On holographic Rényi entropy in some modified theories of gravity

Journal of High Energy Physics, Apr 2018

Abstract We perform a detailed analysis of holographic entanglement Rényi entropy in some modified theories of gravity with four dimensional conformal field theory duals. First, we construct perturbative black hole solutions in a recently proposed model of Einsteinian cubic gravity in five dimensions, and compute the Rényi entropy as well as the scaling dimension of the twist operators in the dual field theory. Consistency of these results are verified 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 modified gravity in five dimensions that include a chemical potential, namely Born-Infeld gravity, charged quasi-topological gravity and a class of Weyl corrected gravity theories with a gauge field, with the last example being treated perturbatively. Some interesting bounds in the dual conformal field theory parameters in quasi-topological gravity are pointed out. We also provide arguments on the validity of our perturbative analysis, whenever applicable.

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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 Born-Infeld gravity, charged quasi-topological 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 quasi-topological gravity are pointed out. We also provide arguments on the validity of our perturbative analysis, whenever applicable. AdS-CFT 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 quasi-topological gravity 4.1 Numerical analysis and results 4.2 Scaling dimension of twist operators 5 Charged Renyi entropy with Born-Infeld and Weyl corrected gravity 5.1 5.2 Charged Renyi entropy with Born-Infeld 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 Weyl-corrected gravity gravitational description arose out of the seminal work by Ryu and Takayanagi [8]. The Ryu-Takayangi 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 Gauss-Bonnet gravity, Lovelock gravity [17], and quasi-topological 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 Einstein-Hilbert 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 Gauss-Bonnet 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 quasi-topological 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 Gauss-Bonnet 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 Born-Infeld 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 Born-Infeld 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 Weyl-corrected gauge eld [39, 40]. We should mention here that our studies of the ERE in Einsteinian cubic gravity and Weyl-corrected 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 Weyl-corrected 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 n-fold 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 Gauss-Bonnet gravity and quasi-topological gravity. In a related work, [43] computed the ERE for four dimensional N = 4 super-Yang-Mills 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 d-dimensional 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 d-dimensional 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 four-derivative Gauss-Bonnet 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 Gauss-Bonnet 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 short-distance cut-o of the dual CFT. Now after expanding V in powers of R , it is straightforward to single out the universal contribution coming from the sub-leading 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 space-time 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 short-distance cut-o 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 cut-o . 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 next-to-leading 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 rst-order 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 quasi-topological gravity with a chemical potential. 4 Charged Renyi entropy with quasi-topological gravity In this section, we will discuss the properties of the charged Renyi Entropy of a four dimensional CFT, dual to ve dimensional charged quasi-topological 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 four-derivative Gauss-Bonnet term, while Z5 is a six-derivative 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 Gauss-Bonnet 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 quasi-topological 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 non-linear 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 Gauss-Bonnet 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 Gauss-Bonnet coupling has to satisfy negative speci c heat in Gauss-Bonnet 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 Gauss-Bonnet 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 quasi-topological 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 Gauss-Bonnet 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 Gauss-Bonnet 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 Gauss-Bonnet gravity ( = 0, c = 0). 5 Charged Renyi entropy with Born-Infeld 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 Born-Infeld 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 Weyl-corrected gravity, we obtain such a black hole solution rst and then proceed to compute the ERE. 5.1 Charged Renyi entropy with Born-Infeld gravity We will start with the standard Einstein-Born-Infeld 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 Born-Infeld parameter. L is the AdS length, and is related to the cosmologg5 we have, g52 = 2``23p . As is well known, the Born-Infeld 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 Born-Infeld 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 Einstein-Born-Infeld 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 Bekenstein-Hawking 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 Born-Infeld gravity with b = 0:05. entropy for Born-Infeld gravity with b = 0:05. Now we can compute the Renyi entropy Sn. Since, Sn contains the volume factor V involving the short-distance cut-o 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 non-linear fashion as the chemical potential c increases. Second, for any value of the Born-Infeld 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 Einstein-Maxwell 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 Weyl-tensor 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 Reissner-Nordstrom 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 Weyl-corrected 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 quasi-topological 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 Gauss-Bonnet 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 Gauss-Bonnet 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 top-down 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 Born-Infeld and Weyl-corrected 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 Weyl-corrected 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 Weyl-corrected gravity Here we note down the Weyl-corrected 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 (CC-BY 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 [hep-th/0405152] [INSPIRE]. (2009) 504005 [arXiv:0905.4013] [INSPIRE]. 110404 [hep-th/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. 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Anshuman Dey, Pratim Roy, Tapobrata Sarkar. On holographic Rényi entropy in some modified theories of gravity, Journal of High Energy Physics, 2018, 98, DOI: 10.1007/JHEP04(2018)098