On holographic Rényi entropy in some modified theories of gravity
HJE
On holographic Renyi entropy in some modi ed theories of gravity
Anshuman Dey 0 1 2 4
Pratim Roy 0 1 3 4
Tapobrata Sarkar 0 1 4
0 Homi Bhabha Rd , Mumbai 400005 , India
1 Kanpur 208016 , India
2 Department of Theoretical Physics, Tata Institute of Fundamental Research
3 School of Physical Sciences , NISER Bhubaneswar
4 Department of Physics, Indian Institute of Technology
We perform a detailed analysis of holographic entanglement Renyi entropy in some modi ed theories of gravity with four dimensional conformal eld theory duals. First, we construct perturbative black hole solutions in a recently proposed model of Einsteinian cubic gravity in ve dimensions, and compute the Renyi entropy as well as the scaling dimension of the twist operators in the dual eld theory. Consistency of these results are veri ed from the AdS/CFT correspondence, via a corresponding computation of the Weyl anomaly on the gravity side. Similar analyses are then carried out for three other examples of modi ed gravity in ve dimensions that include a chemical potential, namely BornInfeld gravity, charged quasitopological gravity and a class of Weyl corrected gravity theories with a gauge eld, with the last example being treated perturbatively. Some interesting bounds in the dual conformal eld theory parameters in quasitopological gravity are pointed out. We also provide arguments on the validity of our perturbative analysis, whenever applicable.
AdSCFT Correspondence; Black Holes in String Theory

1 Introduction
2 Holographic entanglement Renyi entropy
3 ERE for Einsteinian cubic gravity
3.1
3.2
3.3
3.4
Numerical analysis and results
Scaling dimension of twist operators
Weyl anomaly and central charges
Validity of Renyi entropy inequalities
4 Charged Renyi entropy with quasitopological gravity
4.1
Numerical analysis and results
4.2 Scaling dimension of twist operators
5 Charged Renyi entropy with BornInfeld and Weyl corrected gravity
5.1
5.2
Charged Renyi entropy with BornInfeld gravity
Charged Renyi entropy in Weyl corrected gravity
6 Summary and conclusions
A Correction up to O( 2) in Einsteinian cubic gravity
B Correction up to O( 2) in Weylcorrected gravity
gravitational description arose out of the seminal work by Ryu and Takayanagi [8]. The
RyuTakayangi prescription and its generalisations [9{14] (see also [15, 16] for related
discussions) have been at the forefront of research over the last decade and promises to yield
a deep understanding of entanglement in strongly coupled quantum systems. Following up
on this line of research, in recent times, there has been a surge of interest in investigating
higher curvature theories of gravity, like GaussBonnet gravity, Lovelock gravity [17], and
quasitopological gravity [18{20]. We will call such theories generically as modi ed theories
of gravity (in the sense of being modi ed from the standard Einstein gravity). It is well
known that such theories often su er from the problem of negative energy states in their
spectrum, but nonetheless might provide important clues towards the quantization of
gravity. Viewed from a string theory perspective, these theories might be viewed as higher order
corrections to the standard EinsteinHilbert action, the latter being a low energy e ective
theory, and the corrections becoming important at large energy scales. Indeed, CFT duals
to these modi ed theories of gravity have been extensively studied by holographic methods
by now (see, for example, [21{25]).
Understanding entanglement in modi ed theories of gravity is important and
interesting in its own right. Apart from providing insights into the nature of entanglement
in novel examples of strongly coupled quantum
eld theories, these often provide us with
extra tuneable physical parameters that substantially enrich the phase structure of the
system, and is expected to be of use to model realistic situations. Further, these theories
often provide useful insights in terms of novel physical bounds on the parameters of the
where, S denotes the thermal entropy of the CFT living on R
Hd 1. Here, n, called
the order of the ERE is a positive real number. By construction, SEE = limn!1 Sn, and
hence we can obtain the EE from the ERE. The focus of the current work will be on the
{ 2 {
and is by de nition, the von Neumann entropy of subsystem A. The entanglement Renyi
entropy (ERE) [26, 27] on the other hand is known to be the generalisation of the above,
and is believed to be crucial for a complete understanding of quantum states in a physical
system. This is de ned as
As explained in the next section, this can be cast into the form
SEE =
tr ( A log A) ;
1
n
1
n
Sn =
log tr ( nA) :
Sn =
n
work of [28] (for related recent results, see, e.g [29]).
After brie y introducing the necessary notations and conventions in section 2, in the
rst part of this paper, we study in section 3 the ERE associated to the recently discovered
Einsteinian cubic gravity (ECG) [30], focusing on a ve dimensional example. ECG is in general characterised by a Lagrangian density which we schematically write as
L = R
2
+ LGB + Lcubic
(1.4)
HJEP04(218)9
where LGB is the standard GaussBonnet term and Lcubic is a new cubic interaction term
found in [30] that di ers from ones considered in the literature before, for the following
reasons. First, while other higher derivative theories of gravity (for example quasitopological
gravity) that exist in the literature have the characteristic that the coupling constants
associated with di erent terms in the Lagrangian are dimension dependent, in ECG on
the other hand, the coupling constants are dimension independent. Secondly, unlike say
GaussBonnet gravity, the extra cubic terms in the Lagrangian are not topological in four
spacetime dimensions, i.e Einstenian cubic gravity provides an interesting generalisation of
pure Einstein gravity and an useful laboratory to apply the holographic principle in three
dimensional dual CFTs. This novel theory is therefore important and interesting, and has
attracted attention of late. In particular, one might compute useful physical quantities in
ECG in the context of the gauge gravity duality. In the rst part of this paper, we initiate
such a study.
We construct the rst known example of a black hole solution in the theory in ve
dimensions perturbatively, and are naturally led to the study of measures of entanglement
of the resulting CFT, in particular the ERE. We also compute the scaling dimensions of the
twist operators (to be elaborated upon in the next section) and verify the central charges of
the dual CFT via a computation of the Weyl anomaly. We believe that ours is historically
the rst attempt at such a holographic analysis for these types of theories. We should
mention here that after our paper appeared on the arXiv, the work of [31] found a generic
class of black hole solutions in these theories and studied their thermodynamics. Several
papers have appeared on the topic since then, and it is now known that in four dimensions,
Einsteinian cubic gravity belongs to a general class of models called \Generalized
quasitopological gravity" theories [
32
] that has the same linearised spectrum as that of pure
Einstein gravity. For recent developments in holographic studies of ECG, we refer the
reader to [33] and references therein.
Continuing our analysis of the ERE in modi ed gravity theories, in the next part of
this work, we consider a few examples of such theories whose dual CFTs have a conserved
global charge. In this case, we need to understand how the entanglement between the two
subsystems A an B depends on the charge distribution among themselves. For a grand
canonical ensemble, the charged ERE is de ned [34, 35] by simply generalizing eq. (1.2),
,
n
in two other extended gravity theories.
We rst consider BornInfeld gravity [36, 37],
and elaborate upon the behavior of the charged ERE both as a function of the chemical
potential for xed values of the BornInfeld parameter, as well as a function of the
Born
Infeld parameter with xed chemical potential. We then brie y study Einstein's gravity with a Weylcorrected gauge eld [39, 40]. We should mention here that our studies of the ERE in Einsteinian cubic gravity and
Weylcorrected gravity contain perturbative analysis, and in order to justify the numerical
values of the coupling constant that we have chosen, we have to check that the second order
corrections to our linear order results in this paper are indeed small. Two appendices of this
paper address this issue. For ECG, the relevant second order black hole solutions for ECG
are summarised in appendix A, and these have been used in the main text to justify our
linear order results. For Weylcorrected gravity, the second order expressions are detailed
in appendix B, but the analysis become tedious, and the second order corrections could
not be performed satisfactorily, due to numerical issues, as we explain towards the end of
the paper.
A word about the notation used: we consider four di erent theories and consider similar
physical quantities in each of them. However, using di erent symbols for di erent theories
will unnecessarily clutter the notation and we avoid doing it. It has to be thus remembered
that a particular symbol used in a section remains relevant for that section only.
(1.6)
HJEP04(218)9
2
Holographic entanglement Renyi entropy
In this section, we will brie y review the relevant details of the computation of various
quantities associated with holographic ERE. Since the material is quite well established by
now, we will be brief here, and point the reader to the references herein for further details.
The calculation of the Renyi entropy in CFTs depends on the replica trick [1, 2] which
involves computing a path integral over nfold cover consisting of replicas of the original
CFT. The di erent copies are separated by twist operators [41] inserted at branch points.
Recently, holographic methods have been used to compute the ERE for a variety of CFTs
with di erent bulk duals. Extending the idea of [42], the authors of [28] derived a formula
for holographic ERE with spherical entangling surfaces and calculated the ERE explicitly
{ 4 {
for Einstein gravity as well as some higher derivative theories like GaussBonnet gravity and
quasitopological gravity. In a related work, [43] computed the ERE for four dimensional
N = 4 superYangMills holographically, using the same method. This proposal of ERE
was then generalized by [34] to include a chemical potential in the
eld theory. They
constructed the charged entanglement Renyi entropy in a grand canonical ensemble where
one has to compute the Euclidean path integral by inserting a Wilson line encircling the
entangling surface. For some recent interesting works on holographic Renyi entropy we
refer the reader to [44{48].
Let us brie y recapitulate the computation of the ERE, closely following [42] and [28].
We consider a spherical entangling surface Sd 1 in a at ddimensional CFT. From [42],
it can be shown that the CFT can be mapped to a hyperbolic cylinder R
scale R of the cylinder (which is also the radius of the original sphere Sd 1), given by,
Now, using the AdS/CFT correspondence, we can infer that the CFT on the hyperbolic
cylinder R
Hd 1 has a bulk dual which is a black hole with a hyperbolic horizon. The
above argument implies a simple relation between the thermal density matrix and the
reduced density matrix A as,
where U is a unitary transformation and Z(T0) = tr (e H=T0 ) and hence we have,
1)Sn
0 ;
0 ;
Sn
1)Sn
0 ;
0 ;
1Recently, [50] considered corrections to Wald entropy of topological black holes and found logarithmic
corrections to Renyi entropy.
{ 5 {
Now making use of the expression of free energy F in thermodynamics, F (T ) =
T log Z(T ),
the ERE of the CFT can be calculated as,
Sn =
1
n
1
n T0
F (T0)
F
:
T0
n
Since S =
gauge/gravity duality, we can identify the thermal entropy arising in that equation with
the Wald entropy SWald [49] of the hyperbolic black hole and hence we can compute the
Renyi entropy using eq. (1.3).1 An entirely similar analysis can be used to arrive at the
formula of eq. (1.6) for the cases with a global conserved charge.
The Renyi entropy satis es the following inequalities [28, 51],
T0 =
1
where it can be shown from the gravity side that satisfying the inequalities in the rst line
of eq. (2.5) automatically leads to the ones in the second line of that equation [35]. As was
argued in [28], these inequalities will hold for any CFT, as long as we have a stable thermal
ensemble. Nevertheless, in all the examples considered in this paper, we will explicitly
check the above inequalities. This is particularly important for the perturbative solutions
that we consider.
Now, as we have mentioned, the replica trick for computing the Renyi entropy can be
understood as the insertion of a surface operator, known as the twist operator ( n), at the
entangling surface [28, 34, 52]. The scaling dimension of this operator is determined from
the leading singularity in the correlator hT
ni. The leading singular behavior is xed by
the symmetry, tracelessness and the conservation properties of T .
Let us consider a four dimensional CFT where a planar twist operator is positioned
at x
1 = x
2 = 0 in
at Euclidean space. The twist operator extends along the other
two directions x3 and x4. Now, one inserts a stress tensor at (y1; y2; y3; y4) and thus the
orthogonal distance between the twist operator and the stress tensor operator is given by,
l = p(y1)2 + (y2)2. With this notation, the leading singularity in the correlator of the
twist operator and the stress tensor can be calculated as [28, 34],
hTij ni =
hTab ni =
2
hn i4j ;
l
hn 3 ab
2
hTia ni = 0 ;
l
4
4 na nb ;
where, (a; b) = (1; 2) are the normal directions and (i; j) = (3; 4) are the tangential
directions to the twist operator. Here n
a = yla is the orthogonal unit vector from the twist
operator
n to the point where the stress tensor T
one should notice that the leading singularity is now
has been inserted. From eq. (2.6),
xed up to a constant hn, known
as the scaling dimension of n. Following [28, 34], we quote the
nal expression of the
scaling dimension hn (in the four dimensional boundary CFT) in terms of the thermal
energy density,
3
n R
are respectively the thermal entropy density and the charge density of the
where, E (T; c) is the thermal energy density given by,
E (T; c) =
E(T; c) :
R3 V
E(T; c) being the total thermal energy and V is the regulated volume of the hyperbolic
plane H3. We should point out here that the rst term in eq. (2.7) appears due to the
anomalous contribution from the stress tensor under conformal transformations. One can
also express eq. (2.7) in terms of the thermal entropy density S using the rst law of
thermodynamics,
where, S and
Note that h00 = 0, while the authors of [28, 34] showed that the quantity h10 is related to
the central charge C~T for any ddimensional CFT
As mentioned in [34, 35], since we are discussing charged Renyi entropy, one may think
about the correlator of the twist operator n and the current operator J . Here also one
can extract the leading singular behavior of this correlator using the conservation of the
current J as
hJa n( c)i =
2
l
ikn( c) a b3nb ; hJa ni = 0;
where a b stands for the volume form and kn( c) represents the magnetic response
characterizing the response of the current to the magnetic
ux. Finally, one can derive the
magnetic response kn( c) using a conformal mapping [34, 35] as,
kn( c) = 2 nR3 (n; c):
expanded around n = 1 and c = 0,
hn( c) = X
1
! !
h (n
(2.10)
kn( c) = X
1
! !
k (n
1) c
; where k
(2.14)
The conformal dimension hn( c) possesses an interesting universal property, when
In a similar way as argued with the scaling dimension, here one may expand the magnetic
response kn( c) around n = 1 and c = 0,
;
;
(2.11)
In this work, we will broadly study the concepts introduced above. Having set up the
notations and conventions, we will now proceed to the main body of this paper.
3
ERE for Einsteinian cubic gravity
Our rst example is the analysis of ERE in ECG. This is an interesting proposal recently
put forward in [30]. As stated in the introduction, modi ed theories of gravity often
su er from ghost modes in the spectrum. The work of [30] on the other hand attempts to
formulate a theory which at the linearized level has no ghost modes, and where the coupling
constants are independent of spacetime dimensions. This theory has the following action
S =
d x
5 p g R
L2 4 + 1 L4 R
R
R
+
1
12
R
R
R
{ 7 {
R
R
24R
R
12R
R
R + R
3
R
(3.1)
metric ansatz,
ds2 =
r
2
L2 f0(r)
curvature. We will set
where, R is the Ricci scalar and L is the AdS length, related to the cosmological constant
by,
=
L62 . 4 is the standard fourderivative GaussBonnet term,
4 = R
R
4R
R
2 are the coupling constants of two new sets of six derivative terms. As
stated earlier, our goal is to study the ERE of ECG using holographic methods. As in
any other case of extended theories of gravity, this provides a novel dual CFT, which is
interesting to analyze.
It is di cult (if not impossible) to analytically solve the Einstein equations for this
cubic theory taking into account all the three parameters , 1 and 2
. We will be interested
in the cubic terms in the Lagrangian, and hence exclude the GaussBonnet coupling from
our analysis (the latter was studied in [28]). We will thus focus on one of the new coupling
terms
1 and
2
. To simplify the computation, we will set 1 = 32
numerical factor of 32 is just a choice which simpli es some constants that appear in our
analysis in what follows. Any other choice will not alter the essential physics). Of course,
one could alternatively set 1 = 0 to begin with, as well. Now, even with a single coupling,
we found it di cult to solve the system exactly. So, we proceed to solve the system up to
and
2 = 0 (the
linear order in the new coupling constant .
In order to compute the holographic Renyi Entropy dual to this cubic gravity, we need
to construct a hyperbolic black hole solution which can be done by choosing the following
1
1 +
F1(r) N (r)2dt2 +
Lr22 f0(r)
dr2
1 1 +
F2(r)
+ r2d 32 (3.3)
where, d 23 is the line element for the three dimensional hyperbolic plane H3 having unit
where L~ is the AdS curvature scale which is to be determined and R is the curvature of
the hyperbolic spatial slice.
Here f0(r) is the leading order solution which represents a hyperbolic Schwarzschild
black hole in ve dimensional AdS space, and has the form,
The position of the horizon of this unperturbed solution can be obtained by demanding,
f0(rh) = Lr22 . This in turn, xes the constant ! in the leading order solution as,
h
Now varying the action with respect to F1(r) and F2(r) yield the following di erential
equations at order
2 (the absence of O( ) terms signi es that the ansatz satis es the
where we have denoted
from which we obtain the following analytical solutions
where C1 and C2 are constants of integration, to be determined using appropriate boundary
conditions. At this point it is important to note that we have inserted an extra factor of 12
in the metric component gtt. With this inclusion, we make sure that the boundary CFT
dual to this black hole geometry resides on R
has curvature R as mentioned before,
H3, where the hyperbolic spatial slice H3
The constant C2 can be set to zero since it simply produces a rescaling of the time
coordinate, and we demand that the metric of eq. (3.3) be conformally equivalent to the one of
eq. (3.10). Then C1 can be xed by demanding that the position of the horizon rh remains
the same as with the unperturbed solution, and by the fact that the potential singularities
in F1(rh) and F2(rh) need to be avoided. It can be checked that this can be obtained
by setting
C1 =
11L6
r
2
h
54L4 + 75L2r2
h
31rh4
Since the asymptotic form of the bulk metric of eq. (3.3) should represent the
background metric for the dual boundary CFT given by eq. (3.10), we now determine the AdS
curvature scale L~ as,
where we have used the fact that
L~ = p
L
1
p
L
f
1
;
(1 +
F1(r)) f0(r)
L2
r2 jr!1 = 1
f :
1
After determining the perturbed black hole solution up to O( ), we proceed to compute
the Hawking temperature of the black hole. At this stage, it is convenient to make a change
{ 9 {
temperature of the black hole then can be expressed as,
of variables and use a new dimensionless variable, x = rL~h , instead of using rh. The Hawking
T =
2x2
1
2 xR
x
2
Note that there is a O( ) correction in the temperature and as we take the limit
we recover the expression of the temperature with pure Einstein gravity [28], as expected.
To compute the thermal entropy of the hyperbolic black hole, we proceed to evaluate
the Wald entropy following [49],
SWald =
2
Z
horizon
This proposal was put forward to compute the black hole entropy in higher derivative
gravity theories. Here, L represents the Lagrangian of the particular higher derivative
gravity, and
is the binormal Killing vector normalized as
=
expression of the Wald entropy turns out to be
L
`p
3
V
3
2
18
x
The rst term in eq. (3.16) represents the usual thermal entropy in case of a hyperbolic
AdS black hole in pure Einstein gravity, while the second term stands for the correction
in entropy due to the presence of the cubic term in the action. Here V
= RH3 d 3 is the
volume of the hyperbolic plane. As shown in [28], V is divergent and this behavior mimics
the UV divergence of the Renyi Entropy in the dual CFT. Hence, one needs to regularize
the entropy and this is done by integrating out the hyperbolic volume element up to a
maximum radius R , where
is the shortdistance cuto
of the dual CFT. Now after
expanding V in powers of R , it is straightforward to single out the universal contribution
coming from the subleading terms,
V ;univ =
2 log
R
:
(3.17)
Now, writing eq. (1.3) in terms of the dimensionless variable x, the entanglement Renyi
entropy can be expressed as,
(3.14)
! 0,
(3.15)
HJEP04(218)9
Sn =
=
n
n
n
n
where in the second line we have done an integration by parts.
It follows that x1 and xn (the upper and lower limits, respectively, in eq. (3.18)) are
the only two remaining quantities that we need for the computation of the ERE. These
two quantities can be determined by solving the following equation,
T (xn) =
T0
n
(3.19)
given by,
4n
1 + p
Now substituting eq. (3.14) in eq. (3.19) yields a sixth order algebraic equation for xn
which can not be solved exactly for arbitrary n. However, with n = 1, we nd x1 = 1 is
still a solution of the equation. To determine xn for arbitrary n, we solve xn up to linear
HJEP04(218)9
order in ,
xn = x~n
2
(1
x~2 )3
n
x~3n(1 + 2x~2n)
where, x~n is de ned in eq. (3.20). We construct the solution in such a way that xn agrees
with eq. (3.20) when
vanishes.
Finally, we compute the ERE, (3.20) (3.21) (3.23)
We can also calculate the entanglement entropy from the above expression of ERE by
3
2x~2n 1
2
x~
2
n
x~
4n +
x~
2
n
14
56x~2n + 27x~4n
(3.22)
taking the n ! 1 limit, which yields,
Notice that, the above expression in the
! 0 limit corresponds to the entanglement
entropy for any CFT in four spacetime dimensions dual to a ve dimensional pure Einstein
gravity. The second term in the above expression represents the correction in entanglement
entropy due to the presence of the cubic term in the bulk gravity. This correction in
entanglement entropy is related to the central charge a for a four dimensional CFT. As
we will show in section 3.3, this matches with a corresponding calculation from the gravity
side, via a computation of the Weyl anomaly.
3.1
Numerical analysis and results
In this subsection, we present the results for the ERE in ECG, plotted against two
quantities, namely the order of the ERE n and the coupling . Since Sn contains the volume
factor V involving the shortdistance cuto
of the boundary CFT, we plot the ratio of
Renyi entropy Sn to the entanglement entropy S1, instead of plotting Sn. Let us rst
consider the behavior of the ERE as a function of the coupling
for di erent values of n
as shown. This is shown in
gure 1. Since we have treated the problem perturbatively,
we consider small values of
up to
= 0:05 (these numerical values will be justi ed in a
while, towards the end of this section). In gure 2, we have shown the variation of SSn1 with
1.0
0.9
correspond to the values n = 1, 2, 3, 10 and 100, respectively.
the red, orange, green, blue and purple lines de di erent values of , where we follow the same
= 0, 0:01, 0:02, 0:03 and 0:04, respec color coding for
as used in gure 2.
n for di erent values of , and the red, orange, green, blue and purple lines denote
Hence, we notice that the rst inequality of eq. (2.5) is obeyed for any small value of the
= 0,
quantity nn 1 SSn1 with n for di erent values of the coupling . Here again we follow the
same color coding for
as in gure 2. Note that the slope of the lines are positive for all
Renyi entropy in eq. (2.5).
nn 1 Sn
0, which supports the second inequality obeyed by
3.2
Recall that as mentioned in section 2, the eld theoretic method for calculating
entanglement entropy involves performing a path integral over n replicas of the original CFT, and
that twist operators open branch cuts between these copies. In this example, there is no
chemical potential, and hence using eq. (2.9), eq. (2.7) reduces to the result [28],
hn =
n
3T0V
Z x1
xn
dxT (x)S0(x) :
Using the above equation, the dimension of the twist operator can be computed up to
HJEP04(218)9
linear order in
as,
hn =
L 3 1 (1 + p
n3 `p
n3 `p
L 3
As expected, the leading order term in the expression corresponds to a CFT with a bulk
Einstein dual. Also note that, hn at linear order vanishes if one takes the limit n ! 1. We
2
3
L 3
`p
1 +
9
2
+ O
2
=
2
3
c
also nd that
where c = 2 L 3
`p
1+ 92 +O( 2) is the central charge of the four dimensional CFT. The
above relation between the central charge c and the rst order coe cient of the expansion
of the scaling dimension @nhnjn=1 is a general property and we show that this holds for
ECG, as expected. We will compute the central charge c in the next section by studying
holographic Weyl anomaly and show that it matches with the result above. This is of
course expected, as the symmetry of the boundary CFT implies eq. (3.26).
3.3
Weyl anomaly and central charges
In this section, we compute both the central charges c and a characterizing a four
dimensional CFT dual to ECG using the gauge/gravity duality. It is well known that, although
the trace of the energy momentum tensor hT i vanishes in at space, it is no longer zero if
we place the CFT in a curved background. This is known as Weyl anomaly or conformal
anomaly [53, 54], which relates the CFT parameters to the coupling constants of the dual
gravity theory. The trace anomaly for a four dimensional CFT is given by,
Sln v
ln
1
2
Z
d4x pg(0)hT i;
where
is a short distance cuto . This procedure of nding the anomaly coe cients
is straightforward but the calculation becomes complicated in higher derivative gravity
theories. So, instead of following this approach, we follow [56, 57] in order to compute the
central charges.2 We choose the following bulk metric,
ds2 =
L~2
and g(0)ij represents the boundary metric at
= 0. Now substituting the form of the
metric of eq. (3.30) into the gravity action (eq. (3.1)), g(2)ij can be eliminated using the
eld equations. After writing the action in terms of g(0)ij and g(1)ij one can further express
g(1)ij in terms of g(0)ij and write down the action in terms of g(0)ij only. Finally, one can
read the anomaly coe cients by extracting the terms producing a log divergence,
S2. After plugging this metric into the Lagrangian, we extract
the coe cient of 1 which gives rise to a log divergence. We call this term Lln, given by,
Lln =
1
2L2L~5`3
p
AL2uL~6
A2L2uvL~4 + 2ABL2uvL~4
6ABuvL~6
3A L6uL~2
B2L2uvL~4 + 3 BL6vL~2 + BL2vL~6 + 6 L6L~4
3A2 L6uv
6A BL6uv
3 B2L6uv sin
The equations of motion for A and B are given by,
Following the prescription of [54], one can obtain the central charges c and a via the
gauge/gravity duality. Starting with the gravity action and employing the Fe
erman
Graham expansion [55] one can write down the bulk metric as, where,
gij = g(0)ij + g(1)ij + 2
g(2)ij + : : : ;
solving which we can determine A and B. Now the four dimensional quantities I4 and E4
can be computed using the boundary metric g0(ij) as,
2The authors of [58] developed an elegant method to simplify the computation of holographic Weyl
anomaly and obtain the central charges for CFTs dual to higher derivative gravity. Using this approach,
one does not need to solve any equations of motion. Instead, one can expand the action around a \referenced
curvature" and then derive the central charges from the coe cient in the expansion.
Lln = 0;
I4 =
4(u
u v
v)2
;
E4 =
8
u v
(3.30)
(3.31)
(3.32)
(3.34)
(3.35)
(3.36)
Numerical results for SSn1 in ECG
Sn=S1 at O( )
Sn=S1 at O( 2)
0
0:01
0:02
0:03
0:04
right part of the table show results for SSn1 with O( ) and O( 2) corrections, respectively.
Now, by noting the form of I4 and E4 in (eq. (3.36)), it is easy to show that the central
charges can be obtained by taking the following limits,
(3.37)
(3.38)
(3.39)
Finally, using eq. (3.37) along with eq. (3.12), we determine the central charges up to linear
order in
as,
Note that the expressions of the entanglement entropy in section 3 and the scaling
dimension of twist operator in section 3.2 agree with the values for central charges obtained by
the computation of holographic Weyl anomaly.
3.4
Validity of Renyi entropy inequalities
hole solution at O( 2) (see appendix A).
Since we work perturbatively up to linear order in the coupling constant , we should
consider relatively small values of
in all our computations. However, to justify that our
results are reliable (by taking into account the terms up to linear order in
), we should
nd the change in Renyi entropy due to the inclusion of nexttoleading order corrections
and check that it is indeed small. For this purpose, we need to solve the Einstein equations
at O( 2). Following the same strategy as described earlier, we obtained a hyperbolic black
The numerical results for the ERE at di erent values of the cubic coupling with O( )
and O( 2) corrections are shown in table 1. To note the deviation in the values with
increasing n, we have chosen a large value of n (= 1000) to produce the table. However,
even with this large value of n, the second order correction in the entropy ratio is small and
hence the rstorder correction term is su ciently reliable. It can be observed from the
Here we should mention that the inequalities obeyed by Renyi entropy are satis ed in
this regime of the coupling
which are shown in gures 2 and 3.
Having studied ERE and its various features for ECG, we will now move over to
examples with gauge elds. Our next analysis involves a theory of quasitopological gravity
with a chemical potential.
4
Charged Renyi entropy with quasitopological gravity
In this section, we will discuss the properties of the charged Renyi Entropy of a four
dimensional CFT, dual to
ve dimensional charged quasitopological gravity (QTG). The
action of the ve dimensional QTG coupled to a U(1) gauge eld A is given by,
HJEP04(218)9
S =
g R +
12
L2 +
2
L
where ` is related to the ve dimensional gauge coupling g5 as, g52 = 2``23p .
coupling constants, and 4 is the standard fourderivative GaussBonnet term,
while Z5 is a sixderivative interaction term [19, 25] given by,
4 = R
R
R
R
1
56
4R
R
In order to compute the holographic charged Renyi Entropy dual to this bulk gravity
theory, we need to construct charged hyperbolic black hole solutions in this [38]. This can
be done by choosing the following metric and gauge eld ansatz,
ds2 =
r
2
L2 f (r)
A =
(r) dt :
1 N (r)2dt2 +
dr2
Lr22 f (r)
1
;
Substituting the ansatz of eq. (4.4) in the action of eq. (4.1), and varying the action with
respect to f (r), we have N 0(r) = 0, yielding
R
and solving that equation, we get
where L~ is the AdS curvature scale to be determined. Also, note that we have chosen the
12 factor in the same spirit as we did earlier. Now varying the action with respect to (r)
(r) =
c
2 R
and
are
(4.2)
(4.4)
(4.5)
(4.6)
In the above expression, q is an integration constant, related to the charge of the black hole
and rh is the position of the horizon. Note that, we have chosen the chemical potential c
in such a way that the gauge eld
(r) vanishes at the horizon.
The variation of the action with respect to N (r) yields a rst order di erential equation
for f (r), solving which we get a cubic equation for f (r),
1
f (r) + f (r)2 + f (r)3 =
L2m
3r4
L2q2
12r6
;
(4.8)
where m is an integration constant, related to the mass of the black hole. Using the fact
that f (rh) = Lr22 , one can also express the mass parameter in terms of the horizon radius
h
m =
3rh4
L2
r
h
Here we should mention that unlike the Einstein gravity, the AdS curvature scale L~ is not
equal to the length scale L related to the cosmological constant in QTG. These two are
related by L~ = p
solving eq. (4.8) at r ! 1,
Lf1 , where f
1
f
1 + f
As discussed in [25, 59], the 3 + 1 dimensional boundary CFT dual to this cubic gravity,
can be characterized by two central charges c and a, and an extra parameter t4 which is
required to describe certain scattering phenomena. Further, they can be expressed in terms
of the coupling constants in the theory as,
1 is the asymptotic value of f (r) and can be determined by
Here, the chemical potential is given by
c =
L~q
` rh2
= 0 yields t4 = 0, while c and a reduce to the form of the
central charges in GaussBonnet gravity.
From the above equations one can write down the coupling constants in terms of t4
and the central charges as,
t4 =
c =
a =
Using the above equations, one can further write down the form of f
terms of t4 and the central charges as,
1 from eq. (4.10), in
The two coupling constants and hence the central charges along with t4 can be
constrained to avoid negative energy excitations in the boundary CFT [25]:
f
1 = 2 35 aacc ((11 ++ 32tt44))
1
1
(1
c
a
(1
c
a
c
a
As was done earlier with earlier examples, here we rst de ne rh = L~ x and compute the
Hawking temperature as,
T =
1 +
x4`2 c2f 12
24 2L2
f 3
1
x4f
1 + x2f 12 + x6 #
12 2L2f1 (3 f 12 + 2 x2f
1
x4)
where
is the surface gravity of the black hole. It is straightforward to analytically express
the charge parameter q in terms of the chemical potential c and hence in eq. (4.15) we
have written down the nal expression of T in terms of c
.
Also, the thermal entropy of the black hole can be computed using Wald's
prescription [49],
S = 2
V x
In the process of computing the Renyi entropy Sn, we can express eq. (1.6) in terms of the
dimensionless variable x such that it can be rewritten in a form similar to eq. (3.18) as,
where x1 and xn are the integration limits to be determined. Finally, we compute the
charged Renyi entropy,
n
n
1
+ 2(P1Q1
PnQn)
3 x
2
1
x
2
n
1 +
f
f
(4.14)
HJEP04(218)9
(4.13)
(4.15)
(4.16)
(4.18)
(4.19)
Here we de ne xn through T ( c; x) = Tn0 , where with n = 1 we can obtain x1. Hence, the
only remaining task here is to solve the following sixth order algebraic equation,
4x6n
The largest real solution of the above equation determines the value of xn. The analytical
solution for xn is very di cult to obtain, hence we would numerically solve this equation
and compute the Renyi entropy. Here, by considering the limit n ! 1, one can obtain
the entanglement entropy, S1( c). Like we did earlier, we will show here the variation
of the Renyi entropy Sn( c) normalized by the entanglement entropy S1(0), instead of
HJEP04(218)9
plotting Sn( c).
Numerical analysis and results
First, xing the value of the ratio of central charges ac , we analysed the behavior of Renyi
entropy ratio SSn1((0c)) with t4 for di erent values of the chemical potential ( c). A typical
case is exhibited in gure 4. We set ac = 1, for which t4 must be in the following physically
allowed regime from eq. (4.14):
0:00149
t4
We nd a nearly linear behavior of SSn1((0c)) as a function of t4, as can be seen from
where the red, orange, green, blue and purple lines correspond to n = 1, 2, 3, 10 and 100,
gure 4,
respectively. The same holds for other values of the chemical potential as well. However,
we nd that for a relatively small value of chemical potential (say 2 L~
the lines are smaller than the ones seen here. In fact, the authors of [28] considered the
uncharged quasitopological gravity and concluded that the ratio SSn1 is almost independent
of t4 for a xed value of ac in the physically allowed regime and the dependence comes
into play when we are well outside the physical regime. However, by coupling the
quasi1), the slopes of
topological gravity to a U(1) gauge eld, it is evident in
gure 4 that with a higher value
of chemical potential, the Renyi entropy ratio increases linearly with t4. Notice that, as we
increase chemical potential, the ratio SSn1((0c)) reaches its maximum value for the maximum
bound of t4 (i.e., for t4 = 0:00238 with ac = 1). Another important observation is that,
although the slope of the entropy ratio increases as the chemical potential increases, the
c`
slope is independent of n for a xed value of chemical potential with
xed ac .
Next, gure 5 shows the behavior of the entropy ratio with the chemical potential, with
t4 = 0:00238, where we have
xed ac = 1. A similar graph (not shown here) is obtained
for t4 =
0:00149. These values of t4 are the two extreme bounds (i.e., end points of the
physically allowed parameter space for t4) with ac = 1. Here the entropy ratio increases
monotonically and in a nonlinear fashion, as c increases. Here we follow the same color
coding for n as in
gure 4.
Now we go ahead to verify the Renyi entropy inequalities. First, we have veri ed that
the rst inequality of eq. (2.5) holds for any values of t4 and chemical potential for a xed
value of ac . The results become more interesting when we test the second inequality as
0.000
0.001
value of ac = 1 and chemical potential 2c`L~ = 20. tential at xed value of ac = 1 and t4 = 0:00238.
(n  1) Sn (μc )
n
n
S1 (0)
violated with t4 = 0, while the chemical potential obeyed with t4 = 0:0036, while chemical
potenis kept xed at 2c`L~ = 0:45.
tial is kept xed at 2c`L~ = 0:45.
shown in gures 6 and 7. These gures show the variation of (nn 1) SSn1((0c)) with n for xed
value of the chemical potential c`
2 L~ = 0:45 and the central charge ratio ac = 2. For this
chosen value of ac , it is clear from eq. (4.14) that t4 must be in the following physically
allowed regime,
0
t4
Note that with the upper bound t4 = 0:0036, the Renyi entropy obeys the second inequality,
but with the lower bound t4 = 0, it violates the inequality although t4 = 0 is a
wellaccepted physical regime from causality considerations.
This is interesting and let us discuss this further.
As we have mentioned, t4 = 0 reduces our theory to charged GaussBonnet gravity
previously studied in [35]. In that paper, the authors showed that this violation of the Renyi
entropy inequality can occur in CFTs dual to a hyperbolic black hole geometry when the
must satisfy,
potential,
with
black hole possesses negative thermal entropy. Now, the negative thermal entropy of a
topological black hole is a typical behavior of higher derivative gravity theories and is
controlled by the parameters of the higher derivative terms [60, 61]. However, this does
not necessarily mean that the black hole is thermally unstable. In fact, it is known that
the GaussBonnet coupling has to satisfy
negative speci c heat in GaussBonnet gravity. In our case, setting ac = 2 and t4 = 0, is
equivalent to
= 0:09 which is thermally stable.
> 14 for a thermally unstable black hole with
More speci cally, it is clear from eq. (4.16) that the negative entropy black holes
x <
q
3f1( + p 2
):
Now from eq. (4.15), the negative entropy black holes would appear when the chemical
c`
2 L~
2
27p 2
4
2 ;
+ 27
27p 2
4
2 > 0:
Eq. (4.22) can be rewritten in terms of t4 and the central charges of the theory as,
c`
2 L~
2
2
1 2 +
c
a t4
2
3
c
We see from here that negative entropy black holes appear below a certain value of the
chemical potential determined by c, a and t4. For example, with ac = 2 and t4 = 0, from
eq. (4.24), one can see that negative entropy black holes appear when 2c`L~ < 0:49. In
gure 6, since the chosen chemical potential (i.e., 2c`L~ = 0:45) is less than the aforementioned
value, negative entropy black holes would appear in this case and the CFT dual to this
negative entropy hyperbolic black hole would exhibit a violation in the second inequality
c`
obeyed by Renyi entropy. However, with t4 = 0:0036, negative entropy black holes would
appear when 2 L~ < 0:44. Since the chosen potential is greater than this value, the black
hole can not have negative entropy and hence, the corresponding CFT obeys the second
inequality as shown in gure 7.
It is therefore evident that the physically justi ed condition of satisfying the second
ERE inequality of eq. (2.5) constrains the parameter space of t4 from the apparent condition
(4.21)
(4.22)
(4.23)
(4.24)
(4.25)
2 π L
~
of t4 = 0:001 with chemical potential 2c`L~ = 1.
tential at xed value of t4 = 0:001 with ac =
2:183.
physically allowed regime of the t4 parameter space is now
given in eq. (4.14). In fact, it can be checked that, with ac = 2 and 2c`L~ = 0:45, the
0:0029
t4
It is also straightforward to check how the second ERE inequality is controlled by the
chemical potential for xed values of ac and t4. We performed this analysis with ac = 2 and
t4 = 0:001. With these CFT parameters, from our general analysis above, it can be checked
that negative entropy black hole can appear only when 2 L~ < 0:476. Indeed, the second
inequality of eq. (2.5) is violated below this value of the chemical potential, although the
black hole is thermally stable in this region.
Next, we brie y describe the behavior of the entropy ratio SSn1((0c)) as a function of ac
and the chemical potential for a particular value of t4. The plots are similar to those
obtained with GaussBonnet gravity [35]. Figure 8 shows the variation of SSn1((0c)) with the
central charge ratio with t4 = 0:001 for a typical value 2c`L~ = 1. In this gure, the red,
orange, green, blue and purple lines correspond to n = 1, 2, 3, 10 and 100 respectively.
Like gure 4, this behavior is almost linear as long as we are inside the physical regime determined by eq. (4.14). But unlike the former, here the slopes of the lines are di erent for
c`
di erent n. It can be checked that with 2 L~ = 0, the lower bound of ac has the maximum
value of SSn1((0c)) , while with 2c`L~ = 1 i.e the case depicted in gure 8, the maximum value of
SSn1((0c)) is obtained for the upper bound of ac . Hence, as we increase the chemical potential,
the entropy ratio increases and there exists a critical value of the chemical potential beyond
which SSn1((0c)) attains its maximum value at the upper bound of ac for any value of n. For
completeness, we also show a plot of the entropy ratio as a function of the chemical potential
at xed value of t4 = 0:001 for di erent values of ac = 2:183 in gure 9 where we have used
the same color coding as in gure 8.
Now we verify the second inequality obeyed by Renyi entropy for xed values of t4,
and analyze the bound on the central charge ratio ac . We set t4 = 0:002 and the chemical
the physically allowed regime for ac which is, 0:862
a
potential 2c`L~ = 0:45. With this chosen value of t4, using eq. (4.14) one can easily determine
2:404. Now with the lower bound
of ac , it can be checked that the second inequality of Renyi entropy is obeyed while with the
upper bound, this inequality is violated. This is expected if we again connect this result
with the negative entropy condition of the dual black hole geometry (although again the
black hole is thermally stable in the negative entropy region). Following the same logic as
we elaborated before, here one can explain these results with
xed values of t4 using the
same arguments based on eq. (4.24) and we will not go into the details. We just mention
that for this example, we nd that the bound on the central charge ratio imposed by the
inequality of Renyi entropy is modi ed to
a
0:862
< 1:990 :
(4.27)
We should point out here that the constraints on the parameters of charged QTG that
we have presented here follow only from the apparent violation of eq. (2.5). We note here
that the authors of [59] impose bounds on t4 or ac by computing the expectation value
of the energy, and demanding that it should be positive. The bounds on these quantities
that follows from our analysis seem to be more stringent that [59] (the methods developed
there are applicable in the presence of a chemical potential). The black holes that we
consider do appear to be thermally stable in the ranges of parameters in which negative
entropy solutions develop. The constraints on the parameters that we obtain should be
checked from arguments of causality as advocated in the important work of [62]. This is
an interesting question which we leave for a future analysis.
4.2
Scaling dimension of twist operators
Let us compute the scaling dimension hn( c) and the magnetic ux response kn( c) of the
generalized twist operators in the four dimensional CFT dual to charged quasitopological
gravity. Recall that the scaling dimension is given by,
x
4
n
f
1
2 2
`
x2n +
where, E is the energy density given by,
Here, we will use eq. (4.28) with the mass m being given in eq. (4.9),
E ( Tn0
; c)
E =
m
2`3pR3
m = 3L~2
x
4
f
1
x2 +
2 2
`
12 c2L~2 x2 + f
1 +
f
2
x2 1
Using eqs. (2.7), (4.29) and (4.30), it is straightforward to show that the scaling dimension
hn is given by,
(4.28)
(4.29)
(4.30)
(4.31)
Note that with c = 0 and
= 0 we recover the dimension of the twist operator in a CFT
dual to Einstein GaussBonnet gravity, while with c = 0 and
= 0 and
= 0, the results
match with the same in pure Einstein gravity, as expected. We also analyze the rst order
coe cients of the expansion of the scaling dimension h10 and h02 (see section 2) which are
In appropriate limits, the results agree with the corresponding ones in pure Einstein and
On the other hand, the magnetic response can be calculated by computing the charge
given by
and
GaussBonnet gravity. density,
This yields the result
h10 =
2
3
L
~ 3
3 f 2 ) =
1
2
3
h02 =
5
18
L
~ 3
`
2
L~2
(x; c) =
2
R
3 `p
~ 3 `
L
kn( c) = 2 nR3 (x; c) = 2n
~ 3 `
L
2 n
(4.32)
(4.33)
(4.34)
(4.35)
(4.36)
Note that this result of magnetic response depends on the couplings
and
through xn.
One can also extract the expansion coe cients as,
~ 3 `
L
k01( ; ) =
and k11( ; ) =
1
3 `p
~ 3 `
L
2
~
L
The expressions for magnetic response are again in agreement with the case of pure Einstein
( = 0,
= 0, c = 0) and the GaussBonnet gravity (
= 0, c = 0).
5
Charged Renyi entropy with BornInfeld and Weyl corrected gravity
Finally, we brie y address the issue of charged Renyi entropy in two other interesting
theories of modi ed gravity, namely BornInfeld and Weyl corrected gravity. In the former
case, exact hyperbolic black hole solution is known, and computation of the ERE is a
straightforward numerical exercise which we detail below. For Weylcorrected gravity, we
obtain such a black hole solution rst and then proceed to compute the ERE.
5.1
Charged Renyi entropy with BornInfeld gravity
We will start with the standard EinsteinBornInfeld action in ve dimensional AdS space
which is given by,
S =
2`3p
d x
5 p
g R +
12
L2 + b2`2
1
r
1 +
F
F
2b2
!#
;
(5.1)
ical constant
by
Maxwell term ( 14 F
gauge potential.
=
F
where, b is the BornInfeld parameter. L is the AdS length, and is related to the
cosmologg5 we have, g52 = 2``23p . As is well known, the BornInfeld term reduces to the standard
L62 . Here again in terms of the ve dimensional gauge coupling
), when we take the limit b ! 1. On the other hand, with the
limit b ! 0, the BornInfeld term vanishes and we are left with Einstein's gravity with no
The authors of [36] and particularly [37] studied the black hole solutions in these
theories, having horizons with positive(elliptic), zero (planar) and negative (hyperbolic)
constant curvatures. Since we are interested here computing the charged Renyi entropy
holographically, we only need to consider the charged hyperbolic black holes. The
hyperbolic black hole solution of the EinsteinBornInfeld gravity is given by,
ds2 =
2
L2 f (r) 1
L2
R
2 dt2 +
dr2
Lr22 f (r) 1
;
with the gauge eld given by the expression
A = (r) dt :
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
(5.7)
Here, d 23 is the line element for the three dimensional hyperbolic plane H3. The function
f (r) has the form
f (r) = 1
L2m
1 + b2`2r6
q2 !
+
L2q2
8r6 2F1
1 1 4
; ; ;
q
2
gauge eld (r) is given by,
where m and q are constants of integration and 2F1 is the standard hypergeometric
function. The constants m and q are related to the black hole mass M and electric charge
Q, respectively. Also, the position of the horizon rh is given by f (rh) = Lr22 . Further, the
h
dt2 + R2 d 32
where, c is the chemical potential, given by
(r) =
L q
2` R r2 2F1
1 1 4
; ; ;
q
2
2 R
c =
L q
` r2 2F1
h
1 1 4
; ; ;
q
2
b2`2rh6
:
Notice that c is chosen in such a way that the gauge eld (r) vanishes at the horizon
rh in order to prevent the appearance of a conical singularity. Another important point
to note is that, compared to [37], we have inserted an extra factor of L22 in the metric
component gtt. With this inclusion, we make sure that the boundary CFT dual to this black
hole geometry resides on R
H3 where the hyperbolic spatial slice H3 has curvature R,
One can also compute the mass parameter m in terms of the horizon radius rh from,
q2
1 + b2`2rh6
!
+
3q2
; ; ;
b2`2rh6
:
(5.8)
The Hawking temperature of the black hole can be computed as, The black hole entropy computed using the BekensteinHawking formula reads 1
T =
1
2 rh
L
R
+ rh3 f 0(rh)
S = 2
rh
c =
T =
S = 2
L q
` L2x2 2F1
2x2
2 xR
1
+
1 1 4
; ; ;
3 2 3
b2`2L2 "
x3 V :
q
2
b2`2L6 x6
q2
;
1 + b2`2L2x6 ;
#
As was done in the previous section, we now express the chemical potential, temperature
and entropy of the black hole in terms of the dimensionless variable x = rh=L as,
Here we should mention that since we want to compute the Renyi entropy in the
grand canonical ensemble, we should
x the chemical potential c. For this purpose, we
need to express q in terms of c and substitute that in eq. (5.12) to have an expression of
temperature in terms of c. But from eq. (5.11), it is di cult to analytically express q in
2
terms of c (this is because of the factor of xq6 inside the hypergeometric function). Hence,
we numerically evaluate the Renyi entropy at a xed chemical potential.
At this point, we brie y describe the numerical routine for computing the Renyi
entropy. First, we
x the chemical potential
c to a certain value and numerically solve
eq. (5.11) to generate q(x). Note that the entropy S does not explicitly depend on q,
while the temperature T does. But once we numerically generate the function q(x), we can
rewrite the function T (q; x) as T (x).
Since we already have the function q(x) we can straightforwardly determine x1 and xn
(as in the earlier case) by numerically solving the following equations,
T0 =
T0 =
n
1
2x21
2 x1R
1
2x2n
2 xnR
+
1
+
b2`2L2
12 R
x1 1
b2`2L2
12 R
xn 1
s
q2
1 + b2`2L2x61 ;
#
s
q2
1 + b2`2L2x6n
#
:
(5.9)
(5.10)
(5.11)
(5.12)
(5.13)
(5.14)
(5.15)
4
tropy for BornInfeld gravity with b = 0:05.
entropy for BornInfeld gravity with b = 0:05.
Now we can compute the Renyi entropy Sn. Since, Sn contains the volume factor V
involving the shortdistance cuto of the boundary CFT, we should evaluate the ratio of
Renyi entropy Sn to the entanglement entropy S1, as before. Broadly, we nd here the
following behaviour. Firstly, SSn1((0c)) always increases in a nonlinear fashion as the chemical
potential c increases. Second, for any value of the BornInfeld parameter b, the Renyi
entropy decreases as one increases the value of n. Finally, we
nd that for a particular
value of n and c
, SSn1((0c)) decreases as b increases. To end this subsection, we mention that
the inequalities of eq. (2.5) are satis ed. We show this result for xed b = 0:05 in gures 10
and 11. In both these gures, the red, orange, green, blue and purple lines (from bottom
to top) correspond to 2c`L = 0:01, 0:25, 0:5, 0:75 and 1 respectively. An entirely similar
analysis follows for a xed chemical potential.
Here, the mass m is given by,
T0
m
; c
r
4
h
L2
b2`2rh4
4
1
1 +
q2
b2`2rh6
!
+
3q2
1 1 4
; ; ;
3 2 3
q
2
h
Since m(T0; 0) = 0, nally we have the expression of the scaling dimension,
hn =
n L
3`3p m
T0
n
; c
However, from eq. (5.11) it is evident that expressing q in terms of c is di cult because
of the presence of the hypergeometric function. Hence, it is di cult to obtain an
analytical expression of the expansion of the scaling dimension, i.e h10 or h02 are di cult to
calculate. The same di culty also holds for the calculation of the coe cients of magnetic
response. Of course one could have considered various limits of the parameters in order
to (perturbatively) expand the hypergeometric function, but such an analysis may be of
limited interest and we will not belabor upon this here. Alternatively, it is possible to do
the computation in a canonical ( xed charge) ensemble. However we will not undertake
such an analysis here.
(5.16)
(5.17)
and
Next, we brie y consider the EinsteinMaxwell action along with a Weyl coupling. We will
start with the action
S =
d x
R +
12
L2
2
4
F F
+ `2L2C
F F
(5.18)
where C
is the ve dimensional Weyltensor and is the Weyl coupling. The motivation
for this action has been provided in [39, 40] (in which black holes in this theory with planar
and spherical horizon were studied), to which we refer the reader for further details.
The hyperbolic black hole solutions here can be obtained by choosing the following
metric and gauge eld ansatz,
2
L2 f0(r) 1
1 + F1(r) N (r)2dt2 +
dr2
Lr22 f0(r) 1 1 + F2(r)
where f0(r) and
0(r) are the solutions to linear order in
representing a hyperbolic
ReissnerNordstrom black hole solution in ve dimensional AdS space, with
A = ( 0(r) +
H(r)) dt ;
12r6
Solving Einstein's and Maxwell's equation up to linear order in , one can determine
the form of F (r), H(r) and N (r) as
L
R
L2q2 L2 32r2rh8 m + 3r2
q2 7rh8 + r8
+ 96r6 r2
L2q2 L2 64r2rh8 m + 3r2
q2 15rh8 + r8
+ 96r6 r2
12r6rh8 (L2 (q2
12r6rh8 (L2 (q2
4r2 (m + 3r2)) + 12r6)
4r2 (m + 3r2)) + 12r6)
Lq L2 rh8 16mr2
7q2 + 48r8rh4 + 3q2r8
48r8rh6 :
12r8` Rrh8
2rh2 rh6 ;
3rh2 rh6 ;
In the above equations, L~ is the e ective AdS curvature scale. It is straightforward
to show that for Weylcorrected gravity, L~ = L. The Hawking temperature of the
Weylcorrected black hole can be computed up to linear order in
as,
T =
2x2
1
2 xR
q
2
q
4
+
q
2
(5.19)
(5.20)
(5.21)
(5.22)
(5.23)
where again we have set rh = L x. Note that, in the limit
! 0, we get back the
temperature of a hyperbolic RNAdS black hole. Using Wald's prescription one can compute
the linear order correction in entropy as [40],
S = 2
L
V
x
3
L4x3 :
(5.24)
The Renyi entropy can be calculated at this stage by exactly the same methods and
numerical routines given in the previous sections, and we do not repeat the description of the
procedures here. We just mention here that we nd that the ERE increases in an almost
linear fashion with respect to the Weyl coupling, and that the inequalities of eq. (2.5) are
indeed satis ed for small values of . We have checked this up to j j
0:005.
A word regarding the upper limit of the Weyl coupling
is in order. Ideally, the
maximum magnitude of
for which our results are trustworthy should be determined
by computing the corrections to the Renyi entropy that arise by expanding the metric
perturbatively to second order in
and by ensuring that changes in the value of the Renyi
entropy are small for the value of the Weyl coupling chosen. This is what we had done for
the case of Einstein cubic gravity in section 3. In this case, however, due to the complicated
nature of the expressions involved (details can be found in appendix B), such an analysis
could not be performed. This is a caveat in our analysis. Speci cally, to compute the
contribution of the second order correction terms to Renyi entropy, we need to evaluate
the lower limit (xn) and upper limit (x1) (see eq. (4.17)) numerically, using T (xn) = 2 Rn
where the temperature T (x) is given in eq. (B.6). For this computation, one needs to
generate q(x) at a certain value of the chemical potential c
. Unfortunately, unlike the
case with
rst order correction, we were unable to determe q(x) precisely, due to some
issues with numerical stability. However, we should mention that such an analysis was
performed in [40] for black holes with spherical horizons, where it was checked that small
1 ,
values of
like the ones chosen in this paper were indeed trustable.
For completeness, using the methods described earlier, we also computed the scaling
h10 = 23
`p
dimension hn( c) of the twist operators. Since the form of hn is not particularly
illuminating we do not write it here. However, we should mention that the expansion coe cient
L 3 turns out to be the same as with pure Einstein gravity. It is not di cult to
convince oneself that this is due to the fact that in this case the AdS curvature scale L~ = L.
6
Summary and conclusions
In this paper, we have undertaken a detailed study of holographic entanglement Renyi
entropies in various extended theories of gravity. The results of this paper complements the
ones currently available in the literature, and provides novel examples of the computation
of EREs in strongly coupled quantum
eld theories in four dimensions. As mentioned in
the introduction, all the theories that we consider have tuneable parameters, which might
be of interest in understanding eld theories for realistic systems. Let us now summarise
the main
ndings of this paper.
We initiated a holographic study of Einsteinian cubic gravity recently proposed in [30].
Here, we constructed a black hole solution of the theory perturbatively, up to the second
order in a particular parameter, in a
ve dimensional bulk theory.
We computed the
ERE for this case and veri ed the relevant inequalities satis ed by the same. We then
computed the central charges of the dual eld theory, and checked that it matched with
a corresponding calculation of the Weyl anomaly on the gravity side. The validity of
our perturbative analysis was also justi ed. To the best of our knowledge, ours is the
rst holographic study of the ECG, and establishes its physicality as far as the ERE is
concerned. This is the rst main contribution of this paper.
We next considered examples of extended theories of gravity with a chemical potential.
In this context, we studied the ERE for charged quasitopological theories of gravity in ve
dimensions. We showed here that there exists more stringent bounds on the parameters of
the theory than one would expect from those arising out of ruling out negative excitations in
the boundary CFT, when the inequalities involving the ERE are imposed. Speci cally, we
found that such violations are induced by negative entropy black holes in the bulk theory,
as was found in [35] for CFTs dual to charged GaussBonnet gravity. We established such
bounds on the cubic coupling parameter and the ratio of the central charges, in some
speci c examples. This is the second contribution of this work.
We remark here that interestingly, we saw that negative entropy in GaussBonnet black
holes that arose at a certain value of the central charge ratio and the chemical potential
was removed when the coupling to cubic interaction was turned on. This might indication
that in a topdown string theoretic approach, the negative entropy problem in QTG might
as well be cured by possible higher order terms, although this is a mere speculation. As we
have mentioned in the main text, it will be useful to consider the constraints on the CFT
parameters from causality arguments advocated in [62], although we have not attempted
it here.
Finally, we brie y studied ERE in BornInfeld and Weylcorrected gravity, with a
perturbative analysis to establish black hole solutions in the latter. We found that the
ERE inequalities are satis ed to the limit of approximation that we consider. We also
commented upon the scaling dimension of the twist operators.
It might be interesting to investigate the Renyi entropy with an imaginary chemical
potential in the higher derivative charged modi ed gravity theories considered in this paper,
following the line of [34, 35]. It will also be interesting to study the charged ERE in
canonical ensembles where the charge parameter is kept
xed, instead of the chemical
potential. Further, instead of a purturbative analysis, one could numerically solve the
Einstein equations to obtain numerical black hole solutions in Einsteinian cubic gravity
and Einstein gravity with a Weylcorrected gauge
eld. This way of studying the ERE
would help understand better the entropy bounds, and hence the bounds on the coupling
constants of the theories, from the validation of the Renyi entropy inequalities.
Acknowledgments
We sincerely thank P. Bueno and P. Cano for pointing out an error in an ansatz, in a
previous version of this paper. It is also a pleasure to thank Subhash Mahapatra for
helpful discussions. Some of the computations of this paper have been performed with the
MATHEMATICA packages xAct [63] and RGTC [64]. The authors of these packages are
gratefully acknowledged.
A
Correction up to O( 2) in Einsteinian cubic gravity
Here we note down the hyperbolic black hole solution up to O( 2) in Einsteinian cubic
gravity, considering the following metric ansatz,
ds2 =
1 +
1
Lr22 f0 (r) 1 (1 +
F2 (r) + 2G2(r))
F1 (r) + 2G1 (r) N (r)2 dt2
dr2 + r2d 23
(A.1)
(A.3)
(A.4)
Here, f0(r) is the zeroth order solution while F1(r) and F2(r) are the O( ) terms
given in section 3. G1(r) and G2(r) are the O( 2) corrections to be determined. Since, the
recipe to nd out the solution is already discussed in detail in section 3, we just mention
the functions second order in ,
7r16r6
h
G2(r) =
7r16r6
h
L2r2 + r4
!4
(A.2)
A = 7r2!16rh6 14447r2
13314L2 + r4!12r6
h
48000L4 + 104336L2r2
57883r4
+ 14r10!4rh4 12L2
+ 7r8!8rh4 1875L2rh4
19r2
54L4rh2 + 75L2r4
h
31rh6 + 11L6
44926!20r6
h
2rh2 675L4
18L2r2 + 38r4
775rh6 + 275L6 + r16
1501L8rh2 + 4313L6rh4
5610L4rh6 + 3212L2rh8 + 7rh6 r4
88rh4 + 195L10
B = 27r2!16rh6 5285r2
+ 14r10!4rh4 16L2
+ 7r8!8rh4 2775L2rh4
5104L2
23r2
7r4!12rh6 9600L4
19872L2r2 + 10481r4
54L4rh2 + 75L2r4
h
31rh6 + 11L6
70028!20r6
h
2rh2 999L4
24L2r2 + 46r4
1147rh6 + 407L6 + r16
1501L8rh2 + 4313L6rh4
5610L4rh6 + 3212L2rh8 + 7rh6 r4
88rh4 + 195L10
Also, the function N (r) at O( 2) is given by,
N (r) =
L
R
L
p
R f
1
where f
1 is determined as, f1 = 1
+ 3 2.
at O( 2),
For completeness, here we mention the Hawking temperature and the Wald entropy
T =
2x2
2 xR
1
x
2
1 3
x5R
+ 2 x
2
1 3 1834x4
459x2
201
14 x9R
and
`p
3
2
25x3
8
3 2 905x3
3168x +
3996
(A.5)
We also computed the Renyi entropy correction at O( 2), but do not write it explicitly
here, since the expression is unwieldy. Instead we give the expression for the entanglement
entropy which is obtained by taking the n ! 1 limit.
HJEP04(218)9
`p
2
+
8
We also calculated the scaling dimension of the twist operator at O( 2) and computed
the quantity @nhnjn=1, in a similar way as we did with the leading order solution,
2
3
1 +
2
33 2 :
As a further check for our computations with the second order solution, we calculated
the O( 2) correction to the Weyl anomaly, i.e., the correction to the central charges c and
a using the methods outlined in section 3.3. We
nd that the coe cients match exactly
with those that can be read o from the above expressions (A.6) and (A.7),
c = 2
a = 2
`p
1 +
1
9
2
15
2
33 2
8
15 2 :
B
Correction up to O( 2) in Weylcorrected gravity
Here we note down the Weylcorrected hyperbolic black hole solution up to O( 2),
considering the following ansatz for the metric and gauge eld ansatz,
and
ds2 =
+
2
L2 f0 (r) 1
1 + F1 (r) + 2G1 (r) N (r)2 dt2
dr2
Lr22 f0 (r) 1
1 + F2(r) + 2G2(r)
;
A = 0(r) +
H(r) + 2J (r) dt ;
where f0(r) is the zeroth order solution, F1(r) and F2(r) are the rst order solutions of
the metric, while G1(r) and G2(r) represent the O( 2) corrections to the metric. On the
other hand, 0(r) and H(r) are respectively the zeroth order and rst order corrections to
the gauge eld, while J (r) stands for the second order correction to the gauge eld. Since,
(A.6)
(A.7)
(A.8)
(B.1)
(B.2)
1
1
90r12rh14(12r6
4L2mr2 + L2q2
12L2r4)
11rh10 41m + 171r2
+ 384r4rh8 mrh6 11m + 54r2 + 63r10 + q4 5r8rh6
+ 365rh14 + 268r14
12L2r6rh6 384 7mr2rh8 + 18r8rh4 + q2 40r2rh6
L2q2 L4 8q2r2rh4 672r12
1895rh8 + 809r8
45r12rh14(12r6
4L2mr2 + L2q2
12L2r4)
+ 192r4rh8 mrh6 53m + 180r2 + 63r10 + q4 5r8rh6
+ 2025rh14 + 134r14
6L2r6rh6 192r2rh4 35mrh4 + 36r6 + q2 80r2rh6
5625rh8
L2q2 L4 32q2 84r14r4
h
+ 29376r14r12 ;
h
simply write down the functions at O( 2),
the procedure to nd the solution has been described earlier with O( ) correction, here we
(B.3)
HJEP04(218)9
(B.4)
(B.5)
(B.6)
and
and
SWald = 2
V
x
3
L4x3 + 2 6q4
q2
48L4q2x6 + 48L4q2x4
L8x9
:
(B.7)
Open Access.
This article is distributed under the terms of the Creative Commons Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. { 33 {
1
J (r) =
180` Rr14r14 Lq L4 1152 m2r4rh14 9r14r8
h
h
48q2 r2rh14 33m+32r2
+31r14rh4 +q4 5r8rh6 +275rh14 +44r14
+24L2r6rh6 q2
20r2rh6 +75rh8 +71r8
+864r8rh4
10368r14r12 :
h
Also, N (r) is simply given by, N (r) = RL .
computed at O( 2),
T (x) =
For completeness, we also note down the Hawking temperature and Wald entropy
2x2
1
2 xR
+ 2
q
2
49q6
540 L12x17R
32q2
5 L4x9R
7q4
136q2
18 L8x13R
121q4
45 L8x11R
104q2
q
q
2
2 L4x7R
2q2
3 L4x5R
0406 (2004) P06002 [hepth/0405152] [INSPIRE].
(2009) 504005 [arXiv:0905.4013] [INSPIRE].
110404 [hepth/0510092] [INSPIRE].
[1] P. Calabrese and J.L. Cardy, Entanglement entropy and quantum eld theory, J. Stat. Mech.
[2] P. Calabrese and J. Cardy, Entanglement entropy and conformal eld theory, J. Phys. A 42
[4] M. Levin and X.G. Wen, Detecting Topological Order in a Ground State Wave Function,
Phys. Rev. Lett. 96 (2006) 110405 [condmat/0510613].
[5] B. Hsu, M. Mulligan, E. Fradkin and E.A. Kim, Universal entanglement entropy in 2D
conformal quantum critical points, Phys. Rev. B 79 (2009) 115421 [arXiv:0812.0203]
Theor. Phys. 38 (1999) 1113 [hepth/9711200] [INSPIRE].
[hepth/9802150] [INSPIRE].
[8] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT,
Phys. Rev. Lett. 96 (2006) 181602 [hepth/0603001] [INSPIRE].
[9] V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement
entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
[10] L.Y. Hung, R.C. Myers and M. Smolkin, On Holographic Entanglement Entropy and Higher
Curvature Gravity, JHEP 04 (2011) 025 [arXiv:1101.5813] [INSPIRE].
[11] A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090
[arXiv:1304.4926] [INSPIRE].
[12] D.V. Fursaev, A. Patrushev and S.N. Solodukhin, Distributional Geometry of Squashed
Cones, Phys. Rev. D 88 (2013) 044054 [arXiv:1306.4000] [INSPIRE].
[13] X. Dong, Holographic Entanglement Entropy for General Higher Derivative Gravity, JHEP
01 (2014) 044 [arXiv:1310.5713] [INSPIRE].
[14] M.R. Mohammadi Moza ar, A. Mollabashi, M.M. SheikhJabbari and M.H. Vahidinia,
Holographic Entanglement Entropy, Field Rede nition Invariance and Higher Derivative
Gravity Theories, Phys. Rev. D 94 (2016) 046002 [arXiv:1603.05713] [INSPIRE].
[15] D. Allahbakhshi, M. Alishahiha and A. Naseh, Entanglement Thermodynamics, JHEP 08
(2013) 102 [arXiv:1305.2728] [INSPIRE].
(2016) 125015 [arXiv:1607.07899] [INSPIRE].
[16] A. Naseh, Scale versus conformal invariance from entanglement entropy, Phys. Rev. D 94
[17] D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498
[18] J. Oliva and S. Ray, A new cubic theory of gravity in ve dimensions: Black hole, Birkho 's
theorem and Cfunction, Class. Quant. Grav. 27 (2010) 225002 [arXiv:1003.4773]
[arXiv:1003.5357] [INSPIRE].
[20] J. Oliva and S. Ray, Birkho 's Theorem in Higher Derivative Theories of Gravity, Class.
Quant. Grav. 28 (2011) 175007 [arXiv:1104.1205] [INSPIRE].
[21] M. Brigante, H. Liu, R.C. Myers, S. Shenker and S. Yaida, Viscosity Bound Violation in
Higher Derivative Gravity, Phys. Rev. D 77 (2008) 126006 [arXiv:0712.0805] [INSPIRE].
[22] M. Brigante, H. Liu, R.C. Myers, S. Shenker and S. Yaida, The Viscosity Bound and
Causality Violation, Phys. Rev. Lett. 100 (2008) 191601 [arXiv:0802.3318] [INSPIRE].
[23] A. Buchel, J. Escobedo, R.C. Myers, M.F. Paulos, A. Sinha and M. Smolkin, Holographic GB
gravity in arbitrary dimensions, JHEP 03 (2010) 111 [arXiv:0911.4257] [INSPIRE].
[24] J. de Boer, M. Kulaxizi and A. Parnachev, Holographic Lovelock Gravities and Black Holes,
JHEP 06 (2010) 008 [arXiv:0912.1877] [INSPIRE].
JHEP 08 (2010) 035 [arXiv:1004.2055] [INSPIRE].
[25] R.C. Myers, M.F. Paulos and A. Sinha, Holographic studies of quasitopological gravity,
[26] A. Renyi, On measures of information and entropy, in Proceedings of the 4th Berkeley
Symposium on Mathematics, Statistics and Probability, 1, 547, University of California Press,
Berkeley, CA, U.S.A., (1961).
[28] L.Y. Hung, R.C. Myers, M. Smolkin and A. Yale, Holographic Calculations of Renyi
Entropy, JHEP 12 (2011) 047 [arXiv:1110.1084] [INSPIRE].
[29] X. Dong, The Gravity Dual of Renyi Entropy, Nature Commun. 7 (2016) 12472
[30] P. Bueno and P.A. Cano, Einsteinian cubic gravity, Phys. Rev. D 94 (2016) 104005
[31] R.A. Hennigar and R.B. Mann, Black holes in Einsteinian cubic gravity, Phys. Rev. D 95
[arXiv:1601.06788] [INSPIRE].
[arXiv:1607.06463] [INSPIRE].
(2017) 064055 [arXiv:1610.06675] [INSPIRE].
D 95 (2017) 104042 [arXiv:1703.01631] [INSPIRE].
JHEP 03 (2018) 150 [arXiv:1802.00018] [INSPIRE].
[33] P. Bueno, P.A. Cano and A. Ruiperez, Holographic studies of Einsteinian cubic gravity,
[34] A. Belin, L.Y. Hung, A. Maloney, S. Matsuura, R.C. Myers and T. Sierens, Holographic
Charged Renyi Entropies, JHEP 12 (2013) 059 [arXiv:1310.4180] [INSPIRE].
[35] G. Pastras and D. Manolopoulos, Charged Renyi entropies in CFTs with
EinsteinGaussBonnet holographic duals, JHEP 11 (2014) 007 [arXiv:1404.1309]
[INSPIRE].
[36] T.K. Dey, BornInfeld black holes in the presence of a cosmological constant, Phys. Lett. B
595 (2004) 484 [hepth/0406169] [INSPIRE].
70 (2004) 124034 [hepth/0410158] [INSPIRE].
D 86 (2012) 064035 [arXiv:1206.4738] [INSPIRE].
[37] R.G. Cai, D.W. Pang and A. Wang, BornInfeld black holes in (A)dS spaces, Phys. Rev. D
[38] W.G. Brenna and R.B. Mann, Quasitopological ReissnerNordstrom Black Holes, Phys. Rev.
(2013) 063 [arXiv:1305.7191] [INSPIRE].
(2013) 050 [arXiv:1306.2640] [INSPIRE].
[39] A. Dey, S. Mahapatra and T. Sarkar, Holographic Thermalization with Weyl Corrections,
JHEP 01 (2016) 088 [arXiv:1510.00232] [INSPIRE].
[40] A. Dey, S. Mahapatra and T. Sarkar, Thermodynamics and Entanglement Entropy with Weyl
Corrections, Phys. Rev. D 94 (2016) 026006 [arXiv:1512.07117] [INSPIRE].
[41] M. Caraglio and F. Gliozzi, Entanglement Entropy and Twist Fields, JHEP 11 (2008) 076
[arXiv:0808.4094] [INSPIRE].
entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
[42] H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement
[43] D.A. Galante and R.C. Myers, Holographic Renyi entropies at nite coupling, JHEP 08
[44] A. Belin, A. Maloney and S. Matsuura, Holographic Phases of Renyi Entropies, JHEP 12
[45] A. Belin, L.Y. Hung, A. Maloney and S. Matsuura, Charged Renyi entropies and holographic
superconductors, JHEP 01 (2015) 059 [arXiv:1407.5630] [INSPIRE].
[46] T.S. Biro and V.G. Czinner, A qparameter bound for particle spectra based on black hole
thermodynamics with Renyi entropy, Phys. Lett. B 726 (2013) 861 [arXiv:1309.4261]
[INSPIRE].
[47] V.G. Czinner and H. Iguchi, Renyi Entropy and the Thermodynamic Stability of Black Holes,
Phys. Lett. B 752 (2016) 306 [arXiv:1511.06963] [INSPIRE].
[48] C.S. Chu and R.X. Miao, Universality in the shape dependence of holographic Renyi entropy
for general higher derivative gravity, JHEP 12 (2016) 036 [arXiv:1608.00328] [INSPIRE].
[49] R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) R3427
[grqc/9307038] [INSPIRE].
[50] S. Mahapatra, Logarithmic black hole entropy corrections and holographic Renyi entropy,
Eur. Phys. J. C 78 (2018) 23 [arXiv:1609.02850] [INSPIRE].
[51] K. Zyczkowski, Renyi extrapolation of Shannon entropy, Open Syst. Inf. Dyn. 10 (2003) 297
[52] L.Y. Hung, R.C. Myers and M. Smolkin, Twist operators in higher dimensions, JHEP 10
(2014) 178 [arXiv:1407.6429] [INSPIRE].
[hepth/9308075] [INSPIRE].
[hepth/9806087] [INSPIRE].
d'aujourd'hui, Asterisque, (1985), pg. 95.
[55] C. Fe erman and C.R. Graham, Conformal Invariants, in Elie Cartan et les Mathematiques
[56] K. Sen, A. Sinha and N.V. Suryanarayana, Counterterms, critical gravity and holography,
Phys. Rev. D 85 (2012) 124017 [arXiv:1201.1288] [INSPIRE].
Holography, JHEP 10 (2013) 210 [arXiv:1307.0330] [INSPIRE].
[57] M.H. Dehghani and M.H. Vahidinia, Quartic Quasitopological Gravity, Black Holes and
[58] R.X. Miao, A Note on Holographic Weyl Anomaly and Entanglement Entropy, Class.
Quant. Grav. 31 (2014) 065009 [arXiv:1309.0211] [INSPIRE].
JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].
[hepth/0112045] [INSPIRE].
Black Hole with Higher Derivative Gauge Corrections, JHEP 07 (2009) 030
Corrections to the Graviton ThreePoint Coupling, JHEP 02 (2016) 020 [arXiv:1407.5597]
[3] A. Kitaev and J. Preskill , Topological entanglement entropy , Phys. Rev. Lett . 96 ( 2006 ) [7] E. Witten , Antide Sitter space and holography, Adv. Theor. Math. Phys. 2 ( 1998 ) 253 [19] R.C. Myers and B. Robinson , Black Holes in Quasitopological Gravity , JHEP 08 ( 2010 ) 067 [27] A. Renyi , On the foundations of information theory , Rev. Int. Stat. Inst . 33 ( 1965 ) 1 .
[32] R.A. Hennigar , D. Kubiznak and R.B. Mann , Generalized quasitopological gravity , Phys. Rev . [53] M.J. Du , Twenty years of the Weyl anomaly , Class. Quant. Grav . 11 ( 1994 ) 1387 [54] M. Henningson and K. Skenderis , The Holographic Weyl anomaly , JHEP 07 ( 1998 ) 023 [59] D.M. Hofman and J. Maldacena , Conformal collider physics: Energy and charge correlations , [60] M. Cvetic , S. Nojiri and S.D. Odintsov , Black hole thermodynamics and negative entropy in de Sitter and antide Sitter EinsteinGaussBonnet gravity , Nucl. Phys. B 628 ( 2002 ) 295 [61] D. Anninos and G. Pastras , Thermodynamics of the MaxwellGauss Bonnet antide Sitter [62] X.O. Camanho , J.D. Edelstein , J. Maldacena and A. Zhiboedov , Causality Constraints on [63] J.M. MartinGarcia , http://www.xact.es.