Entanglement entropy in cubic gravitational theories

JHEP05(2021)186 Published for SISSA by Springer Received:November 25, 2020 Revised:March 30, 2021 Accepted:April 9, 2021 Published:May 20, 2021 Entanglement entropy in cubic gravitational theories Elena Cáceres,aRodrigo Castillo Vásquezaand Alejandro Vilar Lópezb,c aTheory Group, Department of Physics, University of Texas at Austin, 2515 Speedway, Austin, TX 78712, U.S.A bDepartamento de Física de Partículas, Universidade de Santiago de Compostela, Rúa Xosé María Suárez Núñez s/n, Santiago de Compostela E-15782, Spain cInstituto Galego de Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela, Rúa de Xoaquín Díaz de Rábago s/n, Santiago de Compostela E-15782, Spain E-mail: [email protected],[email protected], [email protected] Abstract: We derive the holographic entanglement entropy functional for a generic gravitational theory whose action contains terms up to cubic order in the Riemann tensor, and in any dimension. This is the simplest case for which the so-called splitting problem manifests itself, and we explicitly show that the two common splittings present in the literature — minimal and non-minimal — produce different functionals. We apply our results to the particular examples of a boundary disk and a boundary strip in a state dual to 4dimensional Poincaré AdS in Einsteinian Cubic Gravity, obtaining the bulk entanglement surface for both functionals and finding that causal wedge inclusion is respected for both splittings and a wide range of values of the cubic coupling. Keywords: AdS-CFT Correspondence, Gauge-gravity correspondence ArXiv ePrint: 2009.11595 Open Access,c The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP05(2021)186 JHEP05(2021)186 Contents 1 Introduction 1 2 Holographic entanglement entropy in higher-derivative gravity 3 2.1 The entropy functional and the splitting problem 5 3 Entanglement entropy functional in cubic gravity 7 4 A concrete example: Einsteinian Cubic Gravity (ECG) 9 4.1 Preliminaries 10 4.2 Entanglement entropy of a disk in 4-dimensional ECG 11 4.3 Entanglement entropy example: a strip in 4-dimensional ECG 13 4.3.1 Numerical results 14 5 Conclusions and future directions 15 A Calculation of the entanglement functional 17 A.1 A first, detailed example 17 A.2 Minimal prescription 19 A.3 Non-minimal prescription 20 B Geometry of the entanglement surface for a boundary disk 22 C Geometry of the entanglement surface for a boundary strip 23 D Details of the HEE in ECG calculation 24 D.1 Minimizing the entropy functional for a strip in ECG 24 D.2 Numerics 26 D.2.1 Series expansion close to the boundary 26 D.2.2 Numerical integration 26 1 Introduction Classical stringy corrections lead to an effective higher-derivative theory of gravity. In such a theory, if the higher-derivative operators are suppressed by powers of `P, we are guaranteed that the theory is well behaved. When the higher-derivative operators are unsuppressed we have to analyze each theory individually; general statements regarding the well-posedness of the theory are, alas, hard to come by. Even a fundamental property as causality has to be re-examined. The generic existence of superluminal modes implies that causality and hyperbolicity of the equations of motion are not guaranteed, and the analysis has to be carried out for each specific theory — or class of theories [1,2]. Despite, or maybe because of, all these characteristics, higher-derivative theories are interesting in more than one way. In holography they provide a testing ground where to – 1 – JHEP05(2021)186 understand more deeply how holography works in theories whose duals are more generic CFTs (different central charges, non-supersymmetric, etc.). Crucial holographic constructs like the holographic entanglement entropy are modified in the case of higher derivative theories [3–5], and certain subtleties arise [6]. We will discuss this in detail in the next sections. The generic presence of superluminal modes in higher-derivative theories implies that previous causal constructs based on null rays have to be reexamined [7]. To add to the multitude of ways in which higher-derivative theories differ from Einstein gravity in the holographic context, there is also the question if a given higher-derivative theory is UV complete and, thus, expected to have a sensible field theory dual, or not. Holographically, relationships like causal wedge inclusion [8] and entanglement wedge nesting [9] are necessary conditions for a background to have a field theory dual and, thus, can be used to rule out certain higher-derivative theories from having unitary relativistic QFT duals. In [7] the authors showed how causal wedge inclusion can be used to arrive to the same conclusion as [10]. In the vast landscape of higher-derivative gravities, Lovelock theories are among the most studied; they have the advantage that they yield second order equations of motion. Among them, the quadratic theory, Gauss-Bonnet, has served as a prototype for many phenomena not present in Einstein gravity. From the violation of the η/s bound [11,12], to recent work related to the information paradox [13], Gauss-Bonnet theory has taught us important lessons for holography. Cubic theories of gravity have been studied in the general relativity community [14,15]. Some of their holographic properties have been explored [16, 17], but the explicit form of the entanglement entropy functional, a fundamental quantity in holography, was not known.1In this paper we advance the understanding of cubic theories in a holographic context by deriving the holographic entanglement functional for a generic cubic gravity theory. This functional can be applied to cubic Lovelock, quasitopological gravity and Einsteinian cubic gravity theories. Our result is applicable to general cubic gravity theories in any dimension, with the only restriction that the action does not involve derivatives of the curvature tensors. Obtaining the functional presents subtleties absent in quadratic theories. In [6,20–22], the authors showed that, in general, there is an ambiguity in the calculation of the entanglement entropy functional. This ambiguity, known as the “splitting problem”, is related to the regularization of the action near the conical singularity that appears in the Lewkowycz-Maldacena prescription [23]. We investigate this issue in detail and present the functional using two different splittings that we refer to as “minimal” and “non-minimal”. The “non-minimal” prescription is known to be correct at the perturbative level. At finite coupling, determining the correct splitting is an open question. However, we illustrate our result calculating the HEE surface in a theory that was built to avoid causality problems at the perturbative level, Einsteinian Cubic Gravity (ECG). The structure of the paper is as follows. In section 2we review the general framework for calculating the holographic entanglement entropy in higher-derivative theories. We pay particular attention to the spliting problem and to the two different proposals that exist in 1There has been previous work on some particular cubic theories, such as quasi-topological gravity [18, 19]. In any case, the general cubic functional was not known, and the “splitting problem” — which we discuss in section 2— was overlooked. – 2 – JHEP05(2021)186 the literature to solve it. Section 3contains our main result: we derive the entanglement entropy functional for a general cubic gravity theory. We explore the result obtained using the two different splitting prescriptions. We point out that quadratic theories are insensitive to the differences between them, as also are Lovelock theories, for which the Jacobson-Myers functional is valid [24]. However, for a generic cubic theory, the minimal and non-minimal prescriptions lead to different answers. As an example of our results, in section 4we work out in detail the entanglement functional for a particular cubic gravity theory, Einstenian Cubic Gravity (ECG) [15], and present some numerical results regarding the minimal surface they produce. Finally, in section 5we summarize our results, its implications, and point out open directions. 2 Holographic entanglement entropy in higher-derivative gravity In a holographic CFT dual to Einstein gravity, the entanglement entropy of a boundary region Ais given by the area of an associated codimension-2 surface [25,26]: S=Area(γA) 4GN .(2.1) The surface γA, also known as the Ryu-Takayangi or RT surface, is defined as the bulk codimension-2 surface which has the minimal area among all those homologous to the region Ain the boundary (it has to end in ∂A if this is not empty). If we assume holography holds, the work of Lewkowycz and Maldacena [23] constitutes a proof that (2.1) indeed gives the entanglement entropy. We will briefly review their argument here in order to set the stage for our future discussion concerning field theories dual to higher-derivative gravities. The computation starts by considering the usual replica trick in the boundary field theory. The entanglement entropy Scan be computed as the limit n→1of the Rényi entropies: Sn(A) = −1 n−1log Tr (ρn A) = −1 n−1(log Zn−nlog Z1),(2.2) where ρAis the reduced density matrix of the subsystem associated with region A, and Zn is the partition function of the field theory in the n-fold cover. This is a manifold consisting of ncopies of the original one, glued cyclically at the spatial region A.Z1is thus just the original partition function. We assume always that an analytic continuation to Euclidean signature has been performed. Notice also that Rényi entropies are defined for n∈N?, therefore an analytic continuation in nis also assumed before taking the limit n→1. So far, all this discussion has been restricted to the field theory, but if this is holographically dual to a gravitational one, it should be possible to find a bulk solution Bn dual to the n-fold cover. Then, log Zn=−I[Bn], where I[Bn]is the on-shell gravitational Euclidean action of this dual geometry. Naturally, log Z1=−I[B1],B1being just the original bulk dual. Now, the n-fold cover boundary manifold has a Znsymmetry, due to the fact that we can do permutations on the ncopies of the original manifold. If we assume that this replica symmetry is respected in the bulk, we can consider the manifold ˆ Bn=Bn/Zn, which has to be regular everywhere except in the codimension-2 submanifold – 3 – JHEP05(2021)186 Cnconsisting of fixed points of the Zn. Notice that, in the boundary, ∂A are precisely the fixed points of Zn. Also, since Bnis a regular bulk solution, the orbifold ˆ Bnhas a conical defect of opening angle 2π/n at Cn. Due to the replica symmetry, we can write: I[Bn] = nI[ˆ Bn],(2.3) where in I[ˆ Bn]we exclude contributions coming from the conical singularity (this is because in the left-hand side of the previous equation there are no such contributions, the geometry is regular).2After doing a suitable analytic continuation of this ˆ Bnto non-integer n, we can finally write: S= lim n→1 n n−1I[ˆ Bn]−I[B1]=∂nI[ˆ Bn]n=1 ,(2.4) where I[ˆ B1] = I[B1]. Once this expression is obtained, the computation of the entanglement entropy of the region Ahas been reduced to a problem in classical gravity, which can be solved in two steps: 1. The geometry with n= 1,ˆ B1, is a regular solution of the equations of motion. In (2.4) we seem to be doing a first order variation away from this solution, so we could naively expect that expression to vanish. This is not so because, when varying, we are changing the opening angle at C1= limn→1Cn, which as mentioned should be excluded from the action integral and that procedure introduces a boundary where conditions are changing if we vary n. This localizes the computation of the entanglement entropy in C1, and in fact in Einstein gravity it is possible to prove from the form of the action (see [23]) that Sis computed as shown in (2.1). γAshould be interpreted at this point as C1, where we have not proven its minimal property yet. 2. The remaining question is how we determine C1. Formally, it is defined by looking for Cnin the analytically continued spacetime ˆ Bn, and then taking the limit n→1. Adopting adapted coordinates at the conical singularity (see appendix of [23]), it is possible to show that the equations of motion derived from Einstein gravity for ˆ Bn impose, in the limit n→1, the minimal area condition: Ka= 0 ,(2.5) where Kaare the traces of the extrinsic curvatures along the transverse directions to C1, and ais an index which runs in these two directions (this notation will be clarified in the following section). This shows that C1is a minimal area surface, which can then be calculated by minimization of the entanglement entropy functional (2.1). With this condition, C1can be characterized as the previously defined surface γAwhich is homologous to the boundary region A. 2Boundary terms at the asymptotic boundary where the field theory lives should be included as usual, since they must appear also in I[Bn]. – 4 – JHEP05(2021)186 2.1 The entropy functional and the splitting problem The previous program can be carried out, with an increasing level of technical difficulty, when the gravitational theory contains higher-derivative corrections to the Einstein-Hilbert action. Following the two steps we have just described, [4] and [5] first obtained the expression for the functional computing entanglement entropy in the presence of higher-derivative terms. Using (2.4), and considering a Lagrangian containing arbitrary contractions of the Riemann tensor (but not its derivatives), one obtains: SEE = 2πZC1 dD−2y√g"∂LE ∂Rz¯zz¯z +X A ∂2LE ∂Rzizj∂R¯zk¯zl !A 8KzijK¯zkl qA+ 1 #.(2.6) where LE=LE(Rµνρσ)is the Euclidean version of the Lagrangian. Let us explain the notation in this expression, which employs the conventions of [4]: •The full manifold has dimension D, and we denote generic coordinates in it by xµ. Indices for this D-dimensional manifold are µ, ν, ρ, σ, . . . The surface on which the previous functional is evaluated, C1, is (D−2)-dimensional. Coordinates in it are yi, and indices will be labelled i, j, k, l, . . . We assume an embedding xµ=xµ(yi), so that we can define tangent vectors (mi)µ≡∂ixµ, and then take two extra orthonormal vectors nato complete the basis, Gµν(na)µ(nb)ν=δab. Indices a, b, c, d, . . . will be used for these two directions. •The functional (2.6) is defined using a particular set of adapted coordinates for C1 (see [4]), where tangent coordinates are xi(y) = yi, and we introduce two extra complex coordinates z, ¯zsuch that the metric factorizes: Gz¯z|C1=1 2, Gij|C1=gij, Gz¯zC1 = 2, GijC1 =gij ,(2.7) with the remaining components vanishing at C1. •Kaij is the extrinsic curvature of C1along the direction na: Kaij ≡(mi)µ(mj)ν∇µ(na)ν=−(na)µh∂i∂jxµ+ Γµ νρ∂ixν∂jxρi,(2.8) where (na)µ=δabGµν(nb)ν. This appears in (2.6) as a spacetime tensor defined in the usual way: Kµνρ ≡Kaij(na)µ(mi)ν(mj)ρ. Notice that, in the adapted coordinates, the orthonormal vectors are going to be taken as: n1=rz ¯z∂z+r¯z z∂¯z, n2=i rz ¯z∂z−r¯z z∂¯z!.(2.9) There remains to explain the sum over Ain the last term of (2.6), usually called the anomaly term. Its origin is subtle, coming from potentially logarithmically divergent terms in the action at C1when taking the limit n→1. For those terms, a careful analysis of the limit has to be performed, and it becomes essential to understand the analytic – 5 – JHEP05(2021)186 continuation of the geometry and the regularization of the conical singularity at C1.3This regularization has to be done guaranteeing that the equations of motion of the theory are satisfied. This was at first overlooked in [4], where a minimal regularization was employed. It is nevertheless useful to quote the result obtained, since it will allow us to see more clearly the differences introduced when other regularizations are used. For the computation of the entropy functional, all the study of the behaviour of the action around C1boils down to the following algorithmic procedure. In a general theory, the second derivative of the Lagrangian in the last term of (2.6) will be a polynomial in curvature tensors. In this polynomial we expand every curvature tensor using the following expressions: Rαβij =˜ Rαβij +gkl [KαjkKβil −KαikKβjl], Rαiβj =˜ Rαiβj +gklKαjkKβil −Qαβij ,(2.10) Rikjl =rikjl +Gαβ [KαilKβjk −KαijKβkl], where indices α, β, γ, . . . denote values zor ¯z,rikjl is the lower-dimensional Riemann tensor, and ˜ Rαβij,˜ Rαiβj and Qαβij are defined in [4] but their particular form will not be needed here. Once the expansion is done, we label each of the individual terms with A. Considering any of them, we associate a value qAto it equal to the number of factors of Qzzij and Q¯z¯zij plus one half the number of factors of Kαij,Rαβγi and Rαijk. Once this is done, we divide by qA+ 1, as indicated in (2.6). This expansion is what we denote by the sum over A, and when it is completed one can rewrite, if desired, everything again in terms of the original curvature tensors using (2.10). As mentioned, this prescription (which we will call minimal prescription) does not take into account the fact that the regularized metric at the conical singularity has to satisfy the equations of motion when computing (2.4). The existence of several ways to regularize the metric (with relevant consequences for the entropy functional) has been called the splitting problem in the literature, and discussions around this issue can be found in [20–22]. For a general higher-derivative theory, it can be quite complicated to impose the onshell condition for the regularization, but one can make a first approach to the problem by studying the constraint imposed by Einstein gravity. This produces a functional which is at least perturbatively correct, since the leading order area term is independent of the splitting chosen. This is done in great detail in [22] (and in [6], although the final result is expressed assuming the surface has Ka= 0, so that it has extremal area), and it is shown that the final effect for the entropy functional is a different prescription for the A expansion (we will call this non-minimal prescription). After expanding all the curvature tensors according to (2.10), we have to rewrite Rz¯zz¯zand Qz¯zij in terms of two new objects: R0 z¯zz¯z=Rz¯zz¯z+1 2KzijK¯zij −KzK¯z,(2.11) Q0 z¯zij =Qz¯zij −KzikK¯zjk −KzjkK¯zik +1 2KzK¯zij +1 2K¯zKzij ,(2.12) 3This is because the computation of (2.4) can be localized at C1, as already mentioned. See [4] for more details. – 6 – JHEP05(2021)186 and associate qA= 1/2to Kαij,Rαβγi, and Rαijk; and qA= 1 to Qzzij and Q¯z¯zij. Then we must divide by qA+ 1 as indicated in the general expression (2.6), and finally, if desired, undo the expansions, writing everything in terms of curvature tensors again. We have presented two ways to define the functional computing entanglement entropy in the presence of higher-derivative corrections to the gravitational action. Let us emphasize that for quadratic theories the functionals obtained using the minimal and non-minimal prescriptions are the same. As shown in [4], this is also the case for Lovelock theories, where the Jacobson-Myers functional is known to be correct. This is a special property of Lovelock theories, for which the entanglement entropy functional depends only on the intrinsic geometry of the surface, rijkl. However, we will show that for general cubic gravities the functionals obtained are different. Until now, we have only completed the first step of the general strategy outlined in the previous section, i.e., we have shown how to obtain the entanglement functional starting from the action of the theory. We still have to find the position of the surface C1, in other words, where to evaluate the functional. In principle, the equations of motion of the theory should determine the location of the surface, in much the same way they determine it to be a minimal area surface in General Relativity. The idea is to explicitly evaluate the equations of motion in the conically singular metric for generic nto linear order in n−1. One does not expect divergences in the stress-energy tensor, but these generically appear in the metric part of the equations of motion, so cancelling them imposes conditions which determine the position of the surface in the limit n→1(in particular, GR equations of motion impose the well-known condition Ka= 0). Further details can be found in [21,23]. In practice, this procedure has the drawback of requiring to deal with the equations of motion of higher-derivative theories, which can be extremely complicated. Fortunately, in [27] it is shown that the same procedure one employs in Einstein gravity is also valid in general: after computing the correct functional, one can minimize it to obtain the surface C1in which it is going to be evaluated.4,5 3 Entanglement entropy functional in cubic gravity This section contains our main result, we will derive the entanglement functional for a generic cubic gravity theory that does not involve explicit derivatives of the curvature tensor. We will use the method diuscussed in section 2to compute the different contributions to the holographic entanglement entropy functional in cubic gravities. As a warm-up exercise, let us revisit the known result for quadratic theories. As explained in section 2, there are different prescriptions to calculate the entanglement entropy 4Notice that the equations of motion are still necessary in principle, because one has to determine the correct splitting. It is only possible to state the correctness of the non-minimal prescription at the perturbative level. 5Previous work on the question of whether minimizing the functional is equivalent to the LewkowyczMaldacena prescription found some issues for certain theories [28]. This may be due to the fact that the splitting problem is being ignored, but it would be interesting to check it explicitly. – 7 – JHEP05(2021)186 functional in higher-derivative theories. In quadratic gravity theories, LE=λ1R2+λ2RµνRµν +λ3RµνρσRµνρσ ,(3.1) any prescription leads to the same result for the holographic entropy functional [29]. The reason for this is that for quadratic theories the expansion in Ais trivial: after two derivatives of the Lagrangian we obtain something which does not contain curvature tensors, so essentially qA= 0 always. This guarantees that, in the case of quadratic gravities, the result obtained using either the minimal or non-minimal splitting is the same, Squad EE =−4πZddy√g2λ1R+λ2Raa−1 2KaKa+ 2λ3Rabab −KaijKaij,(3.2) where Ka≡Kaijgij. This is not the case for cubic gravities. Consider the following generic cubic Lagrangian:6 LE=µ8R3+µ7RµνRµνR+µ6RµνRνρRρµ+µ5RµρRνσRµνρσ +µ4RµνρσRµνρσR +µ3RµνρσRµνρτ Rστ +µ2RµνρσRρσλτ Rλτ µν +µ1RµρνσRρλστRλµτν.(3.3) Note that the second derivative of the Lagrangian (3.3) is linear in the curvature tensor. Therefore, unlike in quadratic gravity, the two different splittings discussed in section 2lead to different entanglement functionals. The details of these highly technical calculations are presented in appendix A. The final result for the entropy functional obtained following the minimal splitting prescription is: Smin EE = 2πZdD−2y√gSR2+SK2R+Smin K4,(3.4) where SR2=−6µ8R2−2µ7(Rµν Rµν +RaaR)−3µ6RaµRaµ −µ52Rµν Raµaν −RabRab +RaaRbb −2µ4RµνρσRµνρσ + 2RRabab−µ3RaµνρRaµνρ + 4RaµRbabµ −6µ2Rabµν Rabµν + 3µ1Raµbν Raνbµ −RaµaνRbµbν ,(3.5) SK2R= + µ7KaKaR+3 2µ6KaKaRbb+ 2µ5KaKaijRij −1 2µ5KaKaRbcbc + 4µ4Kaij Kaij R + 2µ3KaikKajkRij +µ3Kaij KaijRbb+ 2µ3KaKaijRbibj + 12µ2KaikKajkRbibj + 3µ1KaijKaklRikjl −3 2µ1KaijKaijRbcbc + 6 (2µ2+µ1)KaikKbj kRabij ,(3.6) Smin K4= + 1 2µ7KaKaKbKb+µ5KaKaij KbKbij −2KbikKbjk −1 4(6µ7+ 3µ6−8µ4)KaKaKbijKbij −1 2(12µ4+µ3−3µ1)KaijKaijKbklKbkl −3 2(4µ2+ 3µ1)KaijKbj kKaklKbli+ (−2µ3+ 3µ1)KaijKajkKbklKbli.(3.7) 6We do not consider terms with explicit derivatives of the curvature tensors, such as ∇µR∇µR. These could in principle appear also at cubic order, but they complicate considerably the calculations, and many of the known cubic gravity theories like Lovelock [30], quasi-topological gravity [14,16], and ECG [15] do not include them. How to deal with these terms can be found in [20]. – 8 – JHEP05(2021)186 -1.5 -1.0 -0.5 0.5 1.0 1.5 0.5 1.0 1.5 2.0 2.5 (a) µ=−0.002. - 0.10 -0.05 0.05 0.10 2.49 2.50 2.51 2.52 (b) µ=−0.002. Figure 2. Causal wedge (orange, dashed) and entanglement surfaces corresponding to ECG in the case of minimal prescription (blue), ECG in the case of non-minimal prescription (red), and Einstein gravity (green, dashed) [35] for `= 3 and µ=−0.002, a value within the interval −0.00322 ≤µ≤ 0.00312. We include a close-up of the entanglement surfaces near x= 0 to tell them apart better. ary strip in Poincaré AdS does not constrain the allowed values of the coupling in ECG. Note that causal wedge inclusion in a Gauss-Bonnet black hole background does constrain the allowed values of the coupling [7]. A crucial ingredient to arrive at the results of [7] was the modified causal structure of the higher curvature theory due to the presence of superluminal modes. Here we are working in AdS and the causal structure is still governed by null rays. In addition, the disk has the same entanglement surface in both Einstein gravity and ECG, and therefore although causal wedge inclusion is marginally respected, it is so for all values of the coupling. For the strip geometry, in the case of Einstein gravity (µ= 0), causal wedge inclusion is safely respected, as the previous plots show. Thus, to violate causal wedge inclusion, it would be necessary a big modification of the surface due to turning on the coupling µ. This does not happen. Clearly, to explore the effects of causal wedge inclusion in ECG it would be better to study a black hole background, just like the authors did in [7] in Gauss-Bonnet gravity. However, in order to construct the causal wedge in a higher curvature theory, a complete investigation of the superluminal modes that determine the causal structure is required. A different possible avenue to harness the power of causal wedge inclusion is to investigate perturbative modifications of the disk region and solve the Euler-Lagrange equations derived from (4.6) or (4.8) with the perturbed boundary condition, but still in pure AdS, similarly to what [36] did for Einstein gravity. Since causal wedge inclusion is marginally respected there, it could happen that a perturbative violation occurs for some values of the coupling, which would therefore have to be excluded. The complicated form of the Euler-Lagrange equations makes this an extremely involved problem at the technical level, which we therefore leave as an open question. 5 Conclusions and future directions In the present paper we have obtained the entanglement entropy functional for a generic cubic gravitational theory not involving derivatives of the curvature tensor in the action. – 15 – JHEP05(2021)186 This constitutes our main result. We have done this using two different prescriptions: the minimal one, introduced in [4], and the non-minimal one, presented in [6,22] and which is known to be perturbatively valid. The existence of these two alternative functionals is an explicit manifestation of the splitting problem one is forced to face when deriving the entanglement entropy functional for cubic or higher order gravitational theories. Despite knowing that for a particular theory there must be a single, correct functional, we have also performed some consistency checks for both of them in a particular cubic theory. In particular, we investigated whether the functionals obtained for Einsteinian Cubic Gravity produce via minimization entanglement surfaces which satisfy the causal wedge inclusion property for both a boundary disk and a boundary strip in Poincaré AdS. We find that, for all the values of the couplings studied, causal wedge inclusion is respected. Our work makes it possible to investigate several questions that will advance our understanding of the role cubic theories play in a holographic context. Bit threads In [37] the authors put forward an alternative formulation of the holographic entanglement entropy that does not rely on minimizing an area functional but invokes a divergenceless vector field, dubbed bit threads. Many aspects of bit threads have been studied [38–43], but a formulation of bit threads for higher-derivative theories was missing. Recently this gap was closed in [44], where the authors derived a bit thread formulation for a general higher-derivative theory. Now that we have the entanglement functional for cubic theories in the minimal and non-minimal splitting, it would be interesting to investigate if using bit threads we can understand better the splitting problem. Dynamics Holographic entanglement entropy in dynamical backgrounds has been widely studied in Einstein gravity, with and without charge. It led to interesting insights regarding the thermalization time [45–47]. Similar work has been carried out for Gauss-Bonnet backgrounds [48–50]. If a Vaidya type of solution can be written down for black holes in cubic gravity, then dynamical studies in these theories along the lines suggested would be quite interesting. More general boundary regions We have learned valuable lessons studying HEE for different boundary regions: as an example, the divergence structure of the holographic entanglement entropy on regions with corners uncovered cut-off independent coefficients. These coefficients were shown to be universal and to encode important field theory data in Einstein and Gauss-Bonnet theories [51,52]. It would be interesting to study regions with corners in cubic theories. Derive the correct splitting for finite coupling Clearly, an important open question is to formally find the correct splitting for a general cubic theory with finite coupling. This highly technical problem can be approached following [6,22]. – 16 – JHEP05(2021)186 Causal structure of cubic theories In section 4, as an example of the functional we derived, we calculated the entanglement surfaces of a disk and a strip regions in an AdS background solution of ECG. More interesting phenomena regarding the causal and entanglement wedges can be expected if we were to consider black holes in any cubic gravity theory. However, black hole backgrounds in higher-derivative theories will generically have superluminal modes, and their causal structure is no longer determined by null rays. A complete study of the causal structure and hyperbolicity of equations of motion of cubic theories similar to what was done for Gauss Bonnet [1,2,53] is of interest not only for the GR communities but also in a holographic context. Perturbative calculations in field theory The non-minimal functional is known to be the correct one for any perturbative cubic theory which does not include covariant derivatives of the curvature tensors. Furthermore, perturbatively, we know that the entanglement entropy functional can be evaluated in the Ryu-Takayanagi surface obtained in General Relativity. This is because the area term of the functional is minimal for the Ryu-Takayanagi surface, and therefore even if the surface changes at first order, it does not produce a change of that part of the functional. The first order variation of the Ryu-Takayanagi surface can also be neglected for the remaining part of the functional coming from the higherderivative terms. This approach will be explored elsewhere [54]. Acknowledgments We are deeply indebted to Joan Camps for insightful explanations and correspondence. It is a pleasure to thank also Pablo Bueno, José Edelstein and Julio Oliva for useful discussions. EC and RC are supported by the National Science Foundation (NSF) Grant No. PHY1820712. RC is also supported by NSF Grant No. PHY-1620610. The work of AVL is supported by the Spanish MECD fellowship FPU16/06675, and by MINECO FPA2017-84436-P, Xunta de Galicia ED431C 2017/07, Xunta de Galicia (Centro singular de investigación de Galicia accreditation 2019-2022) and the European Union (European Regional Development Fund — ERDF), “María de Maeztu” Units of Excellence MDM2016-0692, and the Spanish Research State Agency. AVL is also pleased to thank the University of Texas at Austin, where part of this work was done, for their warm hospitality. A Calculation of the entanglement functional A.1 A first, detailed example In order to understand better how the entanglement entropy functional is obtained for a generic cubic theory, let us present an example in detail. Consider the following (Euclidean) Lagrangian: LE=λRµνρσRµνρσR , (A.1) – 17 – JHEP05(2021)186 where λis a constant. We will compute separately each of the terms in the general form of the functional (2.6), and for the second one we will do it following the two prescriptions presented in section 2. First of all, for the Wald term, we need the following derivative: ∂LE ∂Rz¯zz¯z =−2λRµνρσRµνρσ + 2λRRz¯zz¯z.(A.2) If we want to use this expression in a situation in which we do not have at our disposal the set of (complex) coordinates adapted to the surface, as will generically be the case, we need to covariantize the last term. This is done by going to the orthonormal basis (2.9): n1=rz ¯z∂z+r¯z z∂¯z, n2=i rz ¯z∂z−r¯z z∂¯z!.(A.3) The simplest way to do this is by first writing the expression as a contraction in the complex coordinates zand ¯z, using the fact that the only non-vanishing components of the metric are Gz¯z=G¯zz = 1/2: Rz¯zz¯z=−4Rz¯zz¯z=−2Rαβαβ ,(A.4) where the antisymmetry of the Riemann tensor in the two pairs of indices has been taken into account. Now we use the fact that a contraction in an α-index can be substituted for a contraction in orthonormal directions, since tangent directions have no zor ¯zcomponents: Tαα=Tab(na)α(nb)α=Taa,(A.5) where Tis any tensor, possibly containing extra indices. The Wald term is then: ∂LE ∂Rz¯zz¯z =−2λRµνρσRµνρσ −4λRRabab .(A.6) This is written in a form which can be evaluated in any set of coordinates just constructing the two orthonormal vectors nato the surface. Consider now the anomaly term. The first step is to obtain the second derivative of the Lagrangian: ∂2LE ∂Rzizj∂R¯zk¯zl = 2λgi(kgl)jR . (A.7) Now we have to expand Rfollowing the two prescriptions. For this we have to write it in terms of Riemann tensor components and expand them following (2.10): R=GαβRαβ +gijRij =GαβGγδRαγβδ + 2GαβgijRαiβj +gijgklRikjl =(A.8) =GαβGγδRαγβδ + 2Gαβgij ˜ Rαiβj −2GαβgijQαβij +gijgklrikjl + 3KαijKαij −KαKα. Now, in the minimal prescription the last two terms have qA= 1, while all the rest have qA= 0 (notice that GαβQαβij involves only Qz¯zij). Therefore, when doing the Aexpansion we have to multiply them by 1/2. Doing that and then rewriting back everything in terms of the Ricci scalar, the minimal prescription gives: X AR 1 + qAmin A =R−3 2KαijKαij +1 2KαKα.(A.9) – 18 – JHEP05(2021)186 For the non-minimal prescription we need to rewrite Rz¯zz¯zand Qz¯zij in terms of the primed versions (2.11). This is done as follows: GαβGγδRαγβδ =−8Rz¯zz¯z=GαβGγδR0 αγβδ +KαijKαij −KαKα,(A.10) GαβgijQαβij = 4gijQz¯zij =GαβgijQ0 αβij + 2KαijKαij −KαKα.(A.11) The Ricci scalar is now written in terms of the basic objects for this prescription as: R=GαβGγδR0 αγβδ + 2Gαβgij ˜ Rαiβj −2GαβgijQ0 αβij +gijgklrikjl ,(A.12) so that there are no terms with non-vanishing qA. We can then immediately rewrite everything in terms of the Ricci scalar, obtaining: X AR 1 + qAnon-min A =R . (A.13) This completes the Aexpansion for the non-minimal prescription. Notice that, in (A.7), the metric tensors are not affected by the expansion, and we can contract them with the extrinsic curvatures appearing in the general formula (2.6) as follows: gi(kgl)jKzijK¯zkl =KzijK¯zij =1 4KαijKαij .(A.14) All contractions in αindices can now be traded for contractions in aindices as explained when discussing the Wald term. We conclude this little example by collecting all contributions, which would produce the following entanglement entropy functionals for the Lagrangian (A.1): Smin EE = 4πλ ZdD−2y√gh−RµνρσRµνρσ −2RRabab + 2KaijKaijR− −3KaijKaijKbklKbkl +KaijKaijKbKbi,(A.15) Snon-min EE = 4πλ ZdD−2y√gh−RµνρσRµνρσ −2RRabab + 2KaijKaijRi.(A.16) The following sections contain the contributions to both the Wald and anomaly terms of the functional using both prescriptions. We include all numerical factors except for the global 2πappearing in (2.6). A.2 Minimal prescription L3,8=R3, Wald :−6R2,(A.17) Anomaly : 0 .(A.18) L3,7=RµνRµνR , Wald :−2RµνRµν −2RaaR , (A.19) Anomaly :KaKaR−3 2KbijKbij +1 2KbKb.(A.20) – 19 – JHEP05(2021)186 L3,6=RµνRνρRρµ, Wald :−3RaµRaµ ,(A.21) Anomaly :3 2KaKaRbb−1 2KbijKbij.(A.22) L3,5=RµρRνσRµνρσ , Wald :−2RµνRaµaν +RabRab −RaaRbb,(A.23) Anomaly : 2KaKaij Rij −KbimKbjm +1 2KbKbij−1 2KaKaRbcbc .(A.24) L3,4=RµνρσRµνρσR , Wald :−2RµνρσRµνρσ −4RRabab ,(A.25) Anomaly : 4KaijKaij R−3 2KbklKbkl +1 2KbKb.(A.26) L3,3=RµνρσRµνρτ Rστ , Wald :−RaµνρRaµνρ −4RaµRbabµ ,(A.27) Anomaly : 2KaikKajkRij +KaijKaijRbb+ 2KaKaijRbibj−(A.28) −2KaijKajkKbklKbli−1 2KaijKaijKbklKbkl . L3,2=RµνρσRρσλτ Rλτ µν , Wald :−6RabµνRabµν ,(A.29) Anomaly : 12KaikKbjkRabij + 12KaikKajkRbibj −6KaijKbjkKaklKbli.(A.30) L3,1=RµρνσRρλστRλµτν, Wald :−3RaµaνRbµbν −RaµbνRaνbµ,(A.31) Anomaly : 3KaijKaklRikjl + 6KaikKbjkRabij −3 2KaijKaijRbcbc+(A.32) +3 2KaijKaijKbklKbkl −9 2KaijKbjkKaklKbli+ 3KaijKajkKbklKbli. A.3 Non-minimal prescription L3,8=R3,(A.33) Wald:−6R2,(A.34) Anomaly:0 .(A.35) – 20 – JHEP05(2021)186 L3,7=Rµν Rµν R , (A.36) Wald:−2Rµν Rµν −2RaaR , (A.37) Anomaly:KaKaR . (A.38) L3,6=RµνRνρRρµ,(A.39) Wald:−3RaµRaµ ,(A.40) Anomaly:3 2KaKaRbb.(A.41) L3,5=RµρRνσRµνρσ ,(A.42) Wald:−2Rµν Raµaν +RabRab −RaaRbb,(A.43) Anomaly:2KaKaij Rij −1 2KaKaRbcbc +1 2KbKb−1 2KbijKbij.(A.44) L3,4=RµνρσRµνρσR , (A.45) Wald:−2RµνρσRµνρσ −4RRabab ,(A.46) Anomaly:4Kaij Kaij R . (A.47) L3,3=RµνρσRµνρτ Rστ ,(A.48) Wald:−RaµνρRaµνρ −4RaµRbabµ ,(A.49) Anomaly:2KaikKajkRij +KaijKaijRbb+ 2KaKaij Rbibj +(A.50) +KaKaijKbikKbjk −KaKaij KbKbij . L3,2=Rµν ρσRρσλτ Rλτ µν ,(A.51) Wald:−6Rabµν Rabµν ,(A.52) Anomaly:12KaikKbjkRabij + 12KaikKajkRbibj −(A.53) −6KaKaijKbikKbjk −6KaijKbjkKaklKbli+ 12KaijKajkKbklKbli. L3,1=RµρνσRρλστRλµτν,(A.54) Wald:−3RaµaνRbµbν −Raµbν Raνbµ,(A.55) Anomaly:3Kaij KaklRikjl + 6KaikKbjkRabij −3 2KaijKaij Rbcbc +1 2KbKb+(A.56) +3 4KaijKaijKbklKbkl +3 2KaijKbklKbij Kakl −9 2KaijKbj kKaklKbli+ 3KaijKajkKbklKbli. – 21 – JHEP05(2021)186 B Geometry of the entanglement surface for a boundary disk Let us collect here the results needed to evaluate the tensors appearing in the entanglement entropy functional for a boundary ball in Poincaré AdS (we consider the case of a 2dimensional disk in the main body of the paper, but results will be presented here for the case of a general (D−2)-ball). Consider then the bulk metric: ds2=L2 ? z2dτ2+ dz2+ dr2+r2dΩ2 D−3,(B.1) where the length scale L?is determined by imposing this to be a solution of the equations of motion for the theory at hand. The ball in the boundary will be parametrized at fixed τas r≤R, so the surface anchored at its boundary and going into the bulk is parametrized as: τ=τ0, r =ξ , z =Z(ξ),Ωp=ψp,(B.2) with ψpcoordinates in the (D−3) unit sphere, ξ∈(0, R), and Z(R)→0. Basis vectors tangent to the surface are then: m1=mξ=∂r+Z0∂z, mp+1 =mψp=∂Ωp.(B.3) This induces a metric on the surface of the form: ds2 C1=L2 ? Z2(ξ)h1 + Z02dξ2+ξ2dΩ2 D−3i.(B.4) We then choose our two normalized vectors orthogonal to the surface to be: n1=Z L? ∂τ, n2=Z L?√1 + Z02Z0∂r−∂z.(B.5) This produces the following extrinsic curvature components: K2ξξ =L? Z2√1 + Z02h1 + Z02+ZZ00i,(B.6) K2ψpψq=L? Z2√1 + Z02ξ+ZZ0ξ γpq ,(B.7) where γpq is the metric on the unit round sphere and the rest of the components of the extrinsic curvature vanish. Transforming the last two indices to coordinate ones: K2zz =Z02 (1 + Z02)2K2ξξ , K2zr =Z0 (1 + Z02)2K2ξξ , K2rr =1 (1 + Z02)2K2ξξ , while K2pq =K2ψpψq. The trace of the second extrinsic curvature is then (clearly K1= 0): K2=1 L?ξ(1 + Z02)3/2hξZZ00 + (D−3)ZZ01 + Z02+ (D−2)ξ1 + Z02i.(B.8) Notice that the RT surface satisfying the prescribed boundary conditions (which is the one solving K2= 0) is given by Z(ξ) = pR2−ξ2. In particular, this implies ZZ0=−ξand ZZ00 =−(1+Z02), showing that not only K2= 0, but the whole tensor satisfies K2µν = 0. This fact is relevant when discussing minimal surfaces for boundary disks in the main body of the paper. – 22 – JHEP05(2021)186 C Geometry of the entanglement surface for a boundary strip The aim of this short appendix is to collect the results needed to evaluate the tensors appearing in the entanglement entropy functional for a boundary strip in Poincaré AdS. Consider then the bulk metric: ds2=L2 ? z2dτ2+ dz2+ dx2+δpq dypdyq,(C.1) where the length scale L?is determined by imposing this to be a solution of the equations of motion for the theory at hand. We have separated the spatial coordinates in the boundary into xand yp(with p= 1,2...,D−3) because we will consider in this spacetime a surface anchored to a boundary strip finite in extent in the x-direction, parametrized as: τ=τ0, z =Z(ξ), x =X(ξ), yp=ψp,(C.2) with ψp∈(−∞,∞),ξ∈(ξi, ξf),X(ξi)→ −`/2,X(ξf)→`/2, and Z(ξi), Z(ξf)→0. Basis vectors tangent to the surface are then: m1=mξ=X0∂x+Z0∂z, mp+1 =mψp=∂yp.(C.3) This induces a metric on the surface of the form: ds2 C1=L2 ? Z2(ξ)hX02+Z02dξ2+δpq dψpdψqi.(C.4) We then choose our two normalized vectors orthogonal to the surface to be: n1=Z L? ∂τ, n2=Z L?√X02+Z02Z0∂x−X0∂z.(C.5) This produces the following extrinsic curvature components: K2ξξ =L?X02 Z2√X02+Z02"X0+ZZ0 X00#,(C.6) K2ψpψq=L?X0 Z2√X02+Z02δpq ,(C.7) with the rest of them vanishing. Transforming the last two indices to coordinate ones: K2zz =Z02 (X02+Z02)2K2ξξ , K2zx =X0Z0 (X02+Z02)2K2ξξ , K2xx =X02 (X02+Z02)2K2ξξ , while K2pq =K2ψpψq. The trace of the second extrinsic curvature is then (clearly K1= 0): K2=X0 L?(X02+Z02)3/2"(D−2) X02+Z02+ZX0Z0 X00#.(C.8) As a final comment, we are employing a generic parametrization, but the results in the main text are presented with z=Z(x). For that case, we can just set X0= 1. For numerical computations we also employed x=X(z), in which case we set Z=zand Z0= 1. – 23 – JHEP05(2021)186 D Details of the HEE in ECG calculation This appendix contains the calculational details of the example presented in section 4. First we will obtain the equations to solve, and then present details of the numerical integration. D.1 Minimizing the entropy functional for a strip in ECG Consider the boundary strip in a state dual to the 4-dimensional vacuum AdS in ECG, with bulk metric given by (4.10). For this setup, the functionals (3.4) and (3.8) have been already obtained in the minimal and non-minimal prescriptions for the z=Z(x)parametrization, eqs. (4.11)–(4.14). Similar expressions are needed for the x=X(z)parametrization, since it will be used in the first two parts of the numerical procedure. With the minimal splitting regularization we obtain Smin strip =L2 ? 4GNZdydz"√1 + X02 z2+3f2 ∞µ 4z2[1 + X02]9/2Smin x#,(D.1) where Smin x≡4−14X06−18X08−6X010 + 28zX03X00 + 32zX05X00 + 12zX07X00 +X046−3z2X002−3X02−4 + z2X002+zX0X00 8 + z2X002.(D.2) Minimizing this functional we obtain a fourth order differential equation for X(z)to solve, 4h2X0+ 2(X0)3−zX00ih1+(X0)2i5−3µf2 ∞h176(X0)9+ 72(X0)11 −6z(X0)10X00 +12(X0)13 + (X0)7224 −60z3X00X(3)+zX00 −4 + z2(X00)2−12z3X00X(3) +(X0)8−38zX00 + 6z3X(4)+ 2(X0)6−41zX00 + 45z3(X00)3+ 9z3X(4)+ +(X0)4−78zX00 + 37z3(X00)3+ 96z4(X00)3X(3) + 18z3X(4)+ +(X0)2−32zX00 −52z3(X00)3+ 84z4(X00)2X(3) + 6z3X(4)+ +2X04 + 27z4(X00)4−3z4(X(3))2−3z3X00 h−2X(3) +zX(4)i −4(X0)3−14 + 36z4(X00)4+ 3z4(X(3))2+ 3z3X00 h3X(3) +zX(4)i −6(X0)5−26 + z4(X(3))2+z3X00 h18X(3) +zX(4)ii= 0 . 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