Shape Dependence of Holographic Rényi Entropy in General Dimensions

Shape Dependence of Holographic Rényi Entropy in General Dimensions

Lorenzo Bianchi, b    Shira Chapman, c    Xi Dong, b,d    Damián A. Galante, b,e    Marco Meineri b    and Robert C. Myers Institut für Theoretische Physik, Universität Hamburg
Luruper Chaussee 149, 22761 Hamburg, GermanyPerimeter Institute for Theoretical Physics
31 Caroline Street North, ON N2L 2Y5, CanadaSchool of Natural Sciences, Institute for Advanced Study
1 Einstein Drive, Princeton, New Jersey 08540, USADepartment of Applied Mathematics, University of Western Ontario,
London, Ontario N6A 5B7, CanadaScuola Normale Superiore, Piazza dei Cavalieri 7 I-56126 Pisa, Italy
and Istituto Nazionale di Fisica Nucleare - sezione di Pisa
Abstract

We present a holographic method for computing the response of Rényi entropies in conformal field theories to small shape deformations around a flat (or spherical) entangling surface. Our strategy employs the stress tensor one-point function in a deformed hyperboloid background and relates it to the coefficient in the two-point function of the displacement operator. We obtain explicit numerical results for spacetime dimensions, and also evaluate analytically the limits where the Rényi index approaches 1 and 0 in general dimensions. We use our results to extend the work of 1602.08493 and disprove a set of conjectures in the literature regarding the relations between the Rényi shape dependence and the conformal weight of the twist operator. We also extend our analysis beyond leading order in derivatives in the bulk theory by studying Gauss-Bonnet gravity.

Keywords:
Rényi entropy, shape deformations, displacement operator, conformal defect

1 Introduction and Summary

Entanglement is one of the key features which distinguishes quantum physics from the classical realm and it is widely recognized as an essential ingredient in shaping many of the physical properties of complex interacting quantum systems. In particular, there is an increasing realization of the important role which entanglement plays in quantum field theory (QFT) Calabrese:2004eu ; Calabrese:2005in ; Calabrese:2005zw and quantum gravity qg1 ; qg2 ; qg3 ; qg4 . While there are a variety of measures of entanglement qibook , two which have received particular attention in the latter fields are entanglement and Rényi entropies. For example, typical calculations begin with some QFT in a (global) state described by the density matrix on a given time slice. Then one restricts the state to a particular region by tracing over the degrees of freedom in the complementary region to produce:

 ρ\tiny A=trB(ρ). (1.1)

The above entanglement measures are constructed from the reduced density matrix as

 S\tiny EE = −Tr(ρ\tiny Alogρ\tiny A), (1.2) Sn = 11−nlogTr(ρn\tiny A). (1.3)

In particular, the Rényi entropies form a one-parameter family labeled by the index , which is often taken to be an integer (with ) renyi1 ; renyi2 . However, when can be continued to real , the entanglement entropy can be recovered with the limit: .

From a certain perspective, the Rényi entropies (1.3) are ‘less complicated’ objects than the entanglement entropy (1.2). One manifestation of this assertion is that can be evaluated as the expectation value of an operator in (a replicated version of) the QFT for integer . In particular, eq. (1.3) can be recast as

 (1.4)

where the twist operator is a codimension-two surface operator with support on the entangling surface (which divides the time slice into regions and ) Calabrese:2004eu ; Cardy:2007mb ; Hung:2014npa . To be precise, the above expectation value is taken in the tensor product of copies of the QFT – see further details in section 2. This reformulation of the Rényi entropies also allows these quantities to be evaluated using quantum Monte Carlo techniques, e.g., roger1 ; roger2 ; roger3 , and even to be measured in the laboratory, e.g., dima ; expEE .

However, turning to holography, the situation is somehow reversed. In the context of the AdS/CFT correspondence, the RT and HRT prescriptions Ryu:2006bv ; Ryu:2006ef ; Hubeny:2007xt provide an elegant geometric tool which can be implemented in a straightforward fashion to evaluate the entanglement entropy in the boundary theory for general situations.111This approach has also been extended to include higher curvature interactions in the gravitational dual Hung:2011xb ; deBoer:2011wk ; Dong:2013qoa ; Camps:2013zua , as well as quantum fluctuations in the bulk Faulkner:2013ana . The recent derivations Lewkowycz:2013nqa ; DLR of these two prescriptions also yield a geometric construction to evaluate holographic Rényi entropies, which can be formulated as evaluating the area of a cosmic brane in a backreacted bulk geometry Dong:2016fnf . Unfortunately, this approach does not yield a practical calculation except in very special situations. One example is the case of a spherical entangling surface in a boundary conformal field theory (CFT) Casini:2011kv ; Hung:2011nu where the backreacted geometry becomes a hyperbolic black hole in AdS, as will be reviewed in section 3. Further, progress in this direction was made recently Dong:2016wcf by studying the variations of for small perturbations of a spherical entangling surface for a four-dimensional boundary CFT – see also Camps:2016gfs . In this paper, we provide a generalization of these calculations Dong:2016wcf to any number of boundary dimensions.

Our investigation relies on the field theoretic approach introduced in Bianchi:2015liz to investigating the shape dependence of Rényi entropies in CFTs.222For previous studies of the shape dependence of entanglement entropy, see mark0 ; Bueno:2015rda ; Bueno:2015lza ; Carmi:2015dla ; Fonda:2014cca ; Fonda:2015nma . In particular, they examined the twist operators as conformal defects, e.g., Cardy:1984bb ; McAvity:1995zd ; Billo:2016cpy . This framework naturally leads to the definition of the displacement operator, which implements small local deformations of the entangling surface. Further, this work allowed a variety of different conjectures, e.g., safdi1 ; Lewkowycz:2014jia ; Bueno:2015qya ; Bueno:2015lza ; Bianchi:2015liz with regards to the shape dependence of to be consolidated in terms of a single simple constraint Bianchi:2015liz :

 CD(n)=dΓ(d+12)(2√π)d−1hn, (1.5)

for a -dimensional CFT. Here, is the coefficient defining the two-point function of the displacement operator and is the conformal weight of the twist operator, which controls the correlator of the stress tensor with the twist operator – see eq. (2.10). This constraint is known to hold for free massless scalars and fermions in Dowker1 ; Dowker2 , as well as free massless scalars in Bianchi:2015liz . Further, the limit of eq. (1.5) was recently proven to hold in general CFTs Faulkner:2015csl .333Of course, both and vanish at . Hence the nontrivial result of Faulkner:2015csl is that the first derivative of eq. (1.5) with respect to holds at . However, eq. (1.5) is not a universal relation for general CFTs at general values of . In particular, the results of Dong:2016wcf imply that this constraint fails for four-dimensional holographic CFTs. With our extension of these holographic calculations to general dimensions, we will explicitly confirm that eq. (1.5) does not hold for holographic CFTs in any dimension.

The paper is organized as follows: In section 2, we review in detail the defect CFT language, and we show how it allows one to generalize the results of Dong:2016wcf to arbitrary number of dimensions. In particular, we show that appears in the expectation value of the stress tensor in the presence of a deformed defect (entangling surface). In section 3, we review the construction of the holographic dual of a deformed planar entangling surface, and the determination of by simply extracting the expectation value of the stress tensor in this background. In section 3.2, we perform this computation numerically in the holographic dual of Einstein gravity in , as well as in an analytic expansion around to order in any number of dimensions. We also examine the limit for general dimensions, which is amenable to analytic result. In section 3.3, we then probe the dependence of on higher derivative corrections in the bulk, by adding a Gauss-Bonnet curvature-squared interaction. We extract numerically in , as well as to second order in an analytic expansion around in . From the latter, we observe that for a special value of the Gauss-Bonnet coupling, eq. (1.5) holds to order . Finally, we obtain analytically the value of in the limit for any number of dimensions, and find that the result is independent of the Gauss-Bonnet coupling and hence, matches the corresponding result for Einstein gravity. We conclude in section 4 with a brief discussion of our results. Some technical details are relegated to the appendices: Appendix A provides the details needed to derive a certain useful representation of the two-point correlators used in section 2. In appendix B, we describe how the expression for the boundary stress tensor used in section 3.2.1 is determined through holographic renormalization with Einstein gravity in the bulk.

Before proceeding, let us finally emphasize that our procedure applies equally well to any other conformal defect: the only place in which the information about Rényi entropy enters the computation is in the specific form of the dual metric. Finally, let us add that while this paper was in the final stages of preparation, ref. again appeared with results similar to those in section 3.2.

2 The CFT Story

The main object of interest here will be the twist operators which appear in evaluating the Rényi entropies as in eq. (1.4). These operators are best understood for two-dimensional CFTs since in this context, they are local primary operators Calabrese:2004eu ; Cardy:2007mb . For CFTs or more generally QFTs in higher dimensions, twist operators are formally defined with the replica method, e.g., see Hung:2011nu ; Hung:2014npa . However, in higher dimensions, they become nonlocal surface operators and their properties are less well understood. The replica method begins with a Euclidean path integral representation of the reduced density matrix where independent boundary conditions are fixed on the region as it is approached from above and below in Euclidean time, i.e., with . To evaluate in eq. (1.3) then, the path integral is extended to a path integral on an -sheeted geometry, where the consecutive sheets are sewn together on cuts running over . The result is often expressed as where is the partition integral on the full -sheeted geometry.444The denominator is introduced here to ensure the correct normalization, i.e., . To introduce the twist operator , this construction is replaced by a path integral over copies of the underlying QFT on a single copy of the background geometry. The twist operator is then defined as the codimension-two surface operator extending over the entangling surface, whose expectation value yields

 ⟨τn[A]⟩=Zn(Z1)n, (2.1)

where the expectation value on the left-hand side is taken in the -fold replicated QFT. Hence eq. (2.1) implies that opens a branch cut over the region which connects consecutive copies of the QFT in the replicated theory. Note that to reduce clutter in the following, we will omit the dependence of the twist operators .

For the remainder of our discussion, we will consider the case where the underlying field theory is a CFT, which allows us to take advantage of the description of the twist operators as conformal defects Bianchi:2015liz . Further we will focus on the special case of a planar entangling surface , which will allow us to take advantage of the symmetry of the background geometry.555Planar and spherical entangling surfaces are conformally equivalent and so the following discussion could equally well be formulated in terms of a spherical entangling surface. With regards to eq. (1.5), we note that both and control short distance singularities in particular correlators involving the twist operators, e.g., see eqs. (2.7) and (2.10), and so these parameters characterize general twist operators, independently of the details of the geometry of the entangling surface. As discussed above and in the introduction, for integer , the computation of is related to the expectation value of a twist operator ,

 (2.2)

where the expectation value is taken in the tensor product theory . The twist operator breaks translational invariance in the directions orthogonal to , and correspondingly the Ward identities of the stress tensor acquire an additional contact term at the location of the defect:666Let us stress that the Ward identity (2.3), as usual, should be interpreted as if both sides were inserted in a correlation function.

 ∂μTμatot(x,y)=δΣ(x)Da(y). (2.3)

Here we split the coordinates of the insertion into orthogonal () and parallel () ones, i.e., the defect sits at . We shall also sometimes regroup them as . The subscript ‘tot’ in eq. (2.3) indicates that the stress tensor is the total stress tensor of the full replicated theory, – equivalently, it is inserted in all the copies of the replicated geometry. In the absence of this subscript, refers to the stress tensor of a single copy of the CFT. The delta function has support on the twist operator. Hence, the operator , which is known as the displacement operator, lives on this defect. If we denote the position of the twist operator in space with i.e., in the present (planar) case, – and the unit vectors orthogonal to the defect with , we can give a definition of the displacement in terms of the correlator of arbitrary insertions in the presence of the twist operator

 (2.4)

In the above expression and throughout the following, expectation values labeled by are implicitly taken in the presence of the twist operator. Furthermore, recall that in the present discussion, has support on the flat entangling surface – in a more general case, eq. (2.4) would compute the connected part of the correlator. The definition (2.4) makes it obvious that, much like a diffeomorphism is equivalent to the insertion of in the path integral, the response of a defect to a displacement

 δXμ=δμafa, (2.5)

is given by repeated insertions of the displacement operator, e.g.,

 (2.6)

In eq. (2.6), we disregarded the insertion of the contributions of a single since the one-point function of vanishes for a flat (or spherical) defect.777The same holds for the three-point function with a flat entangling surface. The two-point function of the displacement operator is fixed up to a single coefficient

 (2.7)

Of course, is the parameter which we wish to determine here. Extracting it from a direct computation of in eq. (2.6) would involve second order perturbation theory around a flat entangling surface. Luckily, appears in other observables, some of which are linear in the displacement operator, and so will require only a leading order perturbation. It is convenient to focus on the correlation function between the displacement operator and the stress tensor. The generic two-point function of primaries (in the presence of a planar defect) with the relevant quantum numbers was given in Billo:2016cpy in terms of three OPE coefficients

 (2.8a) (2.8b) (2.8c)

where we recall that , but we have further fixed in these expressions. In fact, when the operators involved are the displacement operator and the stress tensor, only two of the three coefficients are linearly independent:

 b1 =(d−1)π(d−2)CDn−2d−2π−d+12d2Γ(d+12)d−2hnn (2.9a) b2 =−1π(d−2)CDn+2d−2π−d+12d2Γ(d+12)d−2hnn (2.9b) b3 =2d−1π−d+12dΓ(d+12)hnn. (2.9c)

The coefficient which appears in these expressions is the so-called conformal weight of the twist operator. It is defined by the expectation value of the stress tensor with a planar twist operator888We emphasize again that the stress tensor here acts in single copy of the CFT and hence there is a factor of on the right-hand side, e.g., compare to eq. (2.12) in Hung:2014npa .

 (2.10)

Now we can use eq. (2.4) to compute the same expectation value, but in the presence of the deformed entangling surface ()

 (2.11)

Clearly, for a generic deformation the integral cannot be performed. However, it turns out that the singular terms in the short distance expansion can be written down explicitly. This is due to the following property of the correlation function (2.8). When the limit is taken in the weak sense, i.e., after integration against a test function, the first few coefficients in the expansion are distributions with support at . More precisely, the following formula is proven in appendix A:

 ⟨Da(w)Tij(z)⟩n=xa|x|d[B1(1|x|2+∂22(d−2))δd−2(w)δij   +B4(∂i∂j−δij∂2d−2)δd−2(w)]+…, (2.12a) ⟨Da(w)Tbi(z)⟩n=−∂iδd−2(w)∣∣x|dB1(δab−xaxbx2)−∂i∂2δd−2(w)(d−2)|x|d−2(B12δab+B3xaxbx2)+…, (2.12b) ⟨Da(w)Tbc(z)⟩n= −δd−2(w)|x|d+2B1[2δa(bxc)+xa((d−1)δbc−xbxc|x|2(d+2))] (2.12c) + ∂2δd−2(w)2(d−2)|x|dB1[2δa(bxc)−xa((d−1)δbc−xbxc|x|2(d−2))]+…,

with

 B1 =dhn2πn, B2 =(d−1)Γ(d2−1)πd2−22Γ(d+1)CDn, B3 =B2−dB12(d−2), B4 =B2−B1d−2. (2.13)

The ellipsis in eqs. (2.12a)-(2.12c) stand for terms which are less singular in the distance from the entangling surface, and which we will not need in this work. These terms can however be fully expressed using the formulas of appendix A. The fact that the singular terms are local imply, via eq. (2.11), that in the limit of short distance from the defect depends locally on derivatives of the deformation . One more comment is in order. The one-point function in eq. (2.10) refers to a flat entangling surface. Of course, the one-point function in the presence of a defect obtained from this one via a conformal transformation is still proportional to . Correspondingly, in eqs. (2.12a)-(2.12c), only appears as a coefficient of the traceless part of and of the third derivative . Indeed, recall that at leading order the extrinsic curvature is , e.g., see Bianchi:2015liz , and that conformal transformations map planes into spheres, whose extrinsic curvature is proportional to the identity and constant.

In view of the holographic computation in the next section, it is useful to write the one-point function (2.11) in a coordinate system adapted to the shape of the deformed entangling surface. That is, we wish to introduce a ‘cylindrical’ coordinate system that is centered on the deformed entangling surface. Such coordinates can be constructed perturbatively in the distance from the entangling surface. We will use to denote the extrinsic curvature of and introduce the following notation for the trace and traceless parts:

The new adapted coordinates are related to the previous Cartesian coordinates as follows:

 x′a=xa−fa(y)−1d−2(xaKbxb−12Kax2)+O(ρ4),y′i=yi+∂ifa(y)xa−12(d−2)x2∂iKaxa+O(ρ5), (2.15)

and the metric becomes (to reduce the clutter, we neglect the primes in the following but this metric is understood to be in the new adapted coordinate system)

 ds2 =(1+2Kcxcd−2)(ρ2dτ2+dρ2+[δij+2~Kaijxa]dyidyj+4d−2∂iKbxbρdρdyi)+C, (2.16)

where , and represents the higher order terms with

 C=O(ρ3)dρ2+O(ρ5)dτ2+O(ρ4)dρdτ+O(ρ4)dρdyi+O(ρ5)dτdyi+O(ρ3)dyidyj. (2.17)

Here, we consistently kept track of the corrections to the metric coming from any further change of coordinates allowed by symmetries and linear in the deformation . We do not need to make any assumptions on those terms, because the order at which we work already allows to determine . Of course, the leading order in reduces to the well studied undeformed case. This is obvious from dimensional analysis, and will be useful in section 3.

Notice that the change of coordinates (2.15) simplifies for a traceless extrinsic curvature, i.e., when . When , in order to determine , it is sufficient to consider a deformation of this kind. (While it may not be possible to set everywhere, all of our calculations are local and so this does not matter). However, the choice of the frame defined by eq. (2.15) has two advantages. It is convenient in , where the extrinsic curvature has no traceless component. Further, in higher dimensions, accommodating deformations for which is nonvanishing allows us to perform a consistency check on our computation of , by considering both the traceless and trace contributions.

As a last step, we apply two consecutive Weyl transformations. The first with scale factor to remove the prefactor in the metric (2.16) and the second with in anticipation of our holographic computations. After the first rescaling,999We emphasize that the rescaled metric no longer corresponds to flat space. the metric exhibits an advantage of the change of coordinates (2.15). Indeed if implements a conformal transformation, eq. (2.15) is the inverse transformation. In particular, starting from a planar defect, a conformal transformation maps it to a sphere, whose extrinsic curvature is simply with constant (i.e.,  and ). Hence, after the Weyl rescaling correctly appears in the metric only via and derivatives of , such that, for the map to a sphere, it would trivialize to the flat space metric. Furthermore the position of is fixed by contraction of the indices, whereas the last term in the second line of eq. (2.15) forces the trace of the extrinsic curvature to appear in as few places as possible.

The second Weyl rescaling with does not provide an equivalent simplification, but it will turn out to be useful for the holographic computation in section 3. After the transformation , we find the conformally equivalent metric

 (2.18)

where the higher order corrections now take the form

 C′=O(ρ)dρ2+O(ρ3)dτ2+O(ρ2)dρdτ+O(ρ2)dρdyi+O(ρ3)dτdyi+O(ρ)dyidyj. (2.19)

The metric above describes a slightly deformed version of the manifold , appearing e.g., in Casini:2011kv ; Hung:2011nu – see also section 3. In particular, the deformation decays asymptotically as we approach the asymptotic boundary of the hyperbolic hyperplane with . We denote the new geometry as .

In these coordinates, the stress tensor one-point function looks particularly simple. In order to write it down in even dimensions, we should be careful to include the effect of the conformal anomaly. Under the rescaling (here ), the stress tensor one-point function transforms as follows:

 (2.20)

where is the stress tensor expectation value after the rescaling. The anomalous contributions are the higher dimensional analog of the Schwarzian derivative appearing in and are independent of , because locally the –fold branched cover is identical to the original spacetime manifold Hung:2011nu ; Hung:2014npa . It is therefore possible to subtract this contribution without knowing its explicit form. Using the fact that in flat space the vacuum expectation value of the stress tensor vanishes, i.e., , one easily finds

 (2.21)

Therefore combining the above results, we can write

 ⟨˜Tab(x)⟩n=gnρ2((d−1)δab−dxaxbρ2)+…,⟨˜Tai(x)⟩n=xaxbρ2∂iKbknd−2+…,⟨˜Tij(x)⟩n=1ρ2(−gnδij+kn~Kaijxa)+…, (2.22)

where

 kn−k1 =(d−1)Γ(d2−1)πd2−22Γ(d+1)CDn−3d−4d−2hn2πn gn−g1 =hn2πn. (2.23)

En passant, we note that when the conjecture is satisfied. We emphasize that the anomalous contributions only appear in even dimensions and hence and vanish in odd dimensions. The ellipses stand for higher orders in , which are the same as in the metric (2.18) when written in component form. Eq. (2.22), together with the metric (2.16), are the only ingredients entering the holographic computation. Let us also point out that the one-point function (2.22) and the metric (2.18) have similar structure in terms of the extrinsic curvature. In view of holographic renormalization, this suggests that the bulk metric will preserve the simplicity of the boundary metric.

3 Shape Deformations from Holography

In this section, we use holography to compute the one-point function of the stress tensor and then compare the holographic results to the field theoretic expressions in eqs. (2.22)-(2.23) in order to extract .

As mentioned at the beginning of section 2, the Rényi entropy can be evaluated using the partition function on a branched -fold cover of the original -dimensional spacetime. Implicitly, the latter path integral can be used to define the twist operator using eq. (2.1). For the purposes of our holographic calculations, it turns out that it is most convenient to work with this geometric interpretation. In particular, we will be extending the holographic computations introduced in Casini:2011kv ; Hung:2011nu . The discussion there began by considering how to evaluate the entanglement and Rényi entropies for a spherical or flat entangling surface in the flat space vacuum of a general CFT. By employing an appropriate conformal transformation, this question was then related to understanding the thermal behaviour of the CFT on a hyperbolic hyperplane. That is, the partition function was conformally mapped to the Euclidean path integral on the geometry , i.e., the product of a periodic Euclidean circle and a -dimensional hyperbolic space. Next in the case of a holographic CFT, this thermal partition function is evaluated by considering a so-called ‘topological’ AdS black hole with a hyperbolic horizons. In fact, the latter solutions can be found for a variety of higher derivative theories, as well as Einstein gravity Hung:2011nu ; brandon .

An important element of the conformal mapping in Hung:2011nu is that the conical singularity at the entangling surface in the branched cover of flat space is ‘unwound’ by extending the periodicity on the thermal circle. To make this statement precise, let us consider the metric in eq. (2.18) for the undeformed case, i.e., with . In this case, the geometry is precisely with the radius of curvature on implicitly set to one. Further, beginning with a -fold cover of flat space, the periodicity of the circle is . Considering the path integral of the CFT on this background then yields the corresponding thermal partition function with temperature . However, the important point is that this boundary geometry is completely smooth, which makes the question of finding the dual bulk configuration relatively straightforward. Of course, as noted above, the desired bulk solution corresponds to a hyperbolic black hole in AdS space.

The problem which we face then is to extend this holographic analysis to accommodate deformations away from the very symmetric entangling surfaces considered in the calculations described above. For a generic deformation of a flat or spherical entangling surface, the dual bulk geometry is not known, but the question of small deformations is precisely the one addressed by Dong:2016wcf in . Hence we must only extend this analysis to general dimensions. In fact, we also extend these calculations to a broader class of shape deformations. An essential feature of this approach is that we only solve for the bulk geometry at leading order in the size of the deformation of the entangling surface in the boundary.

With a small deformation, we can solve for the bulk geometry order by order in the distance from the entangling surface . The leading order solution coincides with the black hole geometry described above for an undeformed entangling surface. One can then move to the next order in to compute the bulk metric at first order in the deformation. Once our bulk metric is determined, we can extract . As explained in the introduction, our procedure involves computing the one-point function of the stress tensor, to enhance the appearance of to leading order in the deformation. This one-point function will be computed using standard holographic renormalization techniques deHaro:2000vlm .

3.1 Holographic Setup

Let us start by introducing an ansatz for the bulk metric. We first observe that the parallel components of the metric (2.18) only depend on the traceless part of the extrinsic curvature, while the components contain contributions from the parallel derivatives of the trace of the extrinsic curvature. This was achieved by our choice of coordinates (2.15) (plus the Weyl rescaling) and it is convenient in minimizing the number of unknown functions required for the gravitational ansatz. The bulk metric can then be written as

 ds2bulk=dr2r2L2g(r)−1+(r2L2g(r)−1)L2dτ2+r2ρ2(dρ2+[δij+2k(r)~Kaijxa]dyidyj+4d−2v(r)∂iKbxbρdρdyi)+⋯, (3.1)

where again the ellipsis stand for higher orders in , and denotes the AdS curvature scale. We will refer to the functions and as the traceless and traceful parts of the gravity solution, respectively. Their value will be determined by solving gravitational equations of motion at the first subleading order in . This procedure produces two second-order differential equations for and which we must solve numerically for general values of . We are also able to obtain analytic solutions in the vicinity of , as well as . As boundary conditions, we require and as we approach the AdS boundary () to reproduce the desired boundary metric. We also demand that the geometry is smooth at the ‘horizon’, i.e., where vanishes.

3.2 Einstein Gravity

In this subsection, we extract for the boundary theories whose holographic dual is described by Einstein gravity. The metric function , which is determined by the Einstein equations at zeroth order in , is given by Hung:2011nu ; Dong:2016wcf

 g(r) =1−rdh−L2rd−2hrd, (3.2)

where is the position of the horizon (in Lorentzian signature). It will be useful to define the dimensionless variable . Then is related to by

 n=2xnd(x2n−1)+2. (3.3)

At the next order in , the Einstein equations yield a second order differential equation for ,

 k′′(r)+r3g′(r)+(d+1)r2g(r)−(d−1)L2r3g(r)−L2rk′(r)−L2((d−3)(r2g(r)−L2)+r2)(L2r−r3g(r))2k(r)=0, (3.4)

as well as the algebraic equation

 k(r)=v(r). (3.5)

Note that for , eq. (3.4) correctly reproduces the analogous equation appearing in Dong:2016wcf . Other components of the Einstein equations give additional first and second order equations for , which are automatically solved when eqs. (3.4) and (3.5) are satisfied. To derive eq. (3.5), we used the Gauss-Codazzi relations (at leading order in ). The equality (3.5) provides a nontrivial consistency check of our ansatz (3.1) for the bulk metric. Indeed, eq. (2.22) shows that and appear in the same combination, denoted , in factors multiplying both the traceless and the traceful parts of the deformation. Eq. (3.5) ensures that the holographic solution will match this prediction from the CFT. The case of is slightly different since the traceless part of the extrinsic curvature vanishes. We therefore find that Einstein equations contain only the second order differential equation for

 v′′(r)+r3g′(r)+4r2g(r)−2L2r3g(r)−L2rv′(r)−L2r2(L2r−r3g(r))2v(r)=0, (3.6)

which matches eq. (3.4) upon substituting and .

3.2.1 Holographic Renormalization

Given the bulk metric (3.1), we are interested in evaluating the boundary expectation value of the stress tensor. This computation can be performed using the technique described in deHaro:2000vlm . First, we write the metric in the Fefferman-Graham (FG) form Fefferman:2007rka

 ds2bulk=L2z2(dz2+hμν(x,z)dxμdxν), (3.7)

where

 hμν(x,z)=h(0)μν(x)+z2h(2)μν(x)+⋯+zdh(d)μν(x)+⋯. (3.8)

The expectation value for the stress tensor is then determined by the ’s, with the following general expression

 ⟨Tμν⟩~Hn=d2(Lℓ\tiny P)d−1h(d)μν+Xμν[h(m)μν]m

The subscript indicates that the expectation value is taken in the deformed boundary geometry described by eq. (2.18). Here is a functional of the lower order terms, which are completely fixed by the boundary geometry. This contribution is related to the Weyl anomaly and accordingly, it vanishes with an odd number of boundary dimensions. In even , its explicit expression depends on the dimension. For the cases and , the interested reader is referred to eqs. (3.15) and (3.16) in deHaro:2000vlm . We will see that it is not necessary to compute those contributions in order to obtain . However, for completeness, we show how to obtain the exact expressions for the expectation value of the stress tensor in appendix B.

By comparing eqs. (3.9) and (3.1) with eq. (2.22), we see that the expansions of and near the boundary carry the information about the displacement operator. In this limit, the form of the solution to the equations of motion (3.4)-(3.5) reads

 d=3,k(r)=v(r)=1−L22r2+L3r3βn+…,d=4,k(r)=v(r)=1−L22r2+L4r4βn+…,d=5,k(r)=v(r)=1−L22r2−L48r4+L5r5βn+…,d=6,k(r)=v(r)=1−L22r2−L48r4+L6r6βn+…. (3.10)

Here, is the first coefficient which is not fixed by the boundary conditions at infinity. As one might expect, this coefficient determines , and we obtain it numerically in the next subsection. Matching these expansions with eq. (2.22), we find the following relations:

 kn=(Lℓ% \tiny P)d−1(xdn−xd−2n+dβn+k(d)0),gn=−(Lℓ\tiny P)d−1⎛⎝xdn−xd−2n+g(d)02⎞⎠, (3.11)

where and contain the anomalous contributions. As mentioned before, these vanish for odd dimensions and are independent of in even dimensions101010In particular, one can find that for , and , and for , and . – see appendix B. Note that in order to obtain and from eq. (2.23), we only need to consider the differences and . Then, all the anomalous contributions will cancel.111111Notice that in our conventions the stress tensor has lowered indexes, contrary to the one in Dong:2016wcf . The dictionary between the two conventions is as follows: and , with . This gives precise agreement between both expressions in .

Comparing eqs. (3.11) and (2.23), we find holographic expressions for and ,

 hnπn = (Lℓ\tiny P)d−1(xd−2n−xdn), (3.12) CDn = dΓ(d+1)(d−1)πd/2−2Γ(d/2)⎛⎝(d−2)(Lℓ\tiny P)d−1(βn−β1)+hn2πn⎞⎠. (3.13)

The Planck length can be replaced for CFT data as follows, e.g., see Hung:2014npa :

 CT=(Lℓ\tiny P)d−1(2d−2π−d+12d(d+1)Γ(d−12)), (3.14)

where is the coefficient that appears in the two-point function of the vacuum stress tensor Osborn:1993cr ; Erdmenger:1996yc ,

 ⟨Tμν(x)Tρσ(0)⟩=CTx2dIμν,ρσ(x). (3.15)

In order to obtain , we now only need to solve numerically the equations of motion (3.4) and extract . We will compare with the value in eq. (1.5) related to previous conjectures Bianchi:2015liz

 CconjD(n)=dΓ(d+12)(2√π)d−1hn. (1.5)

We will find that the conjecture is violated for holographic theories in any spacetime dimension. This conclusion will be supported numerically for with arbitrary in section 3.2.2, and also with analytic results near in general dimensions in section 3.2.3. In particular, the expected agreement with eq. (1.5) is reproduced only at linear order in , but we see will depart from eq. (1.5) at order .

3.2.2 Numerical Solutions

To solve the second order differential equation (3.4), we use a shooting method. The two integration constants will be free coefficients in the asymptotic expansions near both limits of integration. Near the asymptotic boundary, we have while regularity of the solution near the horizon fixes a new integration constant. In particular, near the horizon we need , where the proportionality constant will provide the second integration constant. It is useful to consider coordinates in which the extreme values are kept fixed. Hence for our numerical integrations, we defined , so that the AdS boundary is at and the horizon, at . For each value of , we solve the equation numerically both from the boundary and the horizon, fixing the integration constants so that the two curves meet smoothly.

The results for are plotted in fig. 1. In the figure, we chose to normalize by a factor , in order to exhibit that this combination reaches a fixed value at large values of the Rényi index. Notice that, due to the prefactor in the definition of the Rényi entropies (1.3), this normalization quantifies more precisely the shape dependence of at large . As one can see from fig. 2, deviates from away from the linear regime around . Yet, notice that curiously, the relative difference is fairly small for all . Although we are sure that this difference is bigger than our numerical accuracy, the analytic solution of the differential equation (3.4) close to confirms that eq. (1.5) fails (for general dimensions), as does the analytic result for the limit .

3.2.3 Analytic Solutions

It is also possible to produce an analytic treatment of eq. (3.4) near . We can solve the equation analytically order by order in powers of and then fix the integration constants by providing the boundary expansion for and regularity near the horizon. We find that

 k(~r)=k0(~r)+k1(~r)(n−1)+k2(~r)(n−1)2+O(n−1)3, (3.16)

with

 k0(~r)=√1−~r2,k1(~r)=(d−1)(~r2−1)~rd2F1(1,d2;d+22;~r2)+d(~rd−~r2)(d−1)d√1−~r2. (3.17)

For we solve separately for each dimension. These results determine perturbatively around , and the result can be written as

 βdn=βd1+1d(d−1)(n−1)−4d3−8d2+d+22d2(d−1)3(n−1)2+O(n−1)3, (3.18)

with being zero for odd and for even .

Given this expansion for and the corresponding expansion for from eq. (3.3), it is straightforward to compute as a power series in :

 CDCT=2π2d+1(n−1)−2π2(d2−d−1)d3−d(n−1)2+O(n−1)3, (3.19)

which, as expected Faulkner:2015csl , agrees with the conjecture (1.5) at linear order,

 CconjDCT=2π2d+1(n−1)−π2(2d2−4d+1)(d−1)2(d+1)(n−1)2+O(n−1)3, (3.20)

but not at second order. In fact, the relative mismatch between the two expressions can be easily computed and is given by

 CD−CconjDCD=(d−2)2d(d−1)2(n−1)+O(n−1)2. (3.21)

Interestingly, one can also extract the analytic expression for at leading order as . This result follows from the observation that the contribution in eq. (3.13) is subleading with respect to at small . More precisely, one can verify that in this limit. Then, we do not actually need to solve eq. (3.4) but just expand for small to find

 CDCT = −(1dn)d−1(2d−1π2d+1+O(n)), (3.22) CconjDCT = −(1dn)d−1(2dπ2(d−1)d(d+1)+O(n)), (3.23)

which yields

 CD−CconjDCD=−d−2d+O(n). (3.24)

Note that the relative error is order one as goes to zero, contrary to the small differences which were obtained for .

3.3 Gauss-Bonnet Gravity

In this section, we consider holographic CFTs dual to Gauss-Bonnet (GB) gravity. The full gravitational action reads Buchel:2009sk

 I=12ℓd−1p∫dd+1x√−g[d(d−1)L2+R+λL2(d−2)(d−3)X4], (3.25)

where

 X4=RabcdRabcd−4RabRab+R2, (3.26)

and the term (3.26) contributes to the equations of motion only for (note that the bulk theory is dimensional). The coupling is constrained by known unitarity bounds Buchel:2009sk

 −(3d+2)(d−2)4(d+2)2≤λ≤(d−2)(d−3)(d2−d+6)4(d2−3d+6)2. (3.27)

The same constraints can also be derived by excluding the propagation of superluminal modes in thermal backgrounds wok1 ; wok2 ; wok3 . Before proceeding, we should add a word of caution since in fact a detailed analysis indicates that the GB theory (3.25) violates causality unless the spectrum is supplemented by some higher spin modes Camanho:2014apa . However, it remains unclear in which situations these additional degrees of freedom will play an important role. Hence we proceed with the perspective that these holographic theories are amenable to simple calculations and allow us to investigate a broader class of holographic theories. Further, such investigations may still yield interesting insights on universal properties which may hold for general CFTs, beyond the holographic CFTs defined by these toy models. Certainly, this approach has been successful in the past, e.g., in the discovery of the F-theorem Myers:2010xs ; Myers:2010tj .

Conceptually, the procedure here is completely analogous to the one for the Einstein gravity case analyzed in the previous section, although explicit computations can become more tedious due to the -dependence.

In order to have the appropriate AdS asymptotics, we slightly modify the bulk metric ansatz,

 ds2bulk=dr2r2L2g(r)−1 + (r2L2g(r)−1)L2g∞dτ2 +r2ρ2(dρ2+[δij+2k(r)~Kaijxa]dyidyj)+⋯.

Note the additional factor of in the component, which is defined below. The metric for the hyperbolic black holes in GB gravity reads, e.g., brandon

 g(r)=12λ⎡⎢⎣1− ⎷1−4λ(1−rdh−L2rd−2h+λL4rd−4hrd)⎤⎥⎦. (3.29)

It is useful to define the asymptotic limit of as goes to infinity,

 g∞≡limr→∞g(r)=1−√1−4λ2λ