gravity dual, the holographic
Rényi entropy is found to be a complicated function of two
distinct central charges. However, the entanglement entropy
depends only on one central charge. The correction terms in
the entanglement entropy are again found to be proportional
to C times the logarithm of its standard expression.
It is also well known that the Rényi entropy satisfies a few
inequalities involving the derivative with respect to q [
38,39
].
We find that, for higher-dimensional CFTs, these inequalities
are again satisfied even with correction terms provided the
coefficient C is not very large. However, for two-dimensional
CFTs, a few of these inequalities can be violated in the limit
of small central charge. However, for large central charge,
which is the case for the boundary CFT with gravity dual,
these inequalities are again found to be satisfied. Our analysis
therefore provide a strong evidence for their validity at order
G0D.
The RT prescription only provides the leading order result
of the EE (which is of order N 2 or G−D1) and much
attention now a days has been paid to obtained the subleading
corrections to it (of the order of G0D). Recently some work
3 Similar kind of doubly logarithmic structure in the entanglement
entropy has also appeared in a different system in [
33
].
has been done in this direction; see for example [
40,41
].
In [40], it was conjectured that the subleading correction
to the entanglement entropy is given by the entanglement
of the Ryu–Takayanagi minimal-area surface with the rest
of the bulk AdS. This again requires the difficult replica
method in the bulk AdS to obtained the subleading
correction to the entanglement entropy. Moreover, this conjecture
for the subleading order is for the entanglement entropy
only and no such analogous conjecture is known for the
Rényi entropy. In [
41
], subleading corrections to the
entanglement and Rényi entropies were obtained by calculating the
one-loop determinants around the classical solutions using
the Schottky uniformization of q-sheeted Riemann surface.
There are again many difficulties and limitations in
implementing this method, such as that constructing the smooth
bulk solutions is not always straightforward and that
performing analytic continuation of the replica index to non-integer
q is sometimes very difficult. One can use the method of [
41
]
to calculate the corrections for 1 + 1-dimensional CFT with
Einstein gravity in the bulk. Whether this method can be
generalized for higher-dimensional conformal field theory or to
higher derivative gravity theory in the bulk is not clear yet.
In this work we are suggesting another method by which
not only the corrections to entanglement entropy but also
to the Rényi entropies can be obtained. Our idea does not
involve the mathematical complexity of [
40,41
] and is easy
to implement. This method again has the limitation that it
works only for the spherical entangling surface.
The paper is organized as follows: In the next section, we
review the main ideas of [
8,10
] to relate the Rényi entropy
to the entropy of the black hole. In Sect. 3, we highlight the
necessary steps to calculate the asymptotic form of density of
states using the two-dimensional conformal algebra. In Sect.
4, we first construct the Virasoro algebra having a central
extension on the black hole horizon and then use the
expression of the density of states to calculate the the entropy of
the black hole. In the process we calculate the logarithmic
correction to the entropy of AdS-Schwarzschild and Gauss–
Bonnet black hole. In Sect. 5, we analyze the holographic
Rényi entropy in detail and discuss the nature of the
correction terms. Finally, we conclude by summarizing our main
results in Sect. 6.
2 Holographic entanglement and Rényi entropies
In this section, we review some aspects of holographic
entanglement and Rényi entropies which will be the focus of this
paper. As mentioned in the introduction, we follow the
prescription of [
8,10
] to calculate these entropies for a spherical
entangling region in the boundary CFT using the AdS/CFT
correspondence. Let us first briefly discuss the work done in
these papers to set the stage.
Consider a d-dimensional CFT on Minkowski space and
choose a spherical entangling surface of radius R as a
subsystem. The computation of entanglement entropy of this
subsystem with the rest of the system can be performed by
calculating the reduced density matrix ρv. However, the authors in
[
8
], using the conformal structure of the theory, mapped this
problem of the entanglement entropy to the thermal entropy
on a hyperbolic space R × H d−1. They showed that the causal
development of the ball enclosed by the spherical entangling
surface can be mapped to a hyperbolic space R × H d−1; with
curvature of H d−1 space given by the radius of the spherical
entangling region R. An important point of this mapping was
that the vacuum of the original CFT mapped to a thermal bath
with temperature
on the hyperbolic space. Now relating the density matrix
ρtherm in the new spacetime R × H d−1 to the old spacetime
ρv by a unitary transformation, ρv = U −1ρthermU , we get
where Z (T0) = T r [e−H/T0 ]. For the Rényi entropy we also
need the qth power of ρv. From the above equation, we get
Taking the trace of both sides of Eq. (7), we get
as U and its inverse cancel each other upon taking trace. Now,
using the definition of Rényi entropy as in Eq. (2), we arrive
at
1
Sq = 1 − q ln Z (T0/q) − q ln Z (T0) .
The above expression for the Rényi entropy can also be
written in terms of the free energy F (T ) = −T ln Z (T ):
q 1
Sq = 1 − q T0 F (T0) − F (T0/q) ,
and further, using the thermodynamic relation Stherm =
−∂ F/∂ T , we can rewrite the above expression as
q
1
T0
Sq = q − 1 T0 T0/q dT Stherm(T );
here, just to clarify again, Stherm is the thermal entropy of a
d-dimensional CFT on R × H d−1, while Sq is the desired
Rényi entropy. Equation (11) was the main result of [
8,10
],
which relates Rényi (and hence entanglement) entropy of
a spherical entangling region in d-dimensional CFT to the
thermal entropy on a hyperbolic space. As pointed out in
[8], the above analysis just mapped one difficult problem to
another equally difficult problem and is not particularly
useful for practical purposes. However, its true usefulness can be
realized via the AdS/CFT correspondence. In the AdS/CFT
correspondence, the thermal state of the boundary CFT
corresponds to an appropriate non-extremal black hole in the
bulk AdS spacetime, with thermal entropy corresponding
to black hole entropy. Therefore, using the AdS/CFT
correspondence, we can relate Stherm appearing in Eq. (11) to
that of the black hole entropy, which is relatively easy to
compute. Since on the boundary side our CFT is on R × H d−1,
its dual gravity theory will be described by a topological
black hole with hyperbolic event horizon. In any event, in
this AdS/CFT approach, the Rényi entropy is now given by
the horizon entropy of the corresponding hyperbolic black
hole, which can easily be computed using Ward’s standard
formula. However, since we are interested in calculating the
effects of the corrections of the entropy of the black hole on
the Rényi entropy, here we will use an approach based on
symmetry arguments on the horizon to calculate the black
hole entropy, instead of Wald’s formula. This is the topic of
the discussion of the next section.
3 Logarithmic corrections to the entropy of the black hole from the Cardy formula
In this section, we will describe the necessary steps to
calculate the asymptotic form of density of states from a
twodimensional conformal algebra.4 This form of density of
states will be used in a later section to calculate the black hole
entropy and, further, to compute logarithmic corrections. In
this section, we will mostly follow the notations used in [
35
]
and refer the reader to [
35
] for a detailed discussion.5
We start with a standard Virasoro algebra of the two
conformal field theory with central charges c, c¯:
Lm , Ln = (m − n)Lm+n + 1c2 m(m2 − 1)δm+n,0
c
L¯ m , L¯ n = (m − n)L¯ m+n + 1¯2 m(m2 − 1)δm+n,0,
Lm , L¯ n = 0;
(12)
here Lm and L¯ m are the generators of holomorphic and
antiholomorphic diffeomorphisms. If ρ( , ¯ ) denotes the
degeneracy of states carrying L0 = and L¯ 0 = ¯
eigenvalues, then one can define the partition function on the
twotorus of modulus τ = τ1 + i τ2 as
4 See [
42
], for generalization of the Cardy formula in
higherdimensional CFT.
5 Extension of the Cardy formula beyond the subleading order was
performed in [
43
].
Z (τ, τ¯) = T r (e2πiτ L0 e−2πiτ¯ L¯ 0 )
ρ( , ¯ )e2πiτ e−2πiτ¯ ¯ .
Now using q = e2πiτ , q¯ = e−2πiτ¯ and inverting the above
equation, we get
1
ρ( , ¯ ) = (2π i )2
1
1
q +1 q¯ ¯ +1
Z (q, q¯ )dqdq¯ ,
where the integrals are along contours that enclose q = 0 and
q¯ = 0. Therefore, if we know the partition function Z (q, q¯ ),
then we can use Eq. (14) to determine the density of states.
Now modular invariance of the theory implies that
Z0(τ, τ¯) = T r e2πiτ (L0− 2c4 )e−2πiτ¯(L¯ 0− 2c¯4 ) .
Z0 is invariant under large τ −→ −1/τ diffeomorphism.
Cardy has shown that the invariance in Eq. (15) is based on
the general properties of two-dimensional conformal field
theory and therefore expected to be universal. From Eqs.
(13) and (15), we note that
Z (τ, τ¯) = e 2π2i4cτ e− 2π2i4c¯τ¯ Z0(τ, τ¯)
using the modular invariance of Z0, we get
Z (τ, τ¯) = e 2π24ic (τ + τ1 )e− 2π24ic¯ (τ¯+ τ¯ ) Z
1
substituting Eq. (16) into Eq. (14), we obtain
1 1
− τ , − τ¯
,
ρ( , ¯ ) =
dτ dτ¯ e−2πi τ + 2π24ic (τ + τ1 )e2πi ¯ τ¯− 2π24ic¯ (τ¯+ τ1¯ )
×Z
1 1
− τ , − τ¯
.
The above integral has the form
I [a, b] =
dτ e2πiaτ + 2πτib F (τ ),
which can be evaluated by saddle point approximation. For
this purpose, we need to assume that F (τ ) is slowly
varying near the extremum of the phase. As shown in [
35
], this is
indeed the case if one considers the situation where the
imaginary part of τ is large. Now the saddle point, obtained by
extremizing the exponent on the right hand side of the above
equation, is at τ0 = √b/a ≈ i √c/24 , where we have
assumed a large . Expanding around this saddle point, we
find
I [a, b] ≈
=
b
− 4a3
dτ e
e4πi√abF (τ0),
F (τ0)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
and an analogous expression exists for the τ¯ integral. Finally,
one obtains the expression for the density of states as
ρ( , ¯ ) ≈
Since, for most cases of our interest, we have only one
Virasoro algebra instead of two (see the next section for details),
the relevant expression for density of states is
ρ( ) ≈
In the above expression, the exponential term is the standard
Cardy formula. However, an important part, which will play
a significant role in our analysis later on, is the term that has
a power law behavior. In the next section, we will use the
logarithm of ρ to calculate the entropy of the black holes. As
one can anticipate, the exponential part of Eq. (21) will give
the usual Wald entropy and, on the other hand, the power
term will provide a logarithmic correction to the entropy of
the black hole.
4 Black hole entropy and Virasoro algebra from the surface term of the gravitational action
There has been a lot of activity in understanding the
black hole entropy using the symmetry-based horizon CFT
approach. This approach essentially assumes that the
symmetries of a black hole horizon are sufficient to compute the
density of states (hence the black hole entropy) at a given
energy. Many avatars of this approach have appeared in the
literature, and all of them have successfully predicted the
expression for the entropy of the black hole.6 A complete list
of references for later developments can be found in [
44
].
Applying a similar line of symmetry reasoning, a new
approach was recently proposed in [
36
] which is
straightforward and conceptually clearer. Importantly, it does not
require any ad hoc prescription such as shifting the
zeromode energy and is also easy to implement. In this approach,
the Noether currents associated with the diffeomorphism
invariance of the Gibbons–Hawking boundary action, instead
of the bulk gravity action, are used to construct the Virasoro
algebra at the horizon. Here, the diffeomorphisms are
chosen in such a way that they leave the near-horizon structure
of the metric invariant. The Virasoro algebra constructed in
this way is again found to have a central extension, which
upon using the Cardy formula correctly reproduces the black
hole entropy expression. In this work, we will follow this
6 A certain level of arbitrariness is present in the procedure of all
symmetry-based approaches in order to produce the correct black hole
entropy expression. We will further discuss this arbitrariness in the
following.
boundary Noether current procedure of [
36
] to calculate the
black hole entropy. We will first discuss its general
formalism and then apply this formalism to calculate the entropy of
the black hole of two gravity theories, namely Einstein and
Gauss–Bonnet gravity.
We start with the Gibbons–Hawking surface term IGH:
(22)
1
IGH = 8π G D ∂M
d D−1x √
1
−γ L = 8π G D
×
d D x √−g∇a (na L),
M
where ∂M is the boundary of the manifold M, L is related
to the trace of the extrinsic curvature, na is the unit normal
vector to the boundary, gμν denotes the bulk metric and γμν
is the induced metric on the boundary. The expression for
L depends on the gravity theory under consideration. For
example, in Einstein gravity it is equal to trace of the extrinsic
curvature K = −∇μnμ. The conserved current J μ associated
with the diffeomorphism invariance x μ −→ x μ + ξ μ of IGH
can be obtained by considering the variation of both sides of
Eq. (22) as the Lie derivative. After a bit of algebra, one gets
1
J μ[ξ ] = ∇ν J μν [ξ ] = 8π G D ∇ν Lξ μnν − Lξ ν nμ . (23)
The charge corresponding to this conserved current is defined
as
∂Σ
where J μν is called the Noether potential and d μν =
−d D−2x (nμmν − mμnν ) is the surface element of the
(D − 2)-dimensional surface ∂Σ with metric hμν . In order to
discuss black hole physics, we will choose the surface ∂Σ to
be near the horizon. The unit vectors nμ and mμ are chosen
to be spacelike and timelike, respectively.
Following [
36
], we define the Lie bracket of the charges
(24)
as
From the above equation, it is clear that we only need to
know the vector field ξ μ to determine the charge algebra. This
can be done, as explained below, by choosing an appropriate
diffeomorphism which leaves the horizon structure invariant.
However, in order to proceed, let us first write the D(=
d+1)dimensional form of the metric as
ds2 = − f (r )N 2dt 2 + fd(rr2) + r 2d i j (x )dxi dx j ,
where d i j (x )dxi dx j is the line element of the (d −
1)dimensional space. For our discussion in the next section,
(26)
ξ u = S(u, x ),
r
ξ ˜ = −r˜∂u S(u, x ),
7 It is easy to check that the other condition Lξ guu = 0 is trivially
satisfied near the horizon.
dr
˜
du = dt˜ − f (r˜ + rh )
under which the metric in Eq. (27) reduces to
we will take this line element to be hyperbolic but as of
now this metric is completely general. The constant N 2 is
included in gtt to allow us to adjust the normalization of the
time coordinate. This will be useful later on, but will not play
any significant role in the black hole entropy calculation here.
To study the near-horizon structure, defined by f (rh ) = 0,
it is convenient to choose a coordinate r = r˜ + rh , in which
case the above metric reduces to
dr 2
˜
ds2 = − f (r˜ + rh )N 2dt 2 + f (r˜ + rh )
+(r˜ + rh )2d i j (x )dxi dx j .
In the near-horizon region i.e. r˜ → 0 limit, the function
f (r˜ + rh ) can be expanded as
f (r˜ + rh ) = r˜ f (rh ) + 1/2r˜2 f (rh ) + · · ·
Defining the surface gravity κ as κ = N f (rh )/2 and
dropping the terms beyond first order in f (r˜ + rh ), one notice
that the (t − r˜) part of the metric in Eq. (27) reduces to the
standard Rindler metric
dst˜2−r˜ = −2r˜ Nκ dt˜2 + 21r˜ Nκ dr˜2,
where t˜ = N t . The unit normal vectors can be chosen as
nμ = (0,
= (1/N
f (r˜ + rh ), 0, . . . , 0),
f (r˜ + rh ), 0, 0, . . . , 0)
mμ
Further, in order to find ξ μ, it is more useful to first transform
to Bondi like coordinates by the transformation
(27)
(28)
(29)
(30)
(31)
(32)
ds2 = − f (r˜ + rh )du2 − 2dudr˜ + (r˜ + rh )2d i j (x )dxi dx j .
Now we choose our vector field ξ μ by imposing the
condition that the horizon structure remain invariant under
diffeomorphism i.e. the metric coefficients gr˜r˜ and gur˜ remain
unchanged. This implies the following Killing equations,7:
Lξ gr˜r˜ = −2∂r˜ ξ
u = 0
Lξ gur˜ = − f (r˜ + rh )∂r˜ ξ u − ∂u ξ u − ∂r˜ ξ r˜ = 0.
Solving the above two equations, we get
where S is an arbitrary function and x are the coordinates on
the remaining (d − 1)-dimensional space. Now converting
back to the (t, r˜) coordinates, these vector fields take the form
ξ t = T − N f (r˜r˜+ rh ) ∂t T ,
r r
ξ ˜ = − N˜ ∂t T ,
where T (t, r˜, x ) ≡ S(u, x ). Therefore, we can
evaluate the expressions for ξ μ, Q[ξ μ] and the charge algebra
Q[ξ1], Q[ξ2] once the function T is given. Now, we expand
this function T in terms of a set of basis functions Tm , as
T =
m
As is standard in the literature, we choose Tm such that the
resulting ξmμ obeys the algebra isomorphic to Diff S1 i.e.
i {ξm , ξn}μ = (m − n)ξmμ+n,
where {, } is the Lie bracket. A particular choice is
1 eim(αt+g(r˜)+ p.x),
Tm = α
where α is an arbitrary parameter, g(r˜) is a function which
is regular at the horizon and pi are integers. We now have
all the ingredients at our disposal to calculate the leading
Wald as well as the subleading logarithmic correction to the
entropy of the black hole. From now on we will concentrate
on horizons with hyperbolic topology, as this will be required
in the computation of the holographic Rényi entropy in the
later section.
Before going to discuss the entropy of the black hole
explicitly, we like to clarify that the Noether current of Eq.
(23) is associated only with the diffeomorphism invariance of
the surface term and should not be confused with the usual
Noether current which is associated with the bulk gravity
Lagrangian [
45
]. The surface term here is used just to
construct the relevant Virasoro algebra, which we then evaluate
on the static horizon. Once we have the Virasoro algebra the
black hole entropy follows from the Cardy formula. Thus the
surface term is not directly contributing to the entropy of the
black hole, and therefore, not contradicting [
45
].
4.1 Entropy of AdS-Schwarzschild Black hole
We first apply the formalism developed above to a (d +
1)dimensional AdS-Schwarzschild black hole in Einstein
gravity. As mentioned earlier, for Einstein gravity the Gibbons–
Hawking surface term is standard and is given by the
expression L = K = −∇μnμ. The metric is given as
m r 2
f (r ) = −1 − r d−2 + L2 ,
(33)
(34)
(35)
(36)
(37)
Now substituting Eqs. (36) and (37) into Eqs. (39) and (40),
we get the final expressions as
Ad−1 κ
Q[ξm ] = 8π G D α δm,0
Ad−1 κ
Q[ξm ], Q[ξn] = −i (m − n) 8π G D α δm+n,0
−i m3 Ad−1 α
16π G D κ δm+n,0,
where Ad−1 is the area of (d − 1)-dimensional hyperbolic
space (the horizon area). Strikingly, the above charge algebra
is quite similar to the two dimensional Virasoro algebra
discussed in the previous section. From the above expressions,
we obtain the central term in the algebra as
Substituting above expressions into Eq. (21) with Q0 =
and taking log on both side, we obtain the black hole entropy
S = 4AGd−D1 − 23 ln 4AGd−D1 + · · · = SWald − 23 ln SWald + · · · .
where
is the metric on the (d − 1)-dimensional hyperbolic space
with dΩd2−2 being the line element on a unit (d − 2)-sphere.
The exact value of constant N 2 is not required here and will
be specified in the next section. Substituting Eq. (37) into
Eqs. (33), (23), (24) and (25) and taking the near-horizon
limit r˜ → 0, we get
1
Q[ξm ] = 8π G D
1
dd−1x √h κ Tm − 2 ∂t Tm .
H
Similarly, the algebra of the charges corresponding to T =Tm
is given by
1
− 2
1
Q[ξm ], Q[ξn] = 8π G D
We see that the leading term matches exactly with the usual
black hole entropy expression. We also find a logarithmic
correction to it. Interestingly, the coefficient −3/2 of the
logarithmic correction for the hyperbolic horizon is the same as
was found for the BTZ black hole in [
35
]. In order to find
this coefficient we have chosen the arbitrary parameter α to
be such that the central charge c is a universal constant i.e.
independent of the area Ad−1, as in [
35
]. We will say more
about this condition and the coefficient of this logarithmic
term at the end of this section.
4.2 Entropy of Gauss–Bonnet black hole Now we move on to discuss the entropy of Gauss–Bonnet black holes. The surface term of the gravitational action can be found in [46–49] and is given as
L = −∇μnμ + (D − 3)(D − 4)
2λL2
P − 2G˜ μν Kμν ,
(45)
where λ is the coefficient of the Gauss–Bonnet term, P is the
trace of the tensor
1
Pμν = 3 2KKμρ Kρν + Kρσ Kρσ Kμν − 2Kμρ Kρσ Kσν − K2Kμν
and G˜ μν stands for Einstein tensor of d-dimensional
boundary metric. For a Gauss–Bonnet black hole, the
D-dimensional metric is given by
where m is related to the mass of the black hole. Performing
the analogous steps as in the previous subsection to
calculate the charge and Virasoro algebra for Gauss–Bonnet black
hole, we get
(46)
(47)
(48)
.
(49)
1
+ 4κ
1 −
(D − 4)rh2
2(D − 2)λL2 κ
α δm,0
Now substituting Eqs. (36) and (47) into Eqs. (48) and (49),
we obtain
In the above equation, the first term is exactly the Wald
entropy of a Gauss–Bonnet black hole in D-dimensions.
We also find the logarithmic correction to the Wald entropy
with coefficient −3/2, which is the same as in the
AdSSchwarzschild black hole case. As one can see, however, now
in terms of horizon radius, there are two correction terms.
Before ending this section, it is worthwhile to point out
again the debatable nature of the coefficient of logarithmic
term in black hole entropy. The logarithmic correction to
the usual black hole entropy has been discussed in various
quantum gravity models. This correction term has been
successfully computed for asymptotically flat black holes in loop
quantum gravity (LQG), in string theory for both extremal
and near extremal black holes by microscopic counting
methods or by diffeomorphism symmetry arguments which
heavily rely on two dimensional CFT and the Cardy formula (as
we have done in this section). All of these methods have
either predicted different coefficient for the logarithmic term
or have a certain level of arbitrariness in their results. For
example, in LQG the coefficient was found to be −3/2 [
14
].
However, in string theory, this coefficient was found to be
different from −3/2 [
15–18
]. There, the logarithmic
corrections to the entropy of the black hole arise from the one-loop
contribution of the massless fields, and the coefficient was
found to be dependent on the total number of massless fields
as well as on the number of spacetime dimensions.
For the BTZ black hole, Carlip calculated this coefficient
using the Cardy formula and found it to be −3/2 [
35
]. He
also argued that similar logarithmic correction should appear
in all black holes whose microscopic degrees of freedom are
described by an underlying horizon CFT. However, in his
calculation Carlip had assumed, as we have also assumed in this
paper, that the central charge c is universal constant in a sense
that it is independent of the area of the horizon. To achieve
this assumption Carlip chose the parameter α (appearing in
Eq. (36)) in such a way that c becomes independent of area.
It is important to point out here that one could choose α to
be the surface gravity κ as well. The condition α = κ is also
well motivated from Euclidean gravity point of view and is
standard in the literature.8 To see this, let us consider the
Euclidean time (τ → i t ) and take the appropriate ansatz
for Tm as Tm = 1/αeim(ατ +g(r˜)+ p.x). In the Euclidean
formalism our analysis still go through, but now τ must have a
periodicity of 2π/κ to avoid the conical singularity. In order
to maintain this periodicity in τ , we must choose α = κ .
If we choose this condition then its not hard to see that the
coefficient of the logarithmic correction is −1/2 instead of
−3/2.
From above discussion, it is therefore fair to say that as of
now there is no consensus on the coefficient of this
logarithmic correction term and that more work in needed in order
to say anything conclusively. Since we are mostly interested
in studying the effects of this logarithmic correction term on
the holographic Rényi entropy, we adopt here a more neutral
point of view and consider a general logarithmic correction
with arbitrary coefficient C, instead of worrying too much
about its exact magnitude
S = SWald + C ln SWald + · · · .
(54)
The whole purpose of our previous exercise, where we
computed C = −3/2, was to show at-least one method by which
logarithmic correction to the entropy of topological black
holes, not just in Einstein gravity but in other higher
derivative gravity too, can be explicitly calculated and that this
8 In various methods of computing the entropy of the black hole from
diffeomorphism symmetry arguments, the parameter α is generally
chosen to be the surface gravity by hand to get the correct expression for
the entropy of the black hole; see for instance [
31,44,50
].
R
ymax = δ
,
where δ is the short distance cutoff related to the UV cutoff
of the boundary CFT (see [
8
] for more details). Now, the area
V d−1 of the hyperbolic space is calculated as
coefficient is non-zero. It would certainly be interesting to
generalize the method of [
14
] in LQG or of [
17
] in string
theory to calculate logarithmic correction for topological black
holes. Especially, for higher derivative gravity theories it is
not clear (at-least to the author) how to proceed. It would be
an interesting problem in its own right to analyze
similarities and differences in the results of these methods but it is
beyond the scope of this paper.
Let us also point out that the horizon Virasoro algebra
constructed in Eqs. (41) and (50) are not related to the boundary
CFT and its corresponding algebra i.e. to the CFT that appears
in AdS/CFT language. They are two different and separate
things. It is important to keep this in mind. The horizon
Virasoro algebra is constructed only to get the entropy of the
black hole expression and after that it does not appear any
where in the holographic Rènyi entropy calculation, as we
will see in the next section.
(55)
(56)
(57)
5 Calculations and results
In this section, we will present results for the holographic
Rényi entropy by considering logarithmic correction to the
usual entropy of the black hole. We will again concentrate
on two gravity theories: Einstein and Gauss–Bonnet. Our
main focus here will be to see the effects of this logarithmic
correction on the Rényi entropy. However, before going into
the details of each case separately, let us first calculate the area
of the hyperbolic event horizon appearing in Eqs. (44) and
(53). The line element on the (d − 1)-dimensional hyperbolic
space is given in Eq. (38), which we reproduce here again
for convenience
d d2−1 = dθ 2 + sinh2 θ dΩd2−2.
It would be convenient if we make a change of coordinate
θ = cosh−1 y, in which case the above metric reduces to
d d2−1 = y2d y−2 1 + (y2 − 1)dΩd2−2.
The area of the hyperbolic space (and hence the area of the
event horizon) is divergent. In order to regulate this area, we
introduce a upper cutoff by integrating out to a maximum
radius
V d−1 =
d−2
1
ymax
+
2 R
δ
where d−2 = 2π (d−1)/2/Γ ((d − 1)/2) is the area of a unit
(d − 2) sphere. However, for d = 2, we have a logarithmic
behavior
Therefore, the area of the event horizon Ad−1 is given by
Ad−1 = rhd−1V d−1 .
Now we have all the ingredients to compute the holographic
Rényi entropy for spherical entangling surface.
5.1 Rényi entropy from AdS-Schwarzschild black hole
For AdS-Schwarzschild Black hole the metric is given in Eq.
(37). However, in order to make comparison with [
10
], let us
rewrite this in a slightly different form
r 2
ds2 = −[−1 + L2 g(r )]N 2dt 2 +
+ r 2d d2−1 ,
L2
g(r ) = r 2 ( f (r ) + 1)
1
In order to ensure that the boundary spacetime is conformally
equivalent to R × H d−1, i.e.
ds∞2 = −dt 2 + R2d d2−1,
we choose the constant N 2 = L2/(g∞ R2) = L˜ 2/R2, where
g∞ = limr→∞ g = 1. The Hawking temperature of this
black hole is given by
1
T = 4π R
drh
L −
(d − 2)L
,
rh
which is also the temperature of the boundary CFT on
R × H d−1. For computational purposes, it is convenient to
(63)
(64)
(65)
(66)
(68)
consider a coordinate x = rh /L˜ in which case the expression
for the Rényi entropy in Eq. (11) reduces to
q 1 1 dT
Sq = q − 1 T0 xq dx Stherm(x ) dx
q 1
= q − 1 T0
Stherm(x )T (x )|1xq −
1
xq
d
dx T (x ) dx Stherm .
Here, in the above integral, the upper limit x = 1 comes
from the condition T = T0, which implies rh = L˜ . The
lower limit xq , which needs to be determined, corresponds
to the temperature T = T0/q. Using Eq. (63), xq must satisfy
the following equation:
Finally, using the expression for the entropy of the black hole
(Eq. (54)) in place of Stherm in Eq. (64), we get the expression
for Rényi entropy as
( L˜ )d−1, l p is the Planck length related
where B = 2π V d−1 lp
to the gravitation constant G D = ldp−1/8π . We see that Sq
is a complicated function of xq . For C = 0, Sq reduces to
the expression found in [
10
]. However, for C = 0 there are
additional non-trivial correction terms. We now make some
observations:
• For d = 2, we get
B
Sq (d = 2) = 2
c 1
= 6 1 + q
ln
where c = 12π L/ l p is the standard expression of the
central charge in the two dimensional boundary CFT.9 The first
term in Eq. (68) matches with the well known result of the
9 This c should not be confused with the central charge of Eq. (43)
which appears in the Virasoro algebra on the horizon. The c in Eq. (43)
was used to compute the black hole entropy and has nothing to do with
the boundary CFT.
Rényi entropy in a two dimensional CFT for an interval of
length l = 2 R. This is expected as, for d = 2, the spherical
entangling region consists of two points separated by
distance 2 R. However, our main result here is the appearance of
additional corrections to Rényi entropy which are precisely
coming from the coefficient of the logarithmic correction to
the entropy of the black hole. These correction terms to the
Rényi entropy have both additional logarithmic as well as
double logarithmic structure.
• Similarly, the entanglement entropy for d = 2 is obtained
by taking q → 1 limit of Eq. (68)
c
S1(d = 2) = 3 ln
2 R
δ
The leading term again has the same structure as is well
known for the two dimensional CFT. However, now we have
found additional correction too. Interestingly, the correction
term in the entanglement entropy has a simple structure. It is
proportional to C times the logarithm of its standard
expression. Moreover, the same is true in all dimensions. That is,
S1(d) = B + C ln[B]
L
˜
= 2π V d−1 l p
L
˜
= 2π V d−1 l p
In the last line, we have divided the correction term into two
parts. One that depends on (L˜ / l p)d−1, which is related to
the number of degrees of freedom of the boundary CFT and
the other that depends on the size of the system. Let us
concentrate on the term which is of order G0D (last term in Eq.
(70)).10 We observe that at this order there is a logarithmic
correction in all dimensions. In particular, there is a
logarithmic correction in odd d as well. In order to see this, we write
the complete expression for entanglement entropy in d = 3
and 4 dimensions
S1(d = 3) =
2π lL˜p 2 Rδ − 1
+ C ln
L˜ 2
l p
+ C ln 4π 2 R
δ
10 Quantum corrections to the RT formula are expected to occur at the
order G0D [
40
]. The RT formula gives entanglement entropy at the order
1/G D. See the discussion section for more details.
+ C ln
l p
L˜ 3
l p
R2 2 R
δ2 − ln δ
L˜ 3
we note from the last terms of Eq. (71) that at the order
G0D there is a correction to the entanglement entropy which
depends logarithmically on the size of the entangling surface.
The same result holds in all d > 2 dimensions.
• At first sight the results in Eq. (69)–(71) look strange as
they predict additional logarithmic divergence in the
entanglement entropy and seems to violate c-theorem like results.
However, as we will show shortly this pessimism is
unjustified.
As is well known, the entanglement entropy suffers from
UV divergences, and as a result it is ill-defined in the
continuum limit i.e. δ → 0. A normal practice is to subtract
the UV divergent part (in an ad hoc way) by hand, a practice
which is ambiguous and often not unique. In [
37
], a procedure
for “renormalized entanglement entropy” was given which
is intrinsically UV finite and predicts well defined
properties of the entanglement entropy. In particular, depending on
spacetime dimensions, following renormalized expressions
of the entanglement entropy (S1(R)) were proposed,
1 d
S1(R) = (d − 2)!! R d R − 1
d
R d R − 3
d
. . . R d R − (d − 2) S1, for odd d,
1 d
R
= (d − 2)!! d R
d
R d R − 2
d
. . . R d R −(d −2) S1, for even d.
In [
37
], It was shown that (i) the renormalized
entanglement entropy S1(R) is UV finite in the continuum limit, (ii)
it is a monotonic non-negative function that decrease form
UV to IR, therefore satisfying the c-theorem like functions,
and (iii) it may be considered as the “universal part” of the
original entanglement entropy, a part which can be defined
intrinsically in the continuum limit. We can clearly see that
all these properties are explicitly satisfied in the corrected
entanglement entropy expression as well. In particular, for
d = 2 we have
d c C
S1(d = 2) = R d R S1 = 3 + ln (2 R/δ) ,
c
S1(d = 2)|δ=0 = 3 ;
therefore, S1(d = 2) is again UV finite in the continuum
limit. Moreover, the finite/and or universal part of the
renormalized entanglement entropy remains the same even by
taking the corrections to the original entanglement entropy
(72)
(73)
expression. Since c is the central charge of two
dimensional CFT which generally decreases from UV to IR i.e.
c(U V ) > cI R , it indicates that the renormalized
entanglement entropy is also a monotonic non-negative function that
decrease form UV to IR.11 Accordingly, the c-theorem is
again satisfied. Therefore, the additional logarithmic
divergences in the entanglement entropy do not lead to any
nontrivial issues as far as the entropic c-function and universality
is concerned. Moreover, one can also explicitly check that in
higher dimensions as well the renormalized entanglement
entropy is again UV finite and is non-negative monotonic
function. In Sect. 6, we will give another reason, based on
the results of one-loop test of AdS/CFT, why logarithmic
divergences in the entanglement entropy are expected
especially in odd dimensions.
• For d = 2, the ratio of Rényi entropy to entanglement
entropy as a function of q for some reasonable values of C
is plotted in Fig. 1. In order to plot this ratio, we have
chosen R = 1, δ = 10−4 and L = 2l p. We see that the overall
behavior of Sq /S1 is the same for all C. For a fixed q > 1, the
magnitude of the ratio Sq /S1 decreases with decrease in C.
However, we should emphasize here that these results are
cutoff (δ) dependent. For instance, for much smaller cutoff, say
δ = 10−20, differences due to C = 0 are almost negligible.
Similarly, we also found that, for larger and larger values of c
(or L/ l p), the effect of C become smaller and smaller. We can
also note that Sq /S1 > 1 for q < 1 and Sq /S1 < 1 for q > 1.
• Let us also note some useful limits of the Rényi entropy
S∞ = B 1 −
d − 1 d − 2
d d
+ C ln B − (d − 1)2 + (d − 1) d(d − 2)
S0 = B
S1 = B + C ln B
2 d
d
• Now, we discuss Rényi and entanglement entropies for
higher dimensional theories. In Fig. 2, Sq /S1 as a function of
q for various values of d is shown. Here, we have considered
C = −3/2 and again have chosen R = 1, δ = 10−4 and L =
2l p. The overall characteristic features of Sq /S1 is found to
be same as in the d = 2 case. We found that, for fixed C, as we
11 For d = 2, the expression for renormalized entanglement entropy is
similar to the entropic c-function [
51
]. The fact that c-function decreases
monotonically from UV to IR has also been confirmed both in lattice
as well as in holographic studies [
52,53
].
q
q
2
4
6
8
Fig. 1 Sq /S1 as a function of q for various C. Here d = 2 and red,
green, blue and brown curves correspond to C = 0, −1/2, −3/2 and
−2, respectively
Fig. 2 Sq /S1 as a function of q for various d. Here C = −3/2 and red,
green, blue and brown curves correspond to d= 3, 4, 5 and 6, respectively
increase the number of spacetime dimensions the magnitude
of the ratio Sq /S1 increases. Even for large d, we again found
that Sq /S1 > 1 for q < 1 and Sq /S1 < 1 for q > 1.
• Let us also comment on some of the useful inequalities
of the Rényi entropy. It is well known that Rényi entropy
satisfies inequalities involving the derivative with respect q,
such as
(75)
therefore, it is natural to ask whether or not these inequalities
are satisfied with the corrected results. For d > 2, we find that
these inequalities are indeed satisfied provided C is not very
large (which is expected for small corrections). However,
for d = 2, we find few subtleties, especially in the fourth
inequality of Eq. (75). We find that for small central charge
it can be violated. Although for large central charge, which
is the case for boundary theories that have a gravity dual, the
fourth inequality is again found to be satisfied.
• Perhaps, the most important result from our
computation is that the leading UV divergences appearing in Eq.
(67) is same for all q. Indeed, if we note that leading term
Rd−2 d−2 in Eq. (58) gives the area of the spherical
entangling region, then the leading divergent term always has a
“area law” structure for d > 2. For d = 2, we have the usual
logarithmic divergence in the Rényi entropy. As expected,
the correction terms do not change the leading behavior of
the Rényi entropy.
5.2 Rényi entropy from Gauss–Bonnet black hole
In this subsection, we examine the holographic Rényi entropy
for boundary theory which are dual to Gauss–Bonnet
gravity theory. The procedure for calculating Rényi entropy with
Gauss–Bonnet Black hole background is entirely similar to
what has been discussed in the previous subsection, and
therefore we will be brief here.
The metric for Gauss–Bonnet Black hole is given in Eq.
(47), however, we rewrite it in a slightly different form in
order to make comparison with [
10
]
ds2 = −
r 2
−1 + L2 g(r ) N 2dt 2
+
1
.
I(n1 −thi√sc1a−se 4Nλ2)/=(2λL)2/=(g1∞, iRt2im)p=lieL˜s2L/˜ R=2. LS.inTcheisnoiswjugs∞tth=e
statement that the AdS curvature scale is distinct from the
length scale L, which usually appears in the Gauss–Bonnet
gravity action. Now, again using the coordinate x = rh /L˜ ,
the Hawking temperature can be expressed as
1
T = 2π Rx
1 + 2gd∞ x 4 −x 2x −2g∞2λ+g∞λg∞2 .
Similarly, the horizon entropy (Eq. (53)) can be recast as
S = 2π
L
˜
l p
d−1
V d−1 x d−1 1 −
2λg∞ d − 1
x 2 d − 3
+C ln 2π
× 1 −
L
˜
l p
d−1
2λg∞ d − 1
x 2 d − 3
.
V d−1 x d−1
(76)
(77)
(78)
After substituting above results into Eq. (64), we can get
expression for the Rényi entropy for any d. However, for
general d the expressions for Rényi entropy is very
complicated and lengthy and not very illuminating. For this reason
we focus on d = 4 case, in which the Rényi entropy obeys
the following expression:
which is again obtained from the equation T = T0/q, i.e. for
the lower limit of the integral in Eq. (64). λ, which appears in
the argument of logarithmic and inverse hyperbolic functions
in Eq. (79) should be understood as an absolute value. The
first term in Eq. (79) have the same expression for Rényi
entropy as was found in [
10
]. However, now we also have
additional correction terms, which are both logarithmic as
well as non-logarithmic in nature. One can also explicitly
check that, in the limit λ → 0, Eq. (79) reduces to Eq. (67)
for Einstein gravity.
We now make some observations:
• It is well known that the dual four dimensional boundary
CFT of five dimensional Gauss–Bonnet gravity theory has
two distinct central charges12
c = π 2 L˜ 3
l p
(1−2λg∞),
a = π 2 L˜ 3
l p
(1−6λg∞).
(81)
In terms of these central charges the Rényi entropy in Eq.
(79) reduces to
2
V 3 q(1 − xq )
Sq = 4π
q − 1
× (5c − a)xq2 − (13c − 5a) + 16c
2cxq2 − (c − a)
(3c − a)xq2 − (c − a)
12 Again, these central charges should not be confused with the central
charge of Eq. (43).
Cq xq ((5c − a)xq2 − 3c + a)
− (q − 1) ((3c − a)xq2 − c + a)
(3c − a)xq2 − 3c + 3a
Cq
+ (q − 1)
,
2Cq (3c2 + a2 − 6ac) 3(3c − a) tanh−1
+ (q − 1) (3c − a)2 c − a
3c − a
3(c − a)
Fig. 3 Sq /S1 as a function of q for various C. Here λ = 0.08 and
red, green, blue and brown curves correspond to C=0, −1/2, −3/2 and
−2 respectively. For this plot, we have chosen R = 1, δ = 10−4 and
L = 2l p
we see that, as in the Einstein gravity case, correction term to
the entanglement entropy is still given by the logarithmic of
its original expression. This result can be traced back to Eq.
(11), where it is clear that entanglement entropy is nothing but
the entropy of the black hole at temperature T = T0.
Therefore, a logarithmic correction to the entropy of the black hole
implies logarithmic correction to the entanglement entropy.
It is also easy to see that similar results hold in higher
dimensions too.
• Sq /S1 as a function of q for some reasonable values
of C and for λ = 0.08 is shown in Fig. 3. We see that the
which shows that the Rényi entropy is quite a complicated
function of these central charges. It is also clear that the Rényi
entropy is not determined solely by the anomaly coefficient a
as in the case of entanglement entropy (see below). As in the
case of Einstein gravity, here too, the size of the entangling
surface always appears logarithmic in the correction terms.
This feature therefore seems to be universal in nature.
• We get the expression for the entanglement entropy as
2V 3
S1 = a π
+ C ln a 2V 3 ,
π
(82)
(83)
Fig. 4 Sq /S1 as a function of q for various λ. Here C = −3/2 and
red, green, blue, brown and cyan curves correspond to λ= −0.19, −0.1,
−0.05, 0.05 and 0.09, respectively. For this plot, we have chosen R = 1,
δ = 10−4 and L = 2l p
difference due to C is extremely small and that the overall
behavior of Sq /S1 is the same for all C. Similarly, Sq /S1 for
different values of λ and for fixed C = −3/2 is shown in
Fig. 4. We find that Sq /S1, in the region q < 1, increases for
higher and higher values of λ, however, in the region q > 1,
it decreases. Similar results hold for other values of C.
• Before ending this section, let us also note some other
useful limits of the Rényi entropy,
where h is some function of c, a and V 3 but independent
of q. The leading diverging term of S0 matches exactly with
[
10
].
6 Discussions and conclusions
In this paper, we have studied the effects of the
logarithmic correction of the entropy of the black hole on the Rényi
entropy of a spherical entangling surface. We first used the
diffeomorphism symmetry argument at the horizon to
compute the black hole entropy expression. We used the Noether
currents associated with the diffeomorphism invariance of the
Gibbons–Hawking boundary action to construct the Virasoro
algebra at the hyperbolic event horizons and then used this
algebra to calculate the entropy of AdS-Schwarzschild and
Gauss–Bonnet black holes. We found that the leading term
in the expression for the entropy of the black hole is the usual
Wald entropy and that there is a correction to it. This
correction was found to be proportional to the logarithm of the area
of the horizon.
We then applied the prescription of [
8, 10
] to calculate the
holographic Rényi entropy for a spherical entangling surface.
Using the corrected entropy of the black hole expression we
found that there are corrections to the standard expression of
Rényi entropy. These corrections are shown in Eq. (67) for
Einstein and in Eq. (82) for Gauss–Bonnet gravity duals. In
particular, we found that the Rényi entropy is a complicated
function of the index q and the central charges. Interestingly,
the size of the entangling surface always appears
logarithmically in the correction terms of the Rényi entropy. This is true
for both Einstein and Gauss–Bonnet gravity. We found that
the inequalities of the Rényi entropy as well as the entropic
c-functions are satisfied even with the correction terms.
It is important to analyze the nature and significance of
these correction terms in the Rényi entropy. If the corrections
in the Rényi entropy originating from the corrections in black
hole entropy are quantum corrections, then our results can be
useful in many directions. Especially since a lot of work have
recently been appeared in the literature to compute leading
quantum corrections (of order G0D or N 0) to the entanglement
entropy holographically. In this context, a proposal for
quantum corrections to holographic entanglement entropy is given
in [
40
], also see [
54
]. In this proposal, the quantum
corrections are essentially given by the bulk entanglement entropy
between RT minimal-area surface and the rest of the bulk
(see Figure (1) of [
40
]). As an example, the quantum
correction to the entanglement entropy of the Klebanov–Strassler
model in the large N limit was calculated. The correction
was found to be, as in our case, logarithmic in nature and it
depends on the size of the entangling surface. Indeed, one
can also notice from Eq. (70) that at the order G0D (recall
that the factor (L˜ / l p)d−1 measures the number of degrees of
freedom of the dual CFT and it is related to the central charge
of the boundary CFT or to the G D on the gravity side) the
correction to the entanglement entropy is proportional to the
logarithm of the size of the entangling surface. If the above
interpretation is correct, then our results can be useful since
they predict a similar kind of logarithmic correction to the
Rényi entropy.
An indirect hint for the logarithmic correction in the Rényi
entropy can also be found in the following way. In [
18
], a
non-trivial test of the gauge/gravity duality at next-to-leading
order in the 1/N expansion for ABJM theories was
performed. There, it was shown that the subleading logarithmic
correction in the partition function (more correctly in log Z )
of the ABJM theory on a three sphere matches exactly with
the partition function of its 11-dimensional supergravity dual.
The latter partition function at one-loop level was calculated
using the Euclidean quantum gravity method. Now, the
subleading logarithmic term in the partition function can lead
to a logarithmic correction in the Rényi entropy, provided
that an analogous logarithmic term in the partition function
on q-folded cover does not cancel in the definition of the
Rényi entropy. This scenario is partially true, at least in odd
dimensions.
Indeed one can notice, an important result of our
analysis is that there are logarithmic corrections to entanglement
and Rényi entropies in odd dimensions too. At first sight this
seems strange as a logarithmic term in these entropies
generally appears in even dimensions. However, it is also well
known that the entanglement entropy for the sphere in flat
space in odd dimensions is simply the negative of the free
energy on a sphere [
56, 57
] i.e.,
S1 = log Z = −F1
it implies that any finite or divergent corrections in the
partition function will lead to the same corrections in the
entanglement entropy. Now since logarithmic corrections to the
partition function on the sphere do appear at one-loop level,
see for example [18], one can expect the same logarithmic
corrections to appear in the entanglement entropy too.
Therefore, it is possible to have a logarithmic term in the
entanglement entropy in odd dimensions spacetime. However, this
logarithmic term in the entanglement entropy arises only at
one-loop level as compared to the logarithmic term in even
spacetime dimensions which arises at tree level.
Since a similar kind of logarithmic subleading term in the
partition function also arises in other boundary field theories,
see for example [
55
], therefore it appears that logarithmic
correction to the Rényi entropy might be a general feature
of CFTs with gravity duals. Although, in order to
explicitly establish this fact it would be useful if we can calculate
the Rényi entropy at one-loop level on the lines of [
41
].13
It would certainly be interesting to explicitly compare the
results of our calculations with those of [
40, 41
], and find the
similarities and differences between them.
Finally, regardless of the problems associated with the
interpretation of correction terms calculated in this paper, it
is important to carefully examine and explore their physical
13 In [
41
], one-loop bulk corrections to RT formula was systematically
calculated. This was done by calculating the one-loop determinants
around the classical solutions using the Schottky uniformization of
qsheeted Riemann surface. However, there are many difficulties in
implementing this method, such as constructing the smooth bulk solutions and
performing analytic continuation of the replica index to non-integer q.
significance. Especially since in the context of AdS/CFT, it
is well known that on the gravity side the logarithmic
correction to black hole entropy arises only due to the one-loop
contribution of the massless fields. The fact that there are
logarithmic corrections to the black hole entropy has been firmly
established not just in string theory but also in loop quantum
gravity. In [
17
], it was further pointed out by Sen that the
logarithmic corrections to the entropy of the black hole are also
universal in that they depend only on the massless spectrum
of particles and are insensitive to the UV completion of the
quantum gravity theory. Sen suggested that any microscopic
theory of gravity will have a log correction to its entropy.
Once we assume a logarithm correction to the entropy of the
black hole as a well established fact, then this correction must
have some correspondence to the holographic Rényi entropy
via the mapping of [
8, 10
]. What is this correspondence and
how is the structure of the holographic Rényi entropy affected
by this? This is a natural and important question to ask. In
this paper we are suggesting that the logarithmic correction
to the entropy of the black hole (if it exists) naturally leads to
the logarithmic correction to holographic Rényi entropy. We
believe this mapping can be further useful, not just to better
understand the structure of the holographic Rényi entropy
but also to get a better understanding of the coefficient of
the logarithmic correction in the gravity side. We hope to
comment on this issue soon.
Acknowledgements I am very grateful to B. Sathiapalan, N.
Suryanarayana, R. Kaul and D. Dudal for useful discussions and for giving
me valuable comments. I would like to thank A. Dey, Zodinmawia, A.
Sharma and D. Dudal for careful reading of the manuscript and
pointing out the necessary corrections. Part of this work was done while
the author was in The Institute of Mahematical Sciences, Chennai and
was funded by a Postdoctoral fellowship provided by the Department
of Atomic Energy of the Government of India. This work is partially
supported by the postdoctoral grant PDM/15/172 from KU Leuven.
Open Access This article is distributed under the terms of the Creative
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