On holographic entanglement entropy of Horndeski black holes

Journal of High Energy Physics, Oct 2017

We study entanglement entropy in a particular tensor-scalar theory: Horndeski gravity. Our goal is two-fold: investigate the Lewkowycz-Maldacena proposal for entanglement entropy in the presence of a tensor-scalar coupling and address a puzzle existing in the literature regarding the thermal entropy of asymptotically AdS Horndeski black holes. Using the squashed cone method, i.e. turning on a conical singularity in the bulk, we derive the functional for entanglement entropy in Horndeski gravity. We analyze the divergence structure of the bulk equation of motion. Demanding that the leading divergence of the transverse component of the equation of motion vanishes we identify the surface where to evaluate the entanglement functional. We show that the surface obtained is precisely the one that minimizes said functional. By evaluating the entanglement entropy functional on the horizon we obtain the thermal entropy for Horndeski black holes; this result clarifies discrepancies in the literature. As an application of the functional derived we find the minimal surfaces numerically and study the entanglement plateaux.

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On holographic entanglement entropy of Horndeski black holes

HJE On holographic entanglement entropy of Horndeski Elena Caceres 0 1 3 4 Ravi Mohan 0 1 3 4 Phuc H. Nguyen 0 1 2 3 4 0 College Park, MD 20742 , U.S.A 1 Austin , TX 78712 , U.S.A 2 Maryland Center for Fundamental Physics, University of Maryland , USA 3 Theory Group, Department of Physics, University of Texas , USA 4 @ , a particular case of Horndeski gravity 5 To obtain the delta function, we used the identity @ We study entanglement entropy in a particular tensor-scalar theory: Horndeski gravity. Our goal is two-fold: investigate the Lewkowycz-Maldacena proposal for entanglement entropy in the presence of a tensor-scalar coupling and address a puzzle existing in the literature regarding the thermal entropy of asymptotically AdS Horndeski black holes. Using the squashed cone method, i.e. turning on a conical singularity in the bulk, we derive the functional for entanglement entropy in Horndeski gravity. We analyze the divergence structure of the bulk equation of motion. Demanding that the leading divergence of the transverse component of the equation of motion vanishes we identify the surface where to evaluate the entanglement functional. We show that the surface obtained is precisely the one that minimizes said functional. By evaluating the entanglement entropy functional on the horizon we obtain the thermal entropy for Horndeski black holes; this result clari es discrepancies in the literature. As an application of the functional derived we nd the minimal surfaces numerically and study the entanglement plateaux. AdS-CFT Correspondence; Black Holes; Gauge-gravity correspondence - 2 3 4 5 6 A B 3.1 3.2 3.3 4.1 4.2 5.1 5.2 5.3 1 Introduction Review of the squashed cone method Entanglement functional for Horndeski gravity Derivation from Wald entropy formula 3.1.1 Minimization Re-derivation by squashed cone method Canceling divergences of the equation of motion Comments on the thermal entropy Black hole entropy from Iyer-Wald formalism Black hole entropy from conical singularity Numerical applications 3-dimensional planar black hole 3-dimensional, spherical black hole 4-dimensional, spherical black hole Conclusions and future directions Minimal surfaces in 3d Minimal surfaces in 4d proved to be crucial as a tool to prove the Ryu-Takayanagi formula itself [10], as well as to extend the formula to cases where the bulk theory of gravity is a higher derivative or higher curvature theory [11, 12]. In this paper, we study the extension of the Ryu-Takayanagi formula in a di erent direction: the bulk gravity theory includes non-minimally coupled matter. Curiously, while the literature on holographic entanglement in higher curvature theories (such as Lovelock { 1 { gravity) is already extensive [11{17], far less attention has been paid to the case of nonminimal coupling. And the e orts have been concentrated in couplings of the type From the viewpoint of black hole entropy, non-minimal coupling is an interesting and puzzling topic. The niteness of the Bekenstein-Hawking entropy seems to indicate that the UV-divergence of the entanglement entropy can be absorbed into Newton's constant. For minimal coupling, this is indeed true, but non-minimal coupling has the potential to spoil this nice structure [18, 19]. This paper is an attempt to ll in this gap in the literature. We study a theory with a tensor-scalar coupling of the form R Horndeski gravity, despite being discovered a few decades ago [20], fell into oblivion and only received much attention quite recently (for a selection of recent work related to Horndeski gravity, see for example [21{24]). For the purpose of holography, asymptotically AdS black hole solutions of Horndeski gravity have been worked out explicitly [25{27], and that the parameter space of the couplings is reasonably well understood in terms of causality and stability [29, 30]. In this paper, we use the squashed cone method to derive the entanglement entropy functional;1 we nd that it receives a contribution proportional to the gradient-square of the scalar eld. Determining the precise form of the entanglement functional for Horndeski gravity is one of the results of this paper. We also show that demanding that the leading divergence of the tranverse, zz, component of the equations of motion vanishes implies that the entangling surface is a minimal surface. This is our second result. The question of whether the entangling surface minimizes the functional found from the squashed cone method is a topic of recent attention [32]. Until recently, there exist two main arguments that this should be the case: the argument based on the divergences of the equation of motion, and the cosmic-brane argument. Both these arguments are presented in [11] and are based on the equation of motion in the bulk. One of the novelties introduced in [32] is an argument based directly on the action, without going through the equation of motion. More speci cally, [32] considers a double variation of the bulk action (one with respect to the replica index, and a second one which preserves the strength of the conical defect), and derives the stationarity of the entanglement surface from it. In higher derivative theories requiring that the surface is minimal is not enough to cancel all the divergences. This is because in such theories generically all the components of the eld equations diverge, not only the transverse one. Furthermore, there are subleading divergences not present in Einstein gravity. Horndeski gravity is not a higher derivative theory but we do nd a similar pattern of divergences. It is possible that just like in higher derivative theories these divergences cancel if we allow for a more general ansatz [34]. One of the original motivations of the present work was related to a puzzle existing in the literature of Horndeski black holes. In [27] the authors perform a careful calculation of the thermal entropy of some particular Horndeski black holes solutions. They use the Wald formula, Euclidean regularization and the Iyer-Wald formalism and the results do not 1See also [31] for a di erent approach to deriving holographic entanglement entropy via eld rede nition, which could be applicable to Horndeski gravity. { 2 { HJEP10(27)45 agree with each other. Evaluating our result for the entanglement entropy on the horizon we are able to shed some light on this issue. The paper is organized as follows: in section 2, we brie y review the squashed cone method to derive entanglement entropy. In section 3, we apply this machinery to Horndeski gravity, and analyze the divergences of the bulk equation of motion. In section 4, we comment on the thermal entropy and the confusion found in the literature regarding this quantity. In section 5, we nd the RT surfaces numerically and study the phase transition from connected surface to disconnected surface in the case of spherical black holes. We conclude in section 6. We relegate the plots of the RT surfaces to appendices A and B. 2 Review of the squashed cone method In this section, we review the squashed cone method, an application of the replica trick to derive holographic entanglement entropy. Besides reviewing the background material, this section also serves to x the notation for the rest of the paper and to lay out the basic equations to be used later when we apply the formalism to Horndeski gravity. First, recall the microscopic de nition of entanglement entropy: Note that the Riemann surface Mn has a discrete Zn symmetry. If we assume that this discrete symmetry also exists for the bulk geometry Bn, then we can consider the quotient Bn = Bn=Zn. Since Bn has to be regular in the interior, B^n is regular except at the xed ^ { 3 { SEE = Tr( log ) obtain the entanglement entropy, the replica trick tells us to rst nd the nth Renyi entropy, de ned by: The entanglement entropy is the analytical continuation of Sn as n ! 1: SEE = limn!1 Sn. From the path integral representation of , we can express Sn in terms of the partition function Zn on an appropriate n-sheeted Riemann surface Mn as: where Z1 is the original partition function. Next, we use the basic holographic relation ZCF T = e Sbulk between the eld theory partition function ZCF T and the bulk action Sbulk. If the boundary is taken to be Mn, then this relation reads: for an appropriate bulk geometry Bn. Substituting into (2.3), we nd: 1 n 1 points of the Zn symmetry, which now forms a codimension-2 surface around which there is a conical de cit. Moreover, we have: where we do not include any contribution from the conical de cit in S[B^n]. The Renyi entropy can now be written as: S[Bn] = nS[B^n] n=1 =0 SEE = Thus, we can trade the integral outside the tube for the one inside: Again, the entropy only receives contribution from the near-tip region. The fact that the bulk action localizes to the near-tip region means it is enough to work in an approximate metric near the tip. Such a coordinate system, analogous to Riemann normal coordinates, has been worked out in [10] and [11]. The metric takes the form: By taking the limit n ! 1, we nd the following formula for the entanglement entropy: where we introduced the parameter n 1 which characterizes the strength of the conical de cit. Unlike the original replica index n which only makes sense for integer values, the conical de cit can be varied continuously. To summarize: the formula above instructs us to solve the gravity equation of motion with some conical de cit for boundary condition, then evaluate the on-shell action and extract the term rst order in . We stress that the on-shell action does not include any contribution from the conical singularity. In other words, we can integrate over the whole spacetime outside a thin \tube" enclosing the conical defect, then at the end shrink the diameter of the tube to zero. On the face of it, formula (2.8) involves an integral over the whole spacetime (the onshell action). It turns out, however, that the entropy only receives contribution from the region near the tip of the cone. This can be seen in several ways. One way is to argue that the variation of the action with involves the integral of the equation of motion over all of spacetime, but this integral clearly vanishes. This leaves us with a boundary term near the tip of the cone, which contributes to the entropy. Alternatively, one can also argue as follows. Let us call the action evaluated outside the \tube" described above Sout and the action inside the \tube" Sin. Since the total action Sin + Sout is a variation away from an on-shell con guration (due to turning on ), it must be that: ds2 = e2A[dzdz + e2AT (zdz zdz)2] + (hij + 2Kaijxa + Qabijxaxb)dyidyj + 2ie2A(Ui + Vaixa)(zdz zz)dyi + : : : { 4 { (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) Here use polar-like coordinates ( ; ) or complex coordinates z = ei , z = e i for the directions transversal to the surface, and yi for the directions along the surface. The factor e2A is given by: e2A = (zz) and encodes the conical defect. Also, Kzij and Kzij is the extrinsic curvature of the surface. Note that we expand to second order in z, z in the metric (2.11). This is su cient when the gravity equation of motion is second order in the metric, such as Einstein gravity or Horndeski gravity. For such a theory terms at most quadratic in z or z in the metric contribute to the curvature (and the on-shell action) at = 0. For higher derivative theories, it is in general necessary to keep higher order terms in z(z) in the metric (2.11). The Riemann tensor of the metric (2.11) contains the following terms rst order in : The rst one above diverges as a (2-dimensional) delta function, and the other two diverge as z 1.2 Also, the Ricci tensor and Ricci scalar contain the following terms rst order in : (2.12) (2.13) (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) (2.20) Rzzzz Rzizj Rzizj Rzz = Rzz Rzz R 4 e 2A 2(z; z) e2A 2(x; y) 2 z z Kzij Kzij 2(x; y) z z Kz Kz Using the formula (2.10) together with the approximate metric (2.11), [11] derived a formula for holographic entanglement entropy for an arbitrary theory of gravity: SEE = 2 Z d p d y h " + X 8Kzij Kzkl q + 1 # Here h denotes the induced metric on the surface (we will use g for the bulk metric), z and z are complex coordinates transversal to the surface. The rst term above is identical to the Wald entropy, except that it is not evaluated on a black hole horizon (or Killing horizon) here. As for the second term, it is an anomaly-like contribution that only arises in theories of gravity quadratic or higher order in the curvature. For Einstein gravity, only the rst term above contributes and gives (one quarter of) the area, in agreement with the Ryu-Takayanagi formula. Note that the squashed cone method as described above gives us a functional for entanglement entropy, which we are supposed to evaluate on the surface that is the xed (2)(x; y). Here x and y are the real and imaginary parts of z, respectively. { 5 { g R 1 2 (r r (r r )(r r ) + (r r )(r r ) + ) 2 R and the one for the scalar eld is: r (( g G )r ) = 0 1 2 16 1 2 ) 1 2 + 1 2 G 2 point of the Zn symmetry in the bulk. In practice nding this surface from its de nition is di cult, but | as mentioned in the introduction | there exist general arguments that the surface is also the one minimizing the functional.3 One such argument was given in [10] for Einstein gravity, as follows. We go back to the parent space Bn by making the period of the angle in the metric (2.11) 2 n instead of 2 , and extract the leading divergence of the Ricci tensor near z = z = 0. The leading divergences of Rzz and Rzz are as previously found in (2.17) and (2.18), but the delta function divergence of Rzz no longer exists because the parent space is regular at the xed point of the Zn symmetry. Thus, the zz component of Einstein equation diverges as z Kz. But, as argued by [10], we should not expect any divergence on physical grounds. If we demand that the coe cient of the divergence vanishes, then we nd Kz = Kz = 0. This is precisely the condition of a minimal surface (i.e. one which minimizes the area functional). 3 Entanglement functional for Horndeski gravity In this section, we apply the squashed cone method to Horndeski gravity. The Lagrangian of the Horndeski theory is:4 L = (R 2 ) + ( g G Here is the inverse Newton's constant (G 1). The equation of motion for gravity is: 0 = (G + g ) We will derive the functional for holographic entanglement entropy in two ways: the rst way is by using (2.20), and the second way is by going through the squashed cone method. We will of course obtain the same answer in the end. The reason for this two-pronged derivation is as follows: the formula (2.20) was technically derived in the absence of matter elds in [11], but is expected to apply even in the presence of matter elds. By deriving the entanglement functional for Horndeski gravity twice, we verify that (2.20) indeed applies and yields the same answer as the more careful derivation with the matter eld included at the outset of the squashed cone method. 3It is highly desirable that the surface minimizes the functional, since this would immediately imply that the surface satis es quantum-information properties of entanglement entropy such as the concavity [35]. 4A remark about convention here is in order. We have multiplied the Lagrangian as written in the Einstein-Hilbert part of the action into (2.20), we obtain one-quarter of the area times G 1. papers [26, 27] by an overall factor of 161 . This overall factor has been chosen so that when we plug the (3.1) (3.2) (3.3) where ; = g turned o ): simpli es to: g g ) ; = 8 32 ; ; ;z ;z = 8 32 hij ;i ;j SEE = 4 Z d p h d y h 1 4 hij ;i ;j i Upon substituting the metric components into (3.4), the partial derivative @L=@Rzzzz Upon further expanding ; ; = ;z ;z + hij ;i ;j , this further simpli es to: The functional for holographic entanglement entropy then reads: Since the Horndeski gravity action does not involve terms quadratic or higher order in the curvature, only the Wald-like term in (2.20) contributes to the entropy, and the anomalylike term does not. Let us di erentiate the Horndeski Lagrangian with respect to the 128 [g = (g g g g ) ; ; g ; ; + g ; ; g ; . We need to take the zzzz component of the expression above (in the HJEP10(27)45 coordinate system of the metric (2.11)) and evaluate on the surface (where z = z = 0). It is enough to truncate the metric (2.11) to zeroth order in z(z) (with the conical defect Thus, the entropy receives a Wald-like correction proportional to the norm-squared of the gradient of the scalar eld on the surface. Equation (3.8) is one of the results of this paper. By analogy with the RT formula, the entanglement functional is expected to be evaluated in the surface that minimizes its value. In subsection 3.3 we will show that this is indeed the right thing to do. More precisely, we will show that demanding that the divergences of the equations of motion cancel yields a condition that is exactly the equation of the surface that minimizes (3.8)!. This will be another important result of our work. 3.1.1 Minimization Let us now explicitly minimize the functional (3.8) to derive the equation characterizing the surface (the \surface equation"). To do this, we vary the embedding functions x (yi) of the surface (where x denotes the bulk coordinates and yi denotes the coordinates on the surface), and compute the variation S of the functional due to x . When the embedding is varied, the value of the functional is varied due to two e ects: ( 1 ) the change of the induced metric, and (2) the change of the value of the scalar eld on the surface. Thus, we have: S = Z ddy hij + L L hij { 7 { ; ; (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) with hij and related to x as: hij = g = Next, consider the second term in (3.9). Under the z ( z), we have: L = 8 p h hij DiDj ;a a We now adopt the coordinate system x = (yi; z; z) of the metric (2.11). This coordinate system has the nice feature that it splits into coordinates on the surface (yi) and coordinates transversal to the surface (z, z). It is enough to consider the variations z and z (i.e. in the normal direction), since these actually change the embedding, whereas the variations yi merely correspond to coordinate transformations on the surface and should not change the value of the functional. Under the variation z ( z), the rst term in (3.9) can be cast in terms of the extrinsic curvature of the surface as: L hij 1 hij = 2Kaij hij a L where a = z; z (see for example [17] for more details). For our speci c functional (3.8), this becomes after some algebra: L hij hij = p h 4G 4 together, and equating the result to zero, we nally nd the surface equation: 1 4 gravity. Note also that the horizon of a static, spherically symmetric black hole solution also satis es the surface equation above, assuming that it is also a Killing horizon (as is the case with more familiar black holes). Indeed, for a Killing horizon the extrinsic curvature Kij vanishes, and spherical symmetry also implies that the scalar eld is constant on the horizon, so that Di vanishes. Thus, the thermal entropy of the black hole should be found by evaluating the functional (3.8) on the horizon. An immediate interesting consequence of this is that the thermal entropy is equal to the usual Bekenstein-Hawking entropy (assuming again spherical symmetry): where A is the area of the horizon. Indeed, spherical symmetry implies that the Wald-like correction in (3.8) simply vanishes. Sthermal = A 4G { 8 { As pointed out in [27], however, the thermal entropy of Horndeski black holes is surprisingly subtle, with di erent computation methods yielding di erent answers. We will revisit this issue in section 4, but we assume for the rest of this paper that (3.16) is the correct one. We also stress that the Wald-like correction only vanishes for the thermal entropy, and not for entanglement entropy of a boundary subregion as in general the scalar eld is not constant on the Ryu-Takayanagi surface. Re-derivation by squashed cone method We proceed to rederive the functional (3.8) using the squashed cone method, with the scalar eld put in at the outset. We will need to know the expansion of the scalar eld near the surface, in the coordinate system of the metric (2.11). This expansion is -dependent, since the scalar eld has to readjust itself to the conical defect, and this dependence has to be fed into the on-shell action to see if it gives rise to any additional term compared to the functional (3.8). In the end, we will nd that there are no additional terms, thus the expectation that the functional (3.8) applies even with matter elds is borne out. Let us rst recall how the coordinate system of (2.11) is constructed. In the parent space Bn (i.e. before the quotienting by Zn), we set up polar-like coordinate ( ~; ~) in the plane transversal to the surface. Since the bulk has to be regular at the surface and also has the replica Zn symmetry, any ~ dependence has to be through ~ne in~. In the quotient B^n, we rede ne coordinates as follows: where the coe cients 0;0, 0;1 etc. are functions of yi. Let us explain how the di erent powers in the expansion above arise, especially the subleading terms in each of the square brackets above. Consider for example the term with 0;1. This term comes from the power ~2(n 1) in the parent space, which is consistent with regularity and replica symmetry in the bulk. Similarly, the power ( ~2)2n 2 gives rise to (zz)2 etc. Note that at = 0, each square bracket above collapses to a constant and we nd: = 0 + zz + zz + z2 z2 + z2 z2 + zzzz : : : The metric in terms of and then takes the form (2.11). For the scalar eld, we can proceed similarly. Regularity and the replica symmetry dictate that the scalar eld near the surface is a function of ~2 and ~ne in in the parent space Bn. In the quotient space, we nd | to second order in z or z | a rather complicated expansion: = [ 0;0 + 0;1(zz) + 0;2(zz)2 +: : : ]+[ z;0 + z;1(zz) + z;2(zz)2 +: : : ]z +[ z;0 + z;1(zz) + z;2(zz)2 +: : : ]z +[ z2;0 + z2;1(zz) + z2;2(zz)2 +: : : ]z2 +[ z2;0 + z2;1(zz) + z2;2(zz)2 +: : : ]z2 +[ zz;0 + zz;1(zz) + zz;2(zz)2 +: : : ](zz)1 k=0 z;k and so on. In other words, each term in the series at = 0 is a whole in nite series at = 0. This is sometimes dubbed the \splitting problem" [32, 33]. Before proceeding with the squashed cone method, we would like to make two remarks about the expansion (3.19) for the sake of clarity. First, the coe cients 0;0 etc. in this expansion are taken to be independent of . This is not strictly true; for example, the term ~nein~ in the parent space becomes nnz (1 + transformation (3.17) and (3.18). For general n, the + O( 2))z after the coordinate corrections of the coe cients are not ignorable. However, throughout this paper we work with the 0 (or n 1) regime, and these corrections are small. Moreover, even if we keep the corrections, they will not contribute to the entanglement entropy anyway. Note also that this discarding of the corrections is not speci c to the scalar eld expansion: the same treatment was applied to the metric expansion (2.11), i.e. coe cients in this expansion in general receive correction but they have been discarded for the reasons above. Our second remark is that the expansion (3.19) implies that the entanglement entropy functional depends on the scalar eld through the coe cients 0;0, 0;1, z;0, z;1. This may appear peculiar for the following reason. Our experience with the usual Ryu-Takayanagi formula for Einstein gravity, as well as the Jacobson-Myers functional for Gauss-Bonnet gravity, would suggest that the entanglement functional should be a function of the value of the scalar eld or its derivatives at = 0. On the other hand, each of the coe cient 0;0, 0;1 etc. does not really have an intrinsic meaning at P k z;k etc. do: the rst is the value of the scalar eld on the minimal surface, and the second is the derivative of the scalar eld in the z direction on the minimal surface). Fortunately, we nd that the entanglement functional indeed depends on the coe cients in (3.19) only through the sums P k 0;k = 0, P k z;k = z etc., in other words the functional only depends on quantities which \make sense" at = 0. Roughly speaking, this is because we need to look for terms rst order in in the action. While one can extract many terms rst order in from the expansion (3.19), the only terms rst order in that matter come from the curvature at the tip of the cone, i.e. the coupling of the scalar eld to the conical singularity. But the scalar eld that appear in such terms can be evaluated at = 0 since the curvature is already rst order in , therefore only the expansion (3.20) = 0 (only the sums P k 0;k, really matters. Using the scalar eld expansion together with equations (2.17), (2.16), (2.18) and (2.19) for the singular terms of the Ricci tensor and Ricci scalar, we then nd the term rst order in of the non-minimal coupling part of the action: where the subscript N M C stands for non-minimal coupling, and the superscript ( 1 ) means rst order in . We can easily argue that the terms in S( 1 ) NMC containing Kz and Kz vanish. This is because the integral is regular at the origin and we integrate over an in nitesimally small region around the tip of the cone. Thus, we are left with only the term with the delta function. The above then simpli es to: 0;i by ;i since the functional above is to be evaluated on the surface z = z = 0, where those two quantities agree. By the formula (2.10), this contributes to the entropy an amount: SE(NEMC) = 16 Z p hdd 2yhij ;i ;j We now combine the contribution above to the area functional from the Einstein-Hilbert part of the action, to obtain the total functional: SEE = in agreement with the functional previously derived from (2.20) which in this case is equivalent to the Wald entropy formula. 3.3 Canceling divergences of the equation of motion In this subsection, we go back to the parent space Bn and extract the leading order divergence in the equation of motion. We will continue to work with the metric (2.11) but with + 2 n so that there is no conical singularity. Similarly, we will use the expansion (3.19) of the scalar eld, but this expansion is now understood as an expansion in the parent space rather than the quotient space. The divergences of the bulk equation of motion come from two contributions: ( 1 ) rst, there are divergences from the components Rzizj and Rzizj of the Riemann tensor, like in Einstein gravity; (2) Secondly, the components rzrz and rzrz of the second covariant derivative of the scalar eld also contribute divergences. This is because the Christo el symbols zzz and zzz also contain 1=z divergences: Therefore, the following components of r r diverge: z zz z zz rzrz rzrz z z ;z z ;z (3.22) (3.23) (3.24) (3.25) (3.26) (3.27) (3.28) Consider rst the zz component of the gravity equation. Term by term, we nd the following contributions to the =z divergence: Gzz )(rzr Rz z 2 (rzrz ) 1 Kzhij 0;i 0;j Kzij 0;i 0;j z zz 2 z z(4 zz + hij DiDj 0) (3.29) (3.30) (3.31) (3.32) (3.33) and all other terms in the equation of motion are non-singular. Let us sketch out how to obtain some of the expressions above in details. In most of the equations above (namely, equations (3.30), (3.32) and (3.33)), the ( =z) factor comes from the metric (through the curvature tensor or the Christo el symbols), and for the scalar part it su ces to use the scalar eld expansion at = 0 (equation (3.20)).5 In the case of equation (3.31), however, the divergence coming from the metric is actually slightly sublinear, but the scalar expansion (3.19) for a slight enhancement of the divergence from sublinear to linear. To see this, note that when we expand the sum over in (3.31), the only singular term is 2gzz(rzrz )(rzrz ). Expanding the second covariant derivatives into partial derivatives and Christo el symbols, we further nd that the leading divergent part of this expression is 4 2 z ;z ;zz, which is slightly subleading compared to the =z divergence. Using (3.19), however, we nd that 6 = 0 actually results in ;zz secretly contains a factor of Thus, there is an enhancement from a slightly subleading divergence (of the order of 2 to a leading divergence ( z ), and we obtain (3.31). Note also the delicate cancellation of the terms with zz in (3.31) and (3.32). Now we add up (3.29){(3.33) and demanding that the =z divergence cancels. We then nd the condition: 1 + 4 where we dropped the subscript 0 in 0 (and replaced it simply by ), and also replaced z by ;z it is now implicitly understood that this is an equation at = 0 and at z = z = 0. 5More properly, suppose we insist on using the expansion (3.19) at nite . Then equations (3.30), (3.32) and (3.33) each will take the form of the product of ( =z) and a Taylor series in (zz) . In other words, we have a hierarchy of divergences. This is a manifestation of the \splitting problem" on the level of the divergences of the equation of motion. However, we are trying to demonstrate that the coe cient of the ( =z) yields the minimal surface as ! 0. Thus we have simply evaluated the coe cient of ( =z) at = 0, which collapses the hierarchy and amounts to plugging in the scalar eld expansion at zero epsilon at the outset. (3.34) z where : : : stands for subleading divergences (including linear). But the quadratic divergences in these two terms exactly cancel out each other! Finally, we consider the ij component. This one has a genuine quadratic divergence: 4 hij (r r )(r r 2 hij e 4A 2 zz z z Similar, the equation of motion for the scalar eld also has a quadratic divergence: G r r e 4A(Kz z + Kz z) Comparing with the surface equation (3.15), we see that this is precisely the same equation. Thus, the Lewkowycz-Maldacena prescription of xing the surface by the divergence of the equation of motion works at least with respect to the 1=z divergence of the zz component of the equation of motion. We end this section by mentioning the divergences appearing in the other components of the gravity equation of motion as well as the scalar equation. The divergence of the zz component is the same as that of the zz component, except for the substitution z ! z. The zi component has no rst order divergence, but it does have slightly subleading ones (of order z1 ). As for the zz component, two of the terms actually have a quadratic divergence: The presence of the subleading divergences as well as quadratic divergences is somewhat troublesome, but it is a feature that Horndeski theory shares with higher derivative/higher curvature gravity (see for example [15]). In fact, the divergence structure is strikingly similar: the paper [15] shows that the ij component of the equation of motion also su ers from a quadratic divergence in Gauss-Bonnet gravity. We will leave the question of how to get rid of these other divergences to future work, but the work [34] is a promising step in this direction: the authors of [34] show that an ansatz more general than (2.11) is needed to cancel the subleading divergences. We will come back to this point in the Conclusion section. 4 Comments on the thermal entropy In this section, we revisit the issue of the thermal entropy for the black hole solution in section 5. As pointed out in the literature [27], the standard methods of deriving the entropy (Wald's entropy formula, the Iyer-Wald formalism, and the Euclidean method) seem to yield con icting answers. We will make the case that the correct entropy should be the one given in (3.16), i.e. the Wald entropy (which happens to coincide with the usual Bekenstein-Hawking entropy). As shown in [27], Horndeski gravity admits black hole solutions given by: ds2 = h(r)dt2 + where d is the dimension, and = 1; 0; 1 correspond to a hyperbolic, planar and spherical horizon, respectively. Also, the constants g and are related to the couplings and in the action by: = = 1 2 (d 1 2 1)(d 2)g2 (d 1)(d 2)g2 1 + 2 4.1 Black hole entropy from Iyer-Wald formalism Following [27], let us review the computation of black hole entropy used the Iyer-Wald formalism. This formalism gives us 2 statements related to the entropy: ( 1 ) the rst law of black hole mechanics, and (2) the statement relating the integral of the Noether charge (of di eomorphism invariance) over the bifurcation surface of the black hole to the entropy. To obtain the rst statement (the rst law), we compute a closed di erential form Q , where Q is an on-shell perturbation of the Noether charge (of di eomorphism invariance), is the bifurcate timelike Killing vector eld of the black hole, and is the boundary term of the gravity action. The rst law of black hole thermodynamics then comes out as the equation: Z 1 Q = Z H Q where, on the left hand side, we integrate this di erential form on a sphere at in nity, and on the right-hand side we integrate on the bifurcation surface of the black hole. In the case of Einstein gravity, the left-hand side above coincides with the mass perturbation M , and the right-hand side coincides with T S. The second statement of the Iyer-Wald formalism tells us that: In other words, the integral of Q over the bifurcation surface equals the product of the temperature and the entropy. Z H Q = T S (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) Let us now examine what happens to these 2 statements in the case of Horndeski gravity. The Noether charge for a stationary black hole metric of the form (4.1) is given in [27]. If we write it as Q = QEinstein + Q where QEinstein is the contribution from Einstein gravity and the minimal coupling, and Q is the contribution from the nonminimal coupling, we have: QEinstein = From the above, it is easily seen that Q vanishes on the horizon. To see this, we use (4.4) to substitute for ( 0)2, and note the fact that h=f is regular on the horizon. Thus the integral of Q on the bifurcation surface reduces to that of QEinstein, and equals the product T S with S given by the Wald entropy! The fact that the identity (4.8) is consistent with the Wald entropy, i.e. consistent with the squashed cone method, should not be surprising. Indeed, there exist quite rigorous arguments (see for example [36, 37]) that the black hole entropy as derived by the conical singularity method always coincides with the entropy obtained from the integral of the Noether charge. Things do go wrong for the rst law, however: it can be seen from (4.4) that the derivative of the scalar eld in the radial direction diverges at the horizon, and the scalar eld itself has a branch cut singularity there. As the analysis in [27] shows, what happens is that the variational identity (4.7) continues to hold (since it follows from Stokes theorem), but the two sides of this equation can no longer be identi ed as M and T S (with S taken to be the Wald entropy). Explicitly, we obtain [27] for the planar case ( = 0): Z 1 Z H = = (d (d above essentially arise from the singularity of the scalar eld mentioned above. In view of this di culty, the authors of [27] proceed by simply de ning the left-hand side of (4.7) as M and the right-hand side as T S. From (4.11) and (4.12), the authors of [27] then obtain the following de nitions for the mass and the entropy: M~ = S~ = (n 1 4G While the de nitions above for M and S have the virtue that the rst law is automatically satis ed, it is important to keep in mind that they are merely de nitions. In the case of Einstein gravity, there exist indepdendent, nontrivial checks for the mass and the entropy: (4.11) (4.12) (4.13) (4.14) the mass in that case coincides with the Komar integral, and the entropy is of course the Bekenstein-Hawking entropy, which obeys the second law for example. In the case of Horndeski gravity, there are no independent checks of (4.13) and (4.14). On the other hand, we have seen that from the viewpoint of holographic entanglement, it is much more natural to take the thermal entropy to be the usual Wald entropy since this is what we obtain from entanglement entropy as the size of the boundary region approaches the whole boundary. To summarize, for planar black holes we have the following mass and thermal entropy: M = (n 2) The corresponding expressions for = 1 are somewhat more complicated, and can be found in [27]. Black hole entropy from conical singularity Even though the analysis above should assure us that the Wald entropy is correct, there is potentially a loophole because the singular behavior of the scalar eld on the horizon contradicts the assumption of regularity of the scalar eld used in the derivation of the entanglement entropy functional. Indeed, near the horizon, the scalar eld expands as: (4.15) Thus, the assumption of regularity used to derive the expansion (3.19) technically does not apply in the case of the thermal entropy. In this subsection, we take a closer look at this and argue that | despite this singularity of the scalar eld on the horizon | the conical singularity method should still yield the Wald entropy. First, we need to rede ne the radial coordinate from the Schwarzschild-like r to the coordinate used in (3.19) to facilitate comparison between the two near-horizon expansions. Note that the coordinate in (3.19) satis es two properties: the horizon is at = 0, and the near-horizon metric looks like at space in polar coordinates: ds2 = d 2 + 2d 2 when we turn o the conical singularity ( = 0). Note also that the usual steps taken to derive the Hawking temperature of a black hole involves precisely a coordinate rede nition with the two properties above, and this is the procedure we will follow to obtain the transformation from r to . Near the horizon, the functions f (r) and h(r) in the metric expand as: tion: f (r) = f1(r h(r) = h1(r r0) + O(r r0) + O(r r0)2 r0)2 for some coe cients f1 and h1. From the above we obtain the desired coordinate rede niThe near horizon expansion (4.16) in terms of now reads: = p 2 p f1 r r0 = 0 + 1 + : : : (4.17) (4.18) (4.19) (4.20) for some coe cient 1. The expansion above is supposed to replace the expansion (3.19) at = 0, so let us compare the two. The expansion above is a function of alone, whereas (3.19) at = 0 allows for angular dependence. This is expected due to the U( 1 ) symmetry of the black hole (i.e. time translation symmetry) which is not present in (3.19). Secondly, the leading power of is rst order in the expansion above, whereas it is quadratic in (3.19) at = 0. This is, of course, due to the fact that one expansion is regular near = 0 and the other is not. We also remark that, in terms of , the singularity of the scalar eld is only a \kink" near = 0 as opposed to a divergence (4.16). That the nature of this singularity depends on the coordinate used should not be surprising; moreover it was noted in [27] already that the singularity is milder than it seems, in the sense that coordinate-invariant quantities do not su er from divergences across the horizon. The challenge now is to gure out how expansion (4.20) is a ected when we turn on , because we need the expansion at nite to derive entanglement entropy. Equivalently, we need to nd out the leading power of ~ in the parent space. We remark that this is best done by rst computing the parent-space analog of the Horndeski black hole, then go to the quotient space and expanding near the horizon. This is clearly a very nontrivial task in general, and few exact solutions of this type are known (however an explicit parent-space analog of the hyperbolic black hole is known [28]). Fortunately, we can deduce the powers of ~ in the parent space based on (4.20): the smallest power of ~ consistent with a rst power in at = 0 is a rst power in ~. Thus, in the parent space: In the quotient space, this translates to: = 0 + ~1 ~ + : : : = = 0, reduces to (4.20). Of course, there are in nitely many higher powers of ~ which are also consistent with rst order in at = 0. For example, ~n gives (with no dependence) etc. Can the anomalous power of in (4.22) result in a new contribution to the black hole entropy? It is true that one can extract a term rst order in from the expansion above: = 1 log . However, the fact that log vanishes as ! 0 means this term cannot give rise to any new contribution when we plug it into the action and integrate over a small region near the tip of the cone. For example, the minimal coupling term gives: Z 0 Z 0 d d (g ; ; ) / d ( ); = 0 (4.23) similarly for the non-minimally coupled term. To summarize, we think that the correct entropy should be the Wald entropy for 3 reasons: ( 1 ) the Wald entropy is consistent with the Iyer-Wald formalism, in the sense that it is consistent with the integral of Q over the horizon; (2) a rederivation of the black hole entropy from the conical method does not seem to yield any new contribution on top of the Wald entropy, and (3) the Wald entropy coincides with the limit of entanglement entropy as the boundary region approaches the whole boundary. Note that the metric is exactly the BTZ black hole. The horizon is located at r+ = p =g. We will nd it convenient to go to the Fe erman-Graham coordinate: and rescale the boundary coordinates as = t=2 and y = x=(2g), the metric then becomes: ds2 = 1 g2 2 (g2 2)2d 2 + d 2 + (g2 + 2)2dy2 and the scalar eld in terms of z satis es: = gr + pg2r2 g d d = p g (5.1) (5.2) (5.3) (5.4) (5.5) (5.6) Horndeski gravity given in [27]. 3-dimensional planar black hole In this section, we nd numerically the entanglement entropy of black hole solutions of Let us rst specialize to the 3-dimensional, planar black hole (d = 3, = 0). The metric and the scalar eld pro le simplify to: = p g log (gr + p(gr)2 ) + 0 dr2 g2r2 ds2 = (g2r2 )dt2 + + r2dx2 Let us now parametrize the Ryu-Takayanagi surface as X = ( = const; y; (y)). The functional to be minimized then takes the form: SEE = dx Z p( 0)2 + (g2 + 2)2 g 1 + ~ ( 0)2 ( 0)2 + (g2 + 2)2 where we de ned ~ = G4 . We minimized the functional above numerically and plot in gure 1a the entanglement entropy versus the half-width ymax of the boundary interval (which ranges from ymax to ymax). For completeness, we present 3 cases: the Einstein gravity case = 0, a case with > 0 and a case with < 0, even though from the viewpoint of bulk causality is required to be non-positive [29]. In gure 1b, we present the plot of a few RT surfaces for a few di erent values of . Let us comment that all three curves in gure 1a are concave. Concavity is one of the hallmark features of entanglement entropy (in fact, of entropy of any kind), and implies the property of strong subadditivity. From the holographic viewpoint, concavity can be expected from the fact that the holographic entanglement entropy is the minimization of a functional. Indeed, the proof of strong subadditivity in the usual Einstein gravity case can be generalized in a straightforward way to any extensive functional [35]. 2.5 2.0 -0.6 -0.4 -0.2 of the boundary interval. b) Some representative minimal surfaces. In both plots we have used g = = = G = 1, and the values of are: = 0 (black), = 0:05 (light green), = 0:125 (dark green), = 0:05 (light purple) and = 0:125 (dark purple). 3-dimensional, spherical black hole Next, we consider the 3-dimensional spherical black hole (d = 3; = 1). The metric is still HJEP10(27)45 ds2 = 0 + g2r2 dt2 + dr2 ( 0 + g2r2) + r2d 2 0 = and the scalar eld pro le is identical to that of the 3-dimensional planar black hole. As usual, for a given boundary interval of half-width 0 there are two minimal surfaces satisfying the homology constraint: a connected one and a disconnected one which includes the horizon as a connected component. Intuition from Einstein gravity suggests that there exists a phase transition from the connected one to the disconnected as we increase the size 0 of the boundary region. In gure 2a, we vary the size 0 of the boundary interval from 0 to and plot the value of the functional on evaluated on the connected surface and disconnected surface. As expected, the curve for the connected surface increases monotonically while the curve for the disconnected one decreases with increasing 0 . The two curves intersect at an angle c (around 2.8 rad), at which point the phase transition happens (since the disconnected one yields a smaller value of the functional beyond this angle, and the homology constraint instructs us to use whichever surface has the smaller value). Like in Einstein gravity, this phase transition has an interesting consequence for the Araki-Lieb inequality, which tells us that the di erence between the entropy of a subregion A and the entropy of its complement is at most the entropy of the mixed state (the thermal entropy in this case): Sthermal For 0 c, the phase transition implies that the di erence above is exactly equal to the thermal entropy, hence the Araki-Lieb inequality is saturated. Put di erently, if we were (5.7) (5.8) (5.9) to plot jSA for 0 > c . 2.80 2.75 HJEP10(27)45 surfaces as a function of the boundary region 0. We have used 0 = g = = b) Critical angle c versus , also with g = 0 = = G = 1. SAC j=Sthermal as a function of 0, we would see an entanglement plateau Next, we keep the values of the constants g, and xed (to unity) and vary the value of the coupling . For each such we computed the angle c of the phase transition, and we plot in gure 2b the critical angle as a function of . In this plot we focus on the negative regime, since this is the regime consistent with bulk causality (see for instance [29]). Note that keeping g, and xed means both the metric and the scalar eld pro le are kept xed as we vary . However, because of the relations (4.5) and (4.6), this means the value of and are actually varied, so that the di erent curves belong to di erent Horndeski theories. Also, for = 0, gure 2b is consistent with the analytical value of the critical angle which was computed analytically in [38] for the usual BTZ black hole: 1 arccoth(2coth( r+)) 1 (5.10) In terms of the entanglement plateau, this means that the plateau becomes larger and larger as we make more and more negative. 5.3 4-dimensional, spherical black hole In this subsection, we present the phase transition for a higher-dimensional case: the 4-dimensional black hole. We relegate the plots of the minimal surfaces themselves to appendix B. From this appendix, a noteworthy feature of the RT surfaces is that the connected surface stops existing for su ciently large 0 . Equivalently, the disconnected surface does not exist for su ciently small 0. This feature is potentially worrisome, since it implies that the competition between the two kinds of surfaces only exist within a range of 0 smaller than (0; ). The phase transition, thus, must occur within this band. In gure 3b, we plot the angle c of the phase transition as a function of the nonminimal coupling at xed g (which corresponds to the AdS lengthscale at in nity) and xed temperature T . The plot is quite similar to the 3-dimensional one 2b. Like in the 3-dimensional case, we focus on the negative regime. 2.00 1.99 -0.8 -0.6 -0.2 -0.4 (b) surfaces as a function of the boundary region 0 . We have used g = = = 2:711055. b) Critical angle c versus , with g = = = 1. 6 Conclusions and future directions In this paper we obtained the holographic entanglement entropy functional for a particular class of gravity with tensor-scalar coupling, Horndeski gravity. We nd that the entanglement entropy receives a Wald-like contribution proportional to the gradient-square of the scalar eld. We show that, as in Lewkowycz-Maldacena, demanding that the divergence of the zz component of the bulk equation of motion vanishes allows us to identify the surface where to evaluate the entanglement functional. This surface turns out to be the one that minimizes said functional. We also pointed out the existence of other divergences that deserve more study: quadratic divergences in other components of the equation of motion and subleading divergences in the zz component. As an application of the entanglement functional found, we present explicit minimal surfaces for black hole solutions and show that they exhibit similar features to the ones observed in Schwarzchild-AdS: the connected surface ceases to exist for su ciently large boundary region and there exist subdominant saddles. We also study the phase transition due to the exchange of dominance between connected and disconnected surfaces. We show that the size of the entanglement plateau (at xed temperature) depends on the non-minimal coupling. The thermal entropy of the Horndeski black holes we study here was not well established. It has been previously investigated in the literature but di erent methods of calculating it yielded di erent results. We used the entanglement entropy functional derived here to shed light on this issue. We identi ed an oversight in the literature and determined the correct thermal entropy. Let us conclude with some future directions. Other divergences. As previously mentioned in section 3.3, the cancellation of the Tzz divergence implies that the entanglement functional should be evaluated on the surface that minimizes it. However, similar to what occurs in higher derivative theories, there are divergences in other components of the equations of motion. In [34] the authors showed that in the case of Gauss-Bonnet gravity these other divergences cancel if a more general ansatz is taken. This ansatz includes two types of new terms compared to (2.11): terms that break replica-symmetry, and terms that can be gauged away at = 0 (but not at nonzero ). It is the latter kind of new terms that are responsible for the cancellation of subleading divergences in the case of Gauss-Bonnet gravity. It would be very interesting to investigate if a similar cancellation occurs in the case of scalar-tensor gravities. Splitting problem. As mentioned previously, the splitting problem has to do with the fact that the scalar eld on the minimal surface splits into a sum of di erent contributions when we turn on . Investigating this splitting pattern further is of interest. Such an investigation was carried out in the simpler case of dilaton gravity in [32] by solving the equation of motion at 0 near the tip of the cone. We also expect that a resolution of the splitting problem will shed some light on the problem of cancelling divergences of the equation of motion mentioned in the previous point. Field theory dual. Identifying the precise dual of Horndeski gravity is an open question that deserves study. In particular, carrying out the holographic renormalization programme for Horndeski gravity seems an important and attainable goal. Causal wedge. The causal structure of Horndeski gravity has been extensively studied [29, 30]. In the context of holography, it is understood that the RT surface should lie on the causal shadow in order for the boundary theory to be causally well de ned. In [39] it was proven that this is indeed the case if we consider Einstein gravity. It would be interesting to verify if this is also the case in Horndeski or more general scalar-tensor theories. This causality constraint could be used to rule out certain scalar-tensor theories from having QFT duals. Conformally coupled theories Conformally coupled black holes have been knnown for quite some time [40{43]. It would be interesting to derive, following the same approach as in the present work, the entanglement functional relevant for those theories. Quantum corrections 1=N quantum corrections to the entanglement entropy involve Sbulk i.e. the entanglement entropy of the entanglement wedge with the rest of the spacetime, in a manner reminiscent of the generalized entropy of black holes. The scalar-tensor coupling will surely contribute to Sbulk. Acknowledgments We would like to thank Joan Camps for reading the manuscript and giving us useful comments. This material is based upon work supported by the National Science Foundation under Grant Number PHY-1620610 and was performed in part at Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. E.C. would like to thank Centro de Ciencias de Benasque Pedro Pascual for its hospitality while this work was completed. E. C. was also supported by Mexico's National Council of Science and Technology (CONACYT) grant CB-2014-01-238734 and by a grant from the Simons Foundation. d = 4, g = Minimal surfaces in 3d We plot in gure 1b a few Ryu-Takayanagi surfaces for the 3-dimensional planar black hole for various values of at xed g, and . In this appendix, we elaborate further on this plot. For the Einstein case = 0, the surface (black curve in gure 1b) is given by the analytical expression (we set g = = 1): z(y) = s 1 K cosh (2y) 1 + K cosh (2y) (A.1) Here K is an integration constant. Although this may not be apparent from gure 1b, the near-boundary region of these surfaces is qualitatively di erent depending on whether is negative or positive. For positive , the surface is always perpendicular to the boundary, while for negative this is not the case: the surface does not approach the boundary at right angle for negative . This fact can appear surprising, but it comes from the contribution of the scalar eld to the entanglement entropy: the term is basically the norm-squared of the gradient of the scalar eld on the surface, and this term occurs with a negative sign in the functional (3.8). Note that the norm-squared of any vector is positive in a Riemannian metric. Hence, the sign of the term in the functional (3.8) is the opposite of the sign of the coupling itself. For negative , the functional is minimized if the magnitude of hij ;i ;j is minimized on the surface. But since is only a function of z, the quantity hij ;i ;j has maximal magnitude if the surface approaches the boundary perpendicularly. Therefore, in the case of negative , the surface should approach the boundary at some angle less than =2 to keep the magnitude of hij ;i ;j small. On the other hand, for positive , the functional becomes smaller if the magnitude of hij ;i ;j becomes larger. In this case, the surface should approach the boundary perpendicularly because this is when hij ;i ;j is maximal. As for the global black hole, we present in the left panel of gure 4 the connected minimal surface for various boundary interval (from very small to the whole boundary circle) for a negative value of . Like the usual minimal surface of the Einstein-gravity BTZ black hole, the connected surface can wrap around the horizon all the way. G = 1, Minimal surfaces in 4d In this appendix, we present the plots of the RT surfaces for the 3+1 dimensional (spherical) Horndeski black hole, with the boundary region taken to be a disk 0. These surfaces behave qualitatively di erent from the ones in 2+1 dimensions in 2 ways: The connected surface does not exist for all values of 0. There exists a critical value m such that no connected RT surface exists for 0 > m. In general, the threshold m depends on the numerical values of the coupling constants, in particular . In the right-hand side of gure 4, we plot a few connected RT surfaces for various sizes of the boundary region, up to the critical value m. There exists subdominant saddles which come closer to the horizon than the threshold surface 0 = m. For the same boundary region, we may have more than one RT surfaces: the dominant one and the subdominant one. We illustrate this in gure 5. 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Elena Caceres, Ravi Mohan, Phuc H. Nguyen. On holographic entanglement entropy of Horndeski black holes, Journal of High Energy Physics, 2017, 145, DOI: 10.1007/JHEP10(2017)145