Logarithmic Black Hole Entropy Corrections and Holographic Rényi Entropy
aa r X i v : . [ h e p - t h ] S e p Logarithmic Black Hole Entropy Corrections andHolographic R´enyi Entropy
Subhash Mahapatra ∗ The Institute of Mathematical Sciences,Chennai 600113, India&KU Leuven Campus Kortrijk - KULAK, Department of Physics,Etienne Sabbelaan 53 bus 7800, 8500 Kortrijk, Belgium
Abstract
The entanglement and R´enyi entropies for spherical entangling surfacesin CFTs with gravity duals can be explicitly calculated by mapping theseentropies first to the thermal entropy on hyperbolic space and then, usingthe AdS/CFT correspondence, to the Wald entropy of topological blackholes. Here we extend this idea by taking into account corrections to theWald entropy. Using the method based on horizon symmetries and theasymptotic Cardy formula, we calculate corrections to the Wald entropyand find that these corrections are proportional to the logarithm of hori-zon area. With the corrected black hole entropy expression, we then findcorrections to the R´enyi entropies. We calculate these corrections for bothEinstein as well as Gauss-Bonnet gravity duals. Corrections with logarith-mic dependence on the area of the entangling surface naturally occur at theorder G D and it seems to be a general feature of entanglement and R´enyientropies for CFTs with gravity duals. In particular, there is a logarithmiccorrection to the entropy in odd boundary spacetime dimensions as well. ∗ [email protected], [email protected] Introduction
Quantum entanglement is one of the most remarkable properties of quantumsystems which is essential to most of quantum information applications. Thequantification and characterization of entanglement is an important problem inquantum-information science and a number of measures have been suggested forits definition in the literature [1]. Two such measures are entanglement and R´enyientropies. Formally, given a subsystem v and its compliment ¯ v , the entanglemententropy of the subsystem v is defined by the von Neumann entropy of its reduceddensity matrix ρ v S EE = − T r [ ρ v ln ρ v ] (1)where ρ v is obtained by tracing out the degrees of freedom of subsystem ¯ v fromthe full density matrix. Similarly, R´enyi entropy is defined by a one-parametergeneralization of the von Neumann entropy S q = 11 − q ln T r [ ρ qv ] (2)where q is a positive real number. Given the above definition, one finds thatlim q → S q = S EE , i.e , entanglement entropy is recovered from R´enyi entropy bytaking the limit q → q -th power of the den-sity matrix ρ v to the partition function on a singular q -folded Riemann surface.Geometrically, this q -fold space is a flat cone with angle deficit 2 π (1 − q ) at theentangling surface and the Euclidean path integral over fields defined on this q -fold space can be explicitly computed for a few simple cases. However, for genericquantum field theories the computation of the partition function on the singularsurfaces is rather difficult, which limits the usefulness of the replica method.On the other hand, recent developments in the AdS/CFT correspondence [3]have suggested an elegant and geometric way of computing entanglement entropyin conformal field theories (CFT) which have gravity duals. In the seminal workof Ryu and Takayanagi (RT) [4] [5], the entanglement entropy of a d -dimensionalboundary CFT was conjectured to be given by the area of a minimal surface inthe bulk AdS space as S EE = Area ( γ v )4 G D (3)where γ v is the ( d − ∂v of the subsystem v .This proposal has been extensively tested for a variety of systems and by nowthere is a good amount of evidence to support this conjecture [6]. Indeed, further1ndications for the correctness of this conjecture was provided in [7], where thegeneralized replica method was applied in the bulk AdS space to prove the RTconjecture.An alternative approach to calculate the holographic entanglement entropy,which works only for spherical entangling surfaces, was proposed in [8]. In [8], itwas observed that the reduced density matrix for a spherical entangling region inflat space can be conformally mapped to the thermal density matrix on the hy-perbolic space; or equivalently the original entanglement entropy can be mappedto thermal entropy on the hyperbolic space [9]. Then, using the AdS/CFT corre-spondence, the latter thermal entropy can be again mapped on the dual gravityside to black hole entropy of certain topological black holes with hyperbolic hori-zons. This observation therefore also provided an alternative derivation of theRT conjecture for a spherical entangling surface in d -dimensional CFT. Moreoverin [10], it was observed that the idea of [8] can be further extended to study theR´enyi entropy for a spherical entangling region. This is highly useful from thepractical point of view, since no concise prescription to calculate R´enyi entropyholographically is known yet.This procedure of mapping entanglement and R´enyi entropies of the boundaryCFT to black hole entropy have many advantages. The most important advantageperhaps, apart from providing computational simplicity, is that one can easilyextend this procedure to calculate R´enyi entropy for higher dimensional CFTs.This is important since most of the work regarding holographic R´enyi entropyhas been limited to two-dimensional boundary CFTs [11] [12]. Moreover, thismapping of R´enyi entropy can be easily generalized to boundary theories whichare dual to higher derivative gravity theories.An essential point which comes in the derivation of [8] [10] is the notion ofblack hole entropy, which can be computed using the standard Wald formula.Using the Wald entropy, the above procedure indeed reproduces independentknown results of R´enyi entropy. However an important point, which we want tostress here and that will play a significant role in this paper, is that the Waldentropy only gives the leading order contribution to black hole entropy. Thereare additional subleading quantum corrections to the usual Wald entropy, whichare generally proportional to the logarithm of horizon area. Indeed, recent activ-ities in most quantum gravity models, including string theory and loop quantumgravity (LQG) or methods based on diffeomorphism symmetry arguments, havepredicted logarithmic corrections to black hole entropy, with a general expressionlike S = S W ald + C ln S W ald + ... (4)The coefficient C has been calculated for a variety of cases. For example, forthe Schwarzschild black hole the coefficient C was found to be equal to − / C was found2o be dependent on the total number of spacetime dimensions as well as numberof massless fields. A simple comparison showed disagreement on the value of C in these two quantum gravity models .The black hole entropy and its logarithmic correction can also be calculatedusing an approach based on the argument of diffeomorphism symmetry on thehorizon. This approach, using the work of [26], was initiated in [27] and lateremphasized in [28]- [30] to calculate black hole entropy. In this approach onefirst identifies a set of vector fields (based on some physical consideration) andthen construct an algebra for the Fourier modes of the charges (correspondingto an appropriate diffeomorphism symmetry) from these vector fields. Using thisalgebra, which usually has a two dimensional Virasoro algebra like structure, onethen extracts the central charge and the zero mode from it and then finally usethe Cardy formula [31] for the asymptotic density of states to compute black holeentropy. It has been shown in [28]- [30] that this approach does indeed correctlyreproduce the expression for the black hole entropy. Moreover in [32], it wasshown that this symmetry based approach can be further used to calculate thelogarithmic correction to the BTZ black hole and it was found that C = − / .Importantly, the above procedure can also be generalized to compute loga-rithmic corrections to the entropy of hyperbolic black holes which are relevantfor our purpose here, especially since it is unclear (at-least to the author) howto calculate logarithmic corrections to entropy of these black holes in LQG or instring theory context. In this paper, using the methodology of [33], we calculatethese corrections for two gravity theories, namely Einstein and Gauss-Bonnetgravity. We find C = − / C here. Instead,we want to probe the effects of this C on the R´enyi entropy. For this reason, wewill work with arbitrary C in this paper for most of the time.Using the corrected black hole entropy expression (eq.(4)) into the prescrip-tion of [8] [10], we find that there are corrections to the standard results of R´enyientropies. These corrections are both logarithmic as well as non-logarithmic innature. For a two dimensional CFT, we find that the correction terms in theR´enyi entropy are function of the logarithmic of the central charge as well as in-dex q . However, the dependence on the size of the system is doubly logarithmic.For higher dimension CFTs, R´enyi entropy is a complicated function of q . Forthe entanglement entropy our results simplify tremendously. We find that correc-tion term in the entanglement entropy is proportional to C times the logarithmof its standard expression. Moreover, this result is the same in all dimensions.Interestingly, at the order G D , the correction term in the entanglement entropydepends logarithmically on the area of the spherical entangling surface. This re-sult holds for odd boundary dimensions as well. This is important since quantum Black hole entropy can also acquires logarithmic corrections due to thermal fluctuations inthe black hole extensive parameters. More details on this can be found in [21]- [25]. This result was based on certain assumptions. We will discuss these assumptions in moredetails in section 4. G D . Our analysis suggests a similar kind of corrections in the R´enyi entropy too.With a Gauss-Bonnet black hole as a gravity dual, the holographic R´enyi entropyis found to be a complicated function of two distinct central charges. However,the entanglement entropy depends only on one central charge. The correctionterms in the entanglement entropy is again found to be proportional to C timesthe logarithm of its standard expression.It is also well known that R´enyi entropy satisfies few a inequalities involvingthe derivative with respect to q [34]- [35]. We find that for higher dimensionalCFTs, these inequalities are again satisfied even with correction terms providedthe coefficient C is not very large. However for two dimensional CFTs, a few ofthese inequalities can be violated in limit of small central charge. Although forlarge central charge, which is the case for the boundary CFT with gravity dual,these inequalities are again found to be satisfied.The paper is organized as follows : In the next section, we review the mainideas of [8] [10] to relate the R´enyi entropy to black hole entropy. In section 3,we highlight the necessary steps to calculate the asymptotic form of density ofstates using the two dimensional conformal algebra. In section 4, we first con-struct the Virasoro algebra having a central extension on the black hole horizonand then use the expression of the density of states to calculate the black holeentropy. In the process we calculate the logarithmic correction to the entropyof AdS-Schwarzschild and Gauss-Bonnet black hole. In section 5, we analyzethe holographic R´enyi entropy in detail and discuss the nature of the correctionterms. Finally, we conclude by summarizing our main results in section 6. In this section, we review some aspects of holographic entanglement and R´enyientropies which will be the focus of this paper. As mentioned in the introduction,we follow the prescription of [8] [10] to calculate these entropies for a sphericalentangling region in the boundary CFT using the AdS/CFT correspondence. Letus first briefly discuss the work done in these papers to set the stage.Consider a d -dimensional CFT on Minkowski space and choose a sphericalentangling surface of radius R as a subsystem. The computation of entanglemententropy of this subsystem with the rest of the system can be performed by cal-culating the reduced density matrix ρ v . However the authors in [8], using theconformal structure of the theory, mapped this problem of entanglement entropyto the thermal entropy on a hyperbolic space R × H d − . They showed that thecausal development of the ball enclosed by the spherical entangling surface canbe mapped to a hyperbolic space R × H d − ; with curvature of H d − space givenby the radius of the spherical entangling region R . An important point of thismapping was that the vacuum of the original CFT mapped to a thermal bathwith temperature T = 12 πR (5)4n the hyperbolic space. Now relating the density matrix ρ therm in the newspacetime R × H d − to the old spacetime ρ v by the unitary transformation: ρ v = U − ρ therm U , we get ρ v = U − e [ − H/T ] Z ( T ) U (6)where Z ( T ) = T r [ e − H/T ]. For the R´enyi entropy we also need the q ’th power of ρ v . From the above equation, we get ρ qv = U − e [ − qH/T ] Z ( T ) q U (7)Taking the trace of both sides of eq.(7), we get T r [ ρ qv ] = Z ( T /q ) Z ( T ) q (8)as U and its inverse cancels each other upon taking trace. Now, using the defini-tion of R´enyi entropy as in eq.(2), we arrive at S q = 11 − q (cid:2) ln Z ( T /q ) − q ln Z ( T ) (cid:3) (9)The above expression for the R´enyi entropy can also be written in terms of thefree energy F ( T ) = − T ln Z ( T ): S q = q − q T (cid:2) F ( T ) − F ( T /q ) (cid:3) (10)and further, using the thermodynamic relation S therm = − ∂F/∂T , we can rewritethe above expression as S q = qq − T Z T T /q dT S therm ( T ) (11)here, just to clarify again, S therm is the thermal entropy of a d -dimensional CFTon R × H d − while S q is the desired R´enyi entropy. Eq.(11) was the main resultof [8] [10], which relates R´enyi (and hence entanglement) entropy of a sphericalentangling region in d -dimensional CFT to the thermal entropy on a hyperbolicspace. As pointed out in [8], the above analysis just mapped one difficult problemto another equally difficult problem and is not particularly useful for practicalpurposes. However, its true usefulness can be realized via the AdS/CFT corre-spondence. In the AdS/CFT correspondence, the thermal state of the boundaryCFT corresponds to an appropriate non-extremal black hole in the bulk AdSspacetime, with thermal entropy corresponding to black hole entropy. Therefore,using the AdS/CFT correspondence, we can relate S therm appearing in eq.(11)to that of black hole entropy, which is relatively easy to compute. Since on the5oundary side our CFT is on R × H d − , its dual gravity theory will be describedby a topological black hole with hyperbolic event horizon. In any event, in thisAdS/CFT approach, the R´enyi entropy is now given by the horizon entropy ofthe corresponding hyperbolic black hole, which can be easily computed usingWard’s standard formula. However, since we are interested in calculating theeffects of corrections of black hole entropy on the R´enyi entropy, here we will usean approach based on symmetry arguments on the horizon to calculate the blackhole entropy, instead of Wald’s formula. This is the topic of discussion of thenext section. In this section, we will describe the necessary steps to calculate the asymptoticform of density of states from a two dimensional conformal algebra. This formof density of states will be used in a later section to calculate the black holeentropy and, further, to compute logarithmic corrections. In this section, we willmostly follow the notations used in [32] and refer the readers to [32] for a detaileddiscussion.We start with a standard Virasoro algebra of the two conformal field theorywith central charges c , ¯ c : (cid:2) L m , L n (cid:3) = ( m − n ) L m + n + c m ( m − δ m + n, (cid:2) ¯ L m , ¯ L n (cid:3) = ( m − n ) ¯ L m + n + ¯ c m ( m − δ m + n, (cid:2) L m , ¯ L n (cid:3) = 0 (12)here L m and ¯ L m are the generators of holomorphic and antiholomorphic diffeo-morphisms. If ρ (∆ , ¯∆) denotes the degeneracy of states carrying L = ∆ and¯ L = ¯∆ eigenvalues, then one can define the partition function on the two-torusof modulus τ = τ + iτ as Z ( τ, ¯ τ ) = T r ( e πiτL e − πi ¯ τ ¯ L ) = X ∆ , ¯∆ (cid:18) ρ (∆ , ¯∆) e πiτ ∆ e − πi ¯ τ ¯∆ (cid:19) (13)Now using q = e πiτ , ¯ q = e − πi ¯ τ and inverting the above equation, we get ρ (∆ , ¯∆) = 1(2 πi ) Z q ∆+1 q ¯∆+1 Z ( q, ¯ q ) dqd ¯ q (14)where the integrals are along contours that enclose q = 0 and ¯ q = 0. Therefore,if we know the partition function Z ( q, ¯ q ), then we can use eq.(14) to determinethe density of states. Now modular invariance of the theory implies that Z ( τ, ¯ τ ) = T r (cid:20) e πiτ ( L − c ) e − πi ¯ τ (¯ L − ¯ c ) (cid:21) (15)6 is invariant under large τ −→ − /τ diffeomorphism. Cardy has shown thatthe invariance in eq.(15) is based on the general properties of two dimensionalconformal field theory and therefore expected to be universal. From eq.(13) and(15), we note that Z ( τ, ¯ τ ) = e πicτ e − πi ¯ c ¯ τ Z ( τ, ¯ τ )using the modular invariance of Z , we get Z ( τ, ¯ τ ) = e πic ( τ + τ ) e − πi ¯ c (¯ τ + τ ) Z (cid:18) − τ , − τ (cid:19) (16)substituting eq.(16) into eq.(14), we obtain ρ (∆ , ¯∆) = Z dτ d ¯ τ e − πi ∆ τ + πic ( τ + τ ) e πi ¯∆¯ τ − πi ¯ c (¯ τ + τ ) Z (cid:18) − τ , − τ (cid:19) (17)The above integral has the form I [ a, b ] = Z dτ e πiaτ + πibτ F ( τ ) (18)which can be evaluated by saddle point approximation. For this, we need toassume that F ( τ ) is slowly varying near the extremum of the phase. As shownin [32], this is indeed the case if one considers the situation where the imaginarypart of τ is large. Now the saddle point, obtained by extremizing the exponenton the right hand side of the above equation, is at τ = p b/a ≈ i p c/ I [ a, b ] ≈ Z dτ e πi √ ab + πibτ ( τ − τ ) F ( τ ) = (cid:18) − b a (cid:19) / e πi √ ab F ( τ ) (19)and an analogous expression exists for the ¯ τ integral. Finally, one obtains theexpression for the density of states as ρ (∆ , ¯∆) ≈ (cid:18) c (cid:19) (cid:18) ¯ c
96 ¯∆ (cid:19) e π √ c ∆6 e π q ¯ c ¯∆6 (20)Since, for most cases of our interest, we have only one Virasoro algebra instead oftwo (see the next section for details), the relevant expression for density of statesis ρ (∆) ≈ (cid:18) c (cid:19) e π √ c ∆6 (21)In the above expression, the exponential term is the standard Cardy formula.However an important part, which will play a significant role in our analysislater on, is the term that has a power law behavior. In the next section, we willuse the logarithm of ρ to calculate the entropy of the black holes. As one cananticipate, the exponential part of eq.(21) will give the usual Wald entropy and,on the other hand, the power term will provide a logarithmic correction to theblack hole entropy. 7 Black hole entropy and Virasoro algebra fromthe surface term of the gravitational action
There has been a lot of activity in understanding the black hole entropy usingthe symmetry based horizon CFT approach. This approach essentially assumesthat the symmetries of a black hole horizon are enough to compute the densityof states (hence the black hole entropy) at a given energy. Many avatars of thisapproach have been appeared in the literature, and all of them have successfullypredicted the black hole entropy expression . A complete list of references forthe later development can be found in [36].Applying a similar line of symmetry reasoning, a new approach was recentlyproposed in [33] which is straightforward and conceptually more clear. Impor-tantly, it does not require any ad hoc prescription such as shifting the zero-modeenergy and is also easy to implement. In this approach, the Noether currents as-sociated with the diffeomorphism invariance of the Gibbons-Hawking boundaryaction, instead of the bulk gravity action, are used to construct to the Virasoroalgebra at the horizon. Here, the diffeomorphisms are chosen in such a way thatthey leave the near-horizon structure of the metric invariant. The Virasoro al-gebra constructed in this way is again found to have a central extension, whichupon using the Cardy formula correctly reproduces the black hole entropy expres-sion. In this work, we will follow this boundary Noether current procedure of [33]to calculate the black hole entropy. We will first discuss its general formalismand then apply this formalism to calculate the black hole entropy of two gravitytheories, namely Einstein and Gauss-Bonnet gravity.We start with the Gibbons-Hawking surface term I GH : I GH = 18 πG D Z ∂ M d D − x √− γ L = 18 πG D Z M d D x √− g ∇ a ( n a L ) (22)where ∂ M is the boundary of the manifold M , L is related to the trace of theextrinsic curvature, n a is the unit normal vector to the boundary, g µν denotes thebulk metric and γ µν is the induced metric on the boundary. The expression for L depends on the gravity theory under consideration. For example, in Einsteingravity it is equal to trace of the extrinsic curvature K = −∇ µ n µ . The conservedcurrent J µ associated with the diffeomorphism invariance x µ −→ x µ + ξ µ of I GH can be obtained by considering the variation of the both sides of eq.(22) as theLie derivative. After a bit of algebra, one gets J µ [ ξ ] = ∇ ν J µν [ ξ ] = 18 πG D ∇ ν (cid:18) L ξ µ n ν − L ξ ν n µ (cid:19) (23)The charge corresponding to this conserved current is defined as Q [ ξ ] = Z ∂Σ d Σ µν (cid:0) √− hJ µν (cid:1) (24) A certain level of arbitrariness is present in the procedure of all symmetry based approachesin order to produce correct black hole entropy expression. We will discuss more about thesearbitrariness in the following. J µν is called the Noether potential and d Σ µν = − d D − x ( n µ m ν − m µ n ν ) isthe surface element of the ( D − ∂Σ with metric h µν . Inorder to discuss black hole physics, we will chose the surface ∂Σ to be near thehorizon. The unit vectors n µ and m µ are chosen to be spacelike and timelike,respectively.Following [33], we define the Lie bracket of the charges as (cid:2) Q [ ξ ] , Q [ ξ ] (cid:3) ≡ (cid:18) δ ξ Q [ ξ ] − δ ξ Q [ ξ ] (cid:19) = Z ∂Σ d Σ µν √− h (cid:0) ξ µ J ν [ ξ ] − ξ µ J ν [ ξ ] (cid:1) (25)Form the above equation, it is clear that we only need to know the vector field ξ µ to determine the charge algebra. This can be done, as explained below, by choos-ing a appropriate diffeomorphism which leaves the horizon structure invariant.However in order to proceed further, let us first write the D (= d + 1) dimensionalform of the metric as ds = − f ( r ) N dt + dr f ( r ) + r d Ω ij ( x ) dx i dx j (26)where d Ω ij ( x ) dx i dx j is the line element of the ( d − N is included in g tt to allow us to adjust the normalization of the time coordinate. This will beuseful later on, but will not play any significant role in the black hole entropycalculation here. To study the near horizon structure, defined by f ( r h ) = 0, itis convenient to choose a coordinate r = ˜ r + r h , in which case the above metricreduces to ds = − f (˜ r + r h ) N dt + d ˜ r f (˜ r + r h ) + (˜ r + r h ) d Ω ij ( x ) dx i dx j (27)In the near horizon region i.e ˜ r → f (˜ r + r h ) can be expandedas f (˜ r + r h ) = ˜ rf ′ ( r h ) + 1 / r f ′′ ( r h ) + ... Defining the surface gravity κ as κ = N f ′ ( r h ) / f (˜ r + r h ), one notice that the ( t − ˜ r ) part of the metric in eq.(27)reduces to the standard Rindler metric ds t − ˜ r = − r κN d ˜ t + 12˜ r Nκ d ˜ r where ˜ t = N t . The unit normal vectors can be chosen as n µ = (0 , p f (˜ r + r h ) , , ..., , m µ = (1 /N p f (˜ r + r h ) , , , ...,
0) (28)Further, in order to find ξ µ , it is more useful to first transform to Bondi likecoordinates by the transformation du = d ˜ t − d ˜ rf (˜ r + r h ) (29)9nder which the metric in eq.(27) reduces to ds = − f (˜ r + r h ) du − dud ˜ r + (˜ r + r h ) d Ω ij ( x ) dx i dx j (30)Now we choose our vector field ξ µ by imposing the condition that the horizonstructure remain invariant under diffeomorphism i.e the metric coefficients g ˜ r ˜ r and g u ˜ r remain unchanged. This implies the following Killing equations , L ξ g ˜ r ˜ r = − ∂ ˜ r ξ u = 0 L ξ g u ˜ r = − f (˜ r + r h ) ∂ ˜ r ξ u − ∂ u ξ u − ∂ ˜ r ξ ˜ r = 0 (31)Solving the above two equations, we get ξ u = S ( u, ~x ) , ξ ˜ r = − ˜ r∂ u S ( u, ~x ) (32)where S is an arbitrary function and ~x are the coordinates on the remaining( d −
1) dimensional space. Now converting back to the ( t, ˜ r ) coordinates, thesevector fields take the form ξ t = T − ˜ rN f (˜ r + r h ) ∂ t T, ξ ˜ r = − ˜ rN ∂ t T (33)where T ( t, ˜ r, ~x ) ≡ S ( u, ~x ). Therefore, we can evaluate the expressions for ξ µ , Q [ ξ µ ] and the charge algebra (cid:2) Q [ ξ ] , Q [ ξ ] (cid:3) once the function T is given. Now,we expand this function T in terms of a set of basis functions T m , as T = X m A m T m , A ∗ m = A − m (34)As it is standard in the literature, we choose T m such that the resulting ξ µm obeysthe algebra isomorphic to Diff S i.e i { ξ m , ξ n } µ = ( m − n ) ξ µm + n (35)where { , } is the Lie bracket. A particular choice is T m = 1 α e im ( αt + g (˜ r )+ ~p.~x ) (36)where α is an arbitrary parameter, g (˜ r ) is a function which is regular at thehorizon and p i are integers. We now have all the ingredients at our disposal tocalculate the leading Wald as well as the subleading logarithmic correction to theblack hole entropy. From here on we will concentrate on horizons with hyperbolictopology, as this will be required in the computation of holographic R´enyi entropyin the later section. It is easy to check that the other condition L ξ g uu = 0 is trivially satisfied near the horizon. .1 Entropy of AdS-Schwarzschild Black hole We first apply the above developed formalism to a ( d + 1)-dimensional AdS-Schwarzschild black hole in Einstein gravity. As mentioned earlier, for Einsteingravity the Gibbons-Hawking surface term is standard and is given by the ex-pression L = K = −∇ µ n µ . The metric is given as ds = − f ( r ) N dt + 1 f ( r ) dr + r d Σ d − ,f ( r ) = − − mr d − + r L (37)where d Σ d − = dθ + sinh θ dΩ d − (38)is the metric on ( d − dΩ d − being the lineelement on a unit ( d − N is not requiredhere and will be specified in the next section. Substituting eq.(37) into eqs.(33),(23), (24) and (25) and taking the near horizon limit ˜ r →
0, we get Q [ ξ m ] = 18 πG D Z H d d − x √ h (cid:18) κT m − ∂ t T m (cid:19) (39)Similarly, the algebra of the charges corresponding to T = T m , is given by (cid:2) Q [ ξ m ] , Q [ ξ n ] (cid:3) = 18 πG D Z H d d − x √ h (cid:20) κ (cid:18) T m ∂ t T n − T n ∂ t T m (cid:19) − (cid:18) T m ∂ t T n − T n ∂ t T m (cid:19) + 14 κ (cid:18) ∂ t T m ∂ t T n − ∂ t T n ∂ t T m (cid:19)(cid:21) (40)Now substituting eqs.(36) and (37) into eqs.(39) and (40), we get the final ex-pressions as Q [ ξ m ] = A d − πG D κα δ m, (cid:2) Q [ ξ m ] , Q [ ξ n ] (cid:3) = − i ( m − n ) A d − πG D κα δ m + n, − im A d − πG D ακ δ m + n, (41)where A d − is the area of ( d − K (cid:2) ξ m , ξ n (cid:3) = (cid:2) Q [ ξ m ] , Q [ ξ n ] (cid:3) + i ( m − n ) Q [ ξ m + n ]= − im A d − πG D ακ δ m + n, (42)11rom which we can read off the zero mode Q and central charge c as Q = Q [ ξ ] = A d − πG D κα , c
12 = A d − πG D ακ (43)Substituting above expressions into eq.(21) with Q = ∆ and taking log on bothside, we obtain the black hole entropy S = ln ρ ≈ π r c ∆6 + 14 ln c ∆ + ...S = A d − G D −
32 ln A d − G D + ... = S W ald −
32 ln S W ald + ... (44)We see that the leading term matches exactly with the usual black hole entropyexpression. We also find a logarithmic correction to it. Interestingly, the coeffi-cient − / α to be such that the central charge c is auniversal constant i.e independent of the area A d − , as in [32]. We will say moreabout this condition and the coefficient of this logarithmic term at the end of thissection. Now we move on to discuss the entropy of Gauss-Bonnet black holes. The surfaceterm of the gravitational action can be found in [37] [38] and is given as L = −∇ µ n µ + 2 λL ( D − D − (cid:0) P − G µν K µν (cid:1) (45)where λ is the coefficient of the Gauss-Bonnet term, P is the trace of the followingtensor P µν = 13 (cid:0) KK µρ K ρν + K ρσ K ρσ K µν − K µρ K ρσ K σν − K K µν (cid:1) (46)and ˜ G µν stands for Einstein tensor of d -dimensional boundary metric. For aGauss-Bonnet black hole, the D -dimensional metric is given by ds = − f ( r ) N dt + 1 f ( r ) dr + r d Σ d − ,f ( r ) = − r λL (cid:20) − r − λ + 4 λmr D − (cid:21) (47)where m is related to the mass of the black hole. Performing the analogous stepsas in the previous subsection to calculate the charge and Virasoro algebra forGauss-Bonnet black hole, we get Q [ ξ m ] = 18 πG D (cid:18) − D − λL ( D − r h (cid:19) Z H d d − x √ h (cid:18) κT m − ∂ t T m (cid:19) (48)12 Q [ ξ m ] , Q [ ξ n ] (cid:3) = 18 πG D (cid:18) − D − λL ( D − r h (cid:19) Z H d d − x √ h (cid:20) κ (cid:18) T m ∂ t T n − T n ∂ t T m (cid:19) − (cid:18) T m ∂ t T n − T n ∂ t T m (cid:19) + 14 κ (cid:18) ∂ t T m ∂ t T n − ∂ t T n ∂ t T m (cid:19)(cid:21) (49)Now substituting eqs.(36) and (47) into eqs.(48) and (49), we obtain Q [ ξ m ] = A d − πG D (cid:18) − D − λL ( D − r h (cid:19) κα δ m, (cid:2) Q [ ξ m ] , Q [ ξ n ] (cid:3) = − i ( m − n ) A d − πG D κα (cid:18) − D − λL ( D − r h (cid:19) δ m + n, − im A d − πG D ακ (cid:18) − D − λL ( D − r h (cid:19) δ m + n, (50)from which, we obtain the central term K (cid:2) ξ m , ξ n (cid:3) = (cid:2) Q [ ξ m ] , Q [ ξ n ] (cid:3) + i ( m − n ) Q [ ξ m + n ]= − im A d − πG D ακ (cid:18) − D − λL ( D − r h (cid:19) δ m + n, (51)and we can read off the zero mode Q and central charge c as Q = A d − πG D κα (cid:18) − D − λL ( D − r h (cid:19) , c
12 = A d − πG D ακ (cid:18) − D − λL ( D − r h (cid:19) (52)Substituting above expressions into eq.(21) and taking a logarithm, we obtainthe black hole entropy S = ln ρ ≈ π r c ∆6 + 14 ln c ∆ + ...S = A d − G D (cid:18) − D − λL ( D − r h (cid:19) −
32 ln (cid:20) A d − G D (cid:18) − D − λL ( D − r h (cid:19)(cid:21) + ... = S W ald −
32 ln S W ald + ... (53)In the above equation, the first term is exactly the Wald entropy of a Gauss-Bonnet black hole in D -dimensions. We also find the logarithmic correction to theWald entropy with coefficient − /
2, which is same as in the AdS-Schwarzschildblack hole case. As one can see, however now in terms of horizon radius, thereare two correction terms.Before ending this section, it is worthwhile to point out again the debatablenature of the coefficient of logarithmic term in black hole entropy. The loga-rithmic correction to the usual black hole entropy has been discussed in variousquantum gravity models. This correction term has been successfully computedfor asymptotically flat black holes in loop quantum gravity (LQG), in string the-ory for both extremal and near extremal black holes by microscopic counting13ethods or by diffeomorphism symmetry arguments which heavily rely on twodimensional CFT and the Cardy formula (as we have done in this section). All ofthese methods have either predicted different coefficient for the logarithmic termor have a certain level of arbitrariness in their results. For example, in LQG thecoefficient was found to be − / − / − / c isuniversal constant in a sense that it is independent of horizon area. To achievethis assumption Carlip chose the parameter α (appearing in eq.(36)) in such away that c becomes independent of area. It is important to point out here thatone could choose α to be the surface gravity κ as well. The condition α = κ is also well motivated from Euclidean gravity point of view and is standard inthe literature . To see this, let us consider the Euclidean time ( τ → it ) andtake the appropriate ansatz for T m as T m = 1 /αe im ( ατ + g (˜ r )+ ~p.~x ) . In the Euclideanformalism our analysis still go through, but now τ must have a periodicity of2 π/κ to avoid the conical singularity. In order to maintain this periodicity in τ ,we must choose α = κ . If we choose this condition then its not hard to see thatthe coefficient of logarithmic correction is − / − / C , instead of worrying too muchabout its exact magnitude S = S W ald + C ln S W ald + ... (54)The whole purpose of our previous exercise, where we computed C = − /
2, wasto show at-least one method by which logarithmic correction to the entropy oftopological black holes, not just in Einstein gravity but in other higher derivativegravity too, can be explicitly calculated and that this coefficient is non-zero. Itwould certainly be interesting to generalize the method of [13] in LQG or of [16]in string theory to calculate logarithmic correction for topological black holes.Especially, for higher derivative gravity theories it is not clear (at-least to theauthor) how to proceed. It would be an interesting problem in its own rightto analyze similarities and differences in the results of these methods but it isbeyond the scope of this paper. In various methods of computing black hole entropy from diffeomorphism symmetry argu-ments, the parameter α is generally chosen to be the surface gravity by hand to get the correctexpression for black hole entropy, see for instance [29] [36] [39]. Calculations and Results
In this section, we will present results for the holographic R´enyi entropy byconsidering logarithmic correction to the usual black hole entropy. We will againconcentrate on two gravity theories: Einstein and Gauss-Bonnet. Our main focushere will be to see the effects of this logarithmic correction on the R´enyi entropy.However, before going into the details of each case separately, let us first calculatethe area of the hyperbolic event horizon appearing in eqs.(44) and (53). The lineelement on the ( d − d Σ d − = dθ + sinh θ dΩ d − (55)It would be convenient if we make a change of coordinate θ = cosh − y , in whichcase the above metric reduces to d Σ d − = dy y − y − dΩ d − (56)The area of the hyperbolic space (and hence the area of the event horizon) is di-vergent. In order to regulate this area, we introduce a upper cutoff by integratingout to a maximum radius y max = Rδ (57)where δ is the short distance cutoff related to the UV cutoff of the boundaryCFT. Now, the area V Σ d − of the hyperbolic space is calculated as V Σ d − = Ω d − Z y max dy ( y − ( d − / ≃ Ω d − d − (cid:20) ( y d − max − − ( d − d − d −
4) ( y d − max − d − d − d − d −
6) ( y d − max − − ... (cid:21) ≃ Ω d − d − (cid:20) R d − δ d − − ( d − d − d − R d − δ d − + ( d − d − d − d − R d − δ d − − ... (cid:21) (58)where Ω d − = 2 π ( d − / /Γ (( d − /
2) is the area of a unit ( d −
2) sphere. However,for d = 2, we have a logarithmic behaviour V Σ = 2 Z y max dy ( y − − / = 2 ln (2 y max ) = 2 ln (cid:18) Rδ (cid:19) (59)Therefore, the area of the event horizon A d − is given by A d − = r d − h V Σ d − (60)Now we have all the ingredients to compute the holographic R´enyi entropy forspherical entangling surface. 15 .1 R´enyi entropy from AdS-Schwarzschild Black hole For AdS-Schwarzschild Black hole the metric is given in eq.(37). However, inorder to make comparison with [10], let us rewrite this in a slightly different form ds = − [ − r L g ( r )] N dt + 1[ − r L g ( r )] dr + r d Σ d − ,g ( r ) = L r ( f ( r ) + 1) (61)In order to ensure that the boundary spacetime is conformally equivalent to R × H d − , i.e ds ∞ = − dt + R d Σ d − (62)we choose the constant N = L / ( g ∞ R ) = ˜ L /R , where g ∞ = lim r →∞ g = 1.The Hawking temperature of this black hole is given by T = 14 πR (cid:18) dr h L − ( d − Lr h (cid:19) (63)which is also the temperature of the boundary CFT on R × H d − . For computa-tional purposes, it is convenient to consider a coordinate x = r h / ˜ L in which casethe expression for the R´enyi entropy in eq.(11) reduces to S q = qq − T Z x q dx S therm ( x ) dTdx = qq − T (cid:20) S therm ( x ) T ( x ) | x q − Z x q dx T ( x ) ddx S therm (cid:21) (64)Here, in the above integral, the upper limit x = 1 comes from the condition T = T which implies r h = ˜ L . The lower limit x q , which needs to be determined,corresponds to the temperature T = T /q . Using eq.(63), x q must satisfy thefollowing equation dx q − q x q − ( d −
2) = 0 (65)the real and positive root of which is given by x q = 1 qd (cid:20) p d q + 1 − dq (cid:21) (66)Finally, using the black hole entropy expression (eq.(54)) in place of S therm ineq.(64), we get the expression for R´enyi entropy as S q = q B q −
1) (2 − x d − q (1 + x q )) + C qq − B + C q q − x q (cid:20) ( d − x q − dx q + 2 − d ) − ( dx q + 2 − d ) ln( B x d − q ) (cid:21) (67)16here B = 2 πV Σ d − ( ˜ Ll p ) d − , l p is the Plank length related to the gravitation con-stant G D = l d − p / π . We see that S q is a complicated function of x q . For C = 0, S q reduces to the expression found in [10]. However, for C = 0 there are addi-tional nontrivial correction terms. We now make some observations: • For d = 2, we get S q ( d = 2) = B (cid:18) q (cid:19) + C (cid:18) ln( B ) − (cid:19) + C ln qq − c (cid:18) q (cid:19) ln (cid:18) Rδ (cid:19) + C (cid:20) ln (cid:18) c (cid:18) Rδ (cid:19)(cid:19) − (cid:21) + C ln qq − c = 12 πL/l p is the standard expression of the central charge in the twodimensional boundary CFT . The first term in eq.(68) matches with the wellknown result of the R´enyi entropy in a two dimensional CFT for an interval oflength l = 2 R . This is expected as, for d = 2, the spherical entangling regionconsists of two points separated by distance 2 R . However, our main result hereis the appearance of additional corrections to R´enyi entropy which are preciselycoming from the coefficient of the logarithmic correction to black hole entropy.These correction terms to the R´enyi entropy have both additional logarithmic aswell as double logarithmic structure. • Similarly, the entanglement entropy for d = 2 is obtained by taking q → S ( d = 2) = c (cid:18) Rδ (cid:19) + C (cid:20) ln (cid:18) c (cid:18) Rδ (cid:19)(cid:19)(cid:21) (69)The leading term again has the same structure as is well known for the twodimensional CFT. However, now we have found additional correction too. Inter-estingly, the correction term in the entanglement entropy has a simple structure.It is proportional to C times the logarithm of its standard expression. Moreover,the same is true in all dimensions. That is, S ( d ) = B + C ln[ B ] = 2 πV Σ d − ( ˜ Ll p ) d − + C ln[2 πV Σ d − ( ˜ Ll p ) d − ]= 2 πV Σ d − ( ˜ Ll p ) d − + C ln[( ˜ Ll p ) d − ] + C ln[2 πV Σ d − ] (70)In the last line, we have divided the correction term into two parts. One thatdepends on ( ˜ L/l p ) d − , which is related to the number of degrees of freedom ofthe boundary CFT and the other that depends on the size of the system. Let This c should not be confused with the central charge of eq.(43) which appears in theVirasoro algebra on the horizon. The c in eq.(43) was used to compute the black hole entropyand has nothing to do with the boundary CFT.
17s concentrate on the term which is of order G D (last term in eq.(70)) . Weobserve that at this order there is a logarithmic correction in all dimensions. Inparticular, there is a logarithmic correction in odd d as well. In order to seethis, we write the complete expression for entanglement entropy in d = 3 and 4dimensions S ( d = 3) = (cid:18) π ˜ Ll p (cid:19) (cid:18) Rδ − (cid:19) + C ln (cid:18) ˜ Ll p (cid:19) + C ln (cid:18) π Rδ (cid:19) + ...S ( d = 4) = 4 π (cid:18) ˜ Ll p (cid:19) (cid:18) R δ − ln 2 Rδ (cid:19) + C ln (cid:18) ˜ Ll p (cid:19) + C ln (cid:18) π R δ (cid:19) + ... (71)we note from the last terms of eq.(71) that at the order G D there is a correctionto the entanglement entropy which depends logarithmically on the size of theentangling surface. The same result holds in all d > • For d = 2, the ratio of R´enyi entropy to entanglement entropy as a functionof q for some reasonable values of C is plotted in fig.(1). In order to plot thisratio, we have chosen R = 1, δ = 10 − and L = 2 l p . We see that the overallbehaviour of S q /S is same for all C . For a fixed q >
1, the magnitude of theratio S q /S decreases with decrease in C . However, we should emphasize herethat these results are cutoff ( δ ) dependent. For instance, for much smaller cutoff,say δ = 10 − , differences due to C = 0 are almost negligible. Similarly, we alsofound that for larger and larger values of c (or L/l p ), the effect of C becomesmaller and smaller. We can also note that S q /S > q < S q /S < q > • Let us also note some useful limits of the R´enyi entropy S ∞ = B (cid:20) − d − d (cid:18) d − d (cid:19) d − (cid:21) + C (cid:20) ln B − ( d − + ( d − p d ( d − (cid:21) S = B + C ln B S = B (cid:20)(cid:18) d (cid:19) d q d − + (cid:18) d (cid:19) d − q d − (cid:21) + C (cid:20) ln B − ( d − − ( d −
1) ln( dq ) (cid:21) (72) • Now, we discuss R´enyi and entanglement entropies for higher dimensional the-ories. In fig.(2), S q /S as a function of q for various values of d is shown. Here,we have considered C = − / R = 1, δ = 10 − and L = 2 l p . The overall characteristic features of S q /S is found to be same as in the d = 2 case. We found that, for fixed C , as we increase the number of spacetimedimensions the magnitude of the ratio S q /S increases. Even for large d , we againfound that S q /S > q < S q /S < q > Quantum corrections to the RT formula are expected to occur at the order G D [40]. TheRT formula gives entanglement entropy at the order 1 /G D . See the discussion section for moredetails on this. S q (cid:144) S Figure 1: S q /S as a function of q forvarious C . Here d = 2 and red, green,blue and brown curves correspond to C =0, − / − / − S q (cid:144) S Figure 2: S q /S as a function of q for vari-ous d . Here C = − / d = 3, 4, 5and 6 respectively. • Let us also comment on some of the useful inequalities of the R´enyi entropy.It is well known that R´enyi entropy satisfies inequalities involving the derivativewith respect q , such as ∂ q S q ≤ ∂ q [ q − q S q ] ≥ ∂ q [( q − S q ] ≥ ∂ q [( q − S q ] ≤ d >
2, we find that these inequalities are indeed satisfiedprovided C is not very large (which is expected for small corrections). Howeverfor d = 2, we find few subtleties, especially in the fourth inequality of eq.(73).We find that for small central charge it can be violated. Although for large cen-tral charge, which is the case for boundary theories that have a gravity dual, thefourth inequality is again found to be satisfied. • Perhaps, the most important result form our computation is that the leadingUV divergences appearing in eq.(67) is same for all q . Indeed, if we note thatleading term R d − Ω d − in eq.(58) gives the area of the spherical entangling re-gion, then the leading divergent term always has a “area law” structure for d > d = 2, we have the usual logarithmic divergence in the R´enyi entropy. Asexpected, the correction terms do not change the leading behavior of the R´enyientropy. 19 .2 R´enyi entropy from Gauss-Bonnet Black hole In this subsection, we examine the holographic R´enyi entropy for boundary theorywhich are dual to Gauss-Bonnet gravity theory. The procedure for calculatingR´enyi entropy with Gauss-Bonnet Black hole background is entirely similar towhat has been discussed in the previous subsection, and therefore we will be briefhere.The metric for Gauss-Bonnet Black hole is given in eq.(47), however we rewriteit in a slightly different form in order to make comparison with [10] ds = − [ − r L g ( r )] N dt + 1[ − r L g ( r )] dr + r d Σ d − ,g ( r ) = 12 λ (cid:20) − r − λ + 4 λmr d (cid:21) (74)In this case N = L / ( g ∞ R ) = ˜ L /R . Since now g ∞ = (1 − √ − λ ) / (2 λ ) = 1,it implies ˜ L = L . This is just the statement that the AdS curvature scale is dis-tinct from the length scale L which usually appears in the Gauss-Bonnet gravityaction. Now, again using the coordinate x = r h / ˜ L , the Hawking temperature canbe expressed as T = 12 πRx (cid:18) d g ∞ x − x g ∞ + λg ∞ x − λg ∞ (cid:19) (75)Similarly, the horizon entropy (eq.(53)) can be recast as S = 2 π (cid:18) ˜ Ll p (cid:19) d − V Σ d − x d − (cid:18) − λg ∞ x d − d − (cid:19) + C ln (cid:20) π (cid:18) ˜ Ll p (cid:19) d − V Σ d − x d − × (cid:18) − λg ∞ x d − d − (cid:19)(cid:21) (76)After substituting above results into eq.(64), we can get expression for the R´enyientropy for any d . However, for general d the expressions for R´enyi entropy isvery complicated and lengthy and also not very illuminating. For this reason wefocus on d = 4 case, in which the R´enyi entropy has the following expression S q = q B q − (cid:20) − x q g ∞ − − x q ) + 4(1 − λ ) (cid:18) − λg ∞ − x q x q − λg ∞ (cid:19)(cid:21) + C q ( g ∞ − q − g ∞ (2 λg ∞ −
1) ln[ B (1 − λg ∞ )] − C qx q ( g ∞ − x q )( q − g ∞ (2 λg ∞ − x q ) ln[ B x q ( x q − λg ∞ )]+ 6 C q ( q −
1) ( x q − g ∞ + √ C q (12 λ − q − √ λg ∞ (cid:20) tanh − (cid:18) √ λg ∞ (cid:19) − tanh − (cid:18) x q √ λg ∞ (cid:19)(cid:21) (77)where x q corresponds to the positive and real root of the following equation4 qx q − g ∞ x q − qg ∞ x q + 4 λg ∞ x q = 0 (78)20hich is again obtained from the equation T = T /q , i.e. for the lower limitof the integral in eq.(64). The λ which appears in the argument of logarithmicand inverse hyperbolic functions in eq.(77) should be understood as an absolutevalue. The first term in eq.(77) have the same expression for R´enyi entropy aswas found in [10]. However now we also have additional correction terms, whichare both logarithmic as well as non-logarithmic in nature. One can also explicitlycheck that, in the limit λ →
0, eq.(77) reduces to eq.(67) for Einstein gravity.We now make some observations: • It is well known that the dual four dimensional boundary CFT of five dimen-sional Gauss-Bonnet gravity theory has two distinct central charges c = π (cid:18) ˜ Ll p (cid:19) (1 − λg ∞ ) , a = π (cid:18) ˜ Ll p (cid:19) (1 − λg ∞ ) (79)In terms of these central charges the R´enyi entropy in eq.(77) reduces to S q = V Σ π q (1 − x q ) q − (cid:20) (5 c − a ) x q − (13 c − a ) + 16 c cx q − ( c − a )(3 c − a ) x q − ( c − a ) (cid:21) + C q ( q −
1) ln (cid:20) aV Σ π (cid:21) − C qx q ( q −
1) ((5 c − a ) x q − c + a )((3 c − a ) x q − c + a ) ln (cid:20) V Σ x q π (cid:18) (3 c − a ) x q − c + 3 a (cid:19)(cid:21) + 2 C q ( q −
1) (3 c + a − ac )(3 c − a ) r c − a ) c − a (cid:20) tanh − s c − a c − a ) − tanh − s c − a c − a ) x q (cid:21) + C q ( x q − q −
1) (15 c − a )(3 c − a ) (80)which shows that the R´enyi entropy is quite a complicated function of these cen-tral charges. It is also clear that the R´enyi entropy is not determined solely bythe anomaly coefficient a as in the case of entanglement entropy (see below). Asin the case of Einstein gravity, here too, the size of the entangling surface alwaysappears logarithmic in the correction terms. This feature therefore seems to bea universal in nature. • We get the expression for the entanglement entropy as, S = a V Σ π + C ln (cid:20) a V Σ π (cid:21) (81)we see that, as in the Einstein gravity case, correction term to the entanglemententropy is still given by the logarithmic of its original expression. This result canbe traced back to eq.(11), where it is clear that entanglement entropy is nothingbut the black hole entropy at temperature T = T . Therefore, logarithmic correc-tion to the black hole entropy implies logarithmic correction to the entanglemententropy. It is also easy to see that similar results hold in higher dimensions too. Again, these central charges should not be confused with the central charge of eq.(43). S q /S as a function of q forvarious C . Here λ = 0 .
08 and red, green,blue and brown curves correspond to C =0, − / − / − R = 1, δ = 10 − and L = 2 l p . Figure 4: S q /S as a function of q for var-ious λ . Here C = − / λ = − . − . − .
05, 0 .
05 and 0 .
09 re-spectively. For this plot, we have chosen R = 1, δ = 10 − and L = 2 l p . • S q /S as a function of q for some reasonable values of C and for λ = 0 .
08 isshown in fig.(3). We see that difference due to C is extremely small and that theoverall behaviour of S q /S is same for all C . Similarly, S q /S for different valuesof λ and for fixed C = − / S q /S , in the region q <
1, increases for higher and higher values of λ , however in the region q >
1, itdecreases. Similar results holds for other values of C as well. • Before ending this section, let us also note some other useful limits of the R´enyientropy, S = V Σ π (3 c − a ) (5 c − a ) (cid:18) q + 1 q (cid:19) + V Σ πq (3 c − a )(5 a − a c + 159 ac − c )(5 c − a ) − C ln q + h ( c, a, V Σ ) (82) S ∞ = V Σ π (10 ac − c − a )(5 c − a ) + C ln (cid:18) a V Σ π (cid:19) + 3 C (5 c − a )(3 c − a ) (cid:18)r c − a c − a − (cid:19) +2 C (3 c + a − ac )(3 c − a ) r c − a ) c − a (cid:20) tanh − s (3 c − a )3( c − a ) − coth − s − c (3 c − a ) (cid:21) (83)where h is some function of c , a and V Σ but independent of q . The leadingdiverging term of S matches exactly with [10].22 Discussions and Conclusions
In this paper, we have studied the effects of logarithmic correction of the blackhole entropy on the R´enyi entropy of a spherical entangling surface. We first usedthe diffeomorphism symmetry argument at the horizon to compute the black holeentropy expression. We used the Noether currents associated with the diffeo-morphism invariance of the Gibbons-Hawking boundary action to construct theVirasoro algebra at the hyperbolic event horizons and then used this algebra tocalculate the entropy of AdS-Schwarzschild and Gauss-Bonnet black holes. Wefound that the leading term in the black hole entropy expression is the usualWald entropy and that there is correction to it. This correction was found to beproportional to the logarithm of horizon area.We then applied the prescription of [8] [10] to calculate the holographic R´enyientropy for a spherical entangling surface. Using the corrected black hole entropyexpression we found that there are corrections to the standard expression of R´enyientropy. These correction are shown in eq.(67) for Einstein and in eq.(80) forGauss-Bonnet gravity duals. In particular, we found that the R´enyi entropy is acomplicated function of the index q as well as the central charges. Interestingly,the size of the entangling surface always appears logarithmically in the correctionterms of the R´enyi entropy. This is true for both Einstein as well as Gauss-Bonnetgravity. We found that the inequalities of R´enyi entropy are also satisfied evenwith correction terms.It is important to analyze the nature and significance of these correction termsin the R´enyi entropy. If the corrections in the R´enyi entropy originating from thecorrections in black hole entropy are quantum corrections, then our results canbe useful in many directions. Especially since a lot of work have recently beenappeared in the literature to compute leading quantum corrections (of order G D or N ) to the entanglement entropy holographically. In this context, a proposalfor quantum corrections to holographic entanglement entropy is given in [40],also see [41]. In this proposal, the quantum corrections are essentially given bythe bulk entanglement entropy between RT minimal area surface and the restof the bulk ( see fig.(1) of [40]). As an example, the quantum correction to theentanglement entropy of the Klebanov-Strassler model in the large N limit wascalculated. The correction was found to be, as in our case, logarithmic in natureand it depends on the size of the entangling surface. Indeed, one can also noticefrom eq.(70) that at the order G D (recall that the factor ( ˜ L/l p ) d − measures thenumber of degrees of freedom of the dual CFT and it is related to the centralcharge of the boundary CFT or to the G D on the gravity side) the correctionto the entanglement entropy is proportional to the logarithm of the size of theentangling surface. If the above interpretation is correct, then our results canbe useful since they predict a similar kind of logarithmic correction to the R´enyientropy.An indirect hint for the logarithmic correction in the R´enyi entropy can alsobe seen in the following way. In [17], a non-trivial test of the gauge/gravityduality at next-to-leading order in the 1 /N expansion for ABJM theories wasperformed. There, it was shown that the subleading logarithmic correction in thepartition function (more correctly in log Z ) of the ABJM theory on a three sphere23atches exactly with the partition function of its eleven dimensional supergravitydual. The latter partition function at one-loop level was calculated using theEuclidean quantum gravity method. The subleading logarithmic term in thepartition function can lead to a logarithmic correction in the R´enyi entropy,provided that, an analogous logarithmic term in the partition function on q -folded cover does not cancel in the definition of the R´enyi entropy. This scenariois partially true, at-least in odd dimensions.Indeed one can notice, an important result of our analysis is that there arelogarithmic corrections to entanglement and R´enyi entropies in odd dimensionstoo. At first sight this seems strange as a logarithmic term in these entropiesgenerally appears in even dimensions. However, it is also well known that theentanglement entropy for the sphere in flat space in odd dimensions is simplythe negative of free energy on sphere i.e S = log Z [44] [45], which implies thatlogarithmic corrections in the free energy directly lead to logarithmic correctionsin the entanglement entropy. As we have mentioned above, there are exampleswhere logarithmic correction to the free energy indeed occur, therefore, it is notsurprising that we got logarithmic correction in the entanglement entropy in odddimensions as well.Since a similar kind of logarithmic subleading term in the partition functionalso arises in other boundary field theories, see for example [42], therefore itappears that logarithmic correction to the R´enyi entropy might be a generalfeature of CFTs with gravity duals. Although, in order to explicitly establishthis fact it would be useful if we can calculate the R´enyi entropy at one loop levelon the lines of [43] . It would certainly be interesting to explicitly compare theresults of our calculations with those of [40] [43], and find the similarities anddifferences between them.Finally, regardless of the problems associated with the interpretation of cor-rection terms calculated in this paper, it is important to carefully examine andexplore their physical significance. Especially since in the context of AdS/CFT,it is well known that on the gravity side the logarithmic correction to black holeentropy arises only due to one loop contribution of the massless fields. Now,the mapping from the R´enyi entropy to the black hole entropy is exact and isexpected to be valid at all orders. Correspondingly, the logarithmic correction toblack hole entropy must have some meaning in the R´enyi entropy too. Here, wehave taken a small step in this direction and in the process have obtained severalnew results. We believe this mapping can be further useful, not just to betterunderstand the structure of holographic R´enyi entropy but also to get a betterunderstanding of the coefficient of the logarithmic correction in the gravity side.We hope to comment on this issue soon. Acknowledgements