Holographic entanglement entropy under the minimal geometric deformation and extensions

The holographic entanglement entropy (HEE) of the minimal geometrical deformation (MGD) procedure and extensions (EMGD), is scrutinized within the membrane paradigm of AdS/CFT. The HEE corrections of the Schwarzschild and Reissner--Nordstr\"om solutions, due to a finite fluid brane tension, are then derived and discussed in the context of the MGD and the EMGD.

are strongly-coupled. According to the membrane paradigm of AdS/CFT, that has been used to realize the deformation method, a finite brane tension plays the role of the brane energy density, σ. There is a fine-tuning between σ, and the running brane and bulk cosmological parameters [8]. Systems with energy E σ neither feel the self-gravity effects nor the bulk effects, which then allows the recovery of GR in such a regime. An infinitely rigid brane scenario, representing the 4D brane manifold, can be implemented in the σ → ∞ limit. The most strict brane tension bound, σ 2.83 × 10 6 MeV 4 , was derived in the extended MGD (EMGD) context in Ref. [21].
The Gauss-Codazzi equations can be used to represent the brane Ricci tensor to the bulk geometry, when the discontinuity of the extrinsic curvature is related to the brane stress-tensor. Hence, the bulk field equations [17] yield the effective Einstein's field equations on the brane, whose corrections consist of a byproduct of an AdS bulk Weyl fluid. This fluid flow is implemented by the bulk Weyl tensor, whose projection onto the brane, the so-called electric part of the Weyl tensor, reads where hµν denotes the projector operator onto the brane that is orthogonal to the 4-velocity, u µ , associated to the Weyl fluid flow. Besides, U = − 1 6 σEµν u µ u ν is the effective energy density; 3 h ρσ hµν Eµν is the effective non-local anisotropic stress-tensor; and the effective non-local energy flux on the brane, Qµ = − 1 6 σh ρ µ Eρν u ν , is originated from the bulk free gravitational field. Local corrections are encoded in the tensor [17]: where Tµν is the brane matter stress-tensor. Higher-order terms in Eq. (2) are neglected, as the brane matter density is negligible. Denoting by Gµν the Einstein tensor, the 4D effective Einstein's effective field equations read Since Eµν ∼ σ −1 , it is straightforward to notice that in the infinitely rigid brane limit, σ → ∞, GR is recovered and the Einstein's field equations have the standard form Gµν = Tµν . On the other hand, the AdS/CFT setup yields the effective equations on the brane [40,41,42,43,44]: where l = 4/K (here K is the trace of the extrinsic curvature tensor) and Γ CFT corresponds to the effective action of CFT in the boundary, whose trace anomaly reads [43,44]: where Rµν and R are the Ricci tensor and scalar of the four-dimensional metric. The quantity Sct encodes R 2 terms of the counter-term, making the action finite, and δSct/δgµν is traceless, Then, the trace part of Eq. (4) reads R = −8πG 4 T − l 2 4 Rµν R µν − 1 3 R 2 . Hence, in the linear order the energy-momentum tensor of CFT is governed by the electric part of the Weyl tensor [41,42,43]: The effective Einstein's equations read where G N = p/mp, with mp and p the four-dimensional Planck mass and scale, respectively and Λ is the cosmological constant, which will be neglected hereafter. The effective stress tensor in Eq. (8) contains the matter energy-momentum tensor on the brane, the electric component of the Weyl tensor and the projection of the bulk energy-momentum tensor onto the brane [8]. For static and spherically symmetric metrics, compact stellar distributions in 4D, which must be solutions of Eq. (3), can be described in Schwarzschild-like coordinates as ds 2 = −e ν(r) dt 2 + e λ(r) dr 2 + r 2 dΩ 2 , The MGD provides a solution to Eqs. (8) by deforming the radial metric component of the corresponding GR solution [12,13]. For the GR Schwarzschild metric, and dismissing terms of order σ −2 or higher, one obtains [12] e ν(r) = 1 so thatr > r − for any > 0. For studying the Hawking radiation, one is interested in the region outsider, that effectively acts as the event horizon, and just note that r − is not a (Cauchy) horizon [12]. We just mention in passing that an explicit expression for in terms of σ −1 can be obtained by first considering a compact source of finite size r 0 and proper mass M 0 [12,11], and then letting the radius r 0 decrease belowr. However, for practical purposes, it is more convenient and general to show the dependence on the length . For example, observational data impose bounds on the length , from which bounds on σ can be straightforwardly inferred according to the underlying model [16,20]. The MGD and EMGD black holes were respectively used in Refs. [6] and [34] to explore the observational signatures of SU(N ) dark glueball condensates and their gravitational waves.
A more general solution for the exterior radial metric component was derived in Ref. [5], under the extended minimal geometric deformation, EMGD, with where k is a constant known as the exponential deformation parameter. Naturally, k = 0 results no temporal geometric deformation, being directly associated with the Schwarzschild metric when σ → ∞. For k = 1, one has [5] e ν(r) = 1 − 4M . Now, in order to the radial metric component asymptotically approach the Schwarzschild behavior with ADM mass M 1 = 2M , e −λ(r) ∼ 1− 2M1 r +O(r −2 ), one must necessarily have κ 1 = −2M . In this case, the temporal and spatial components of the metric will be inversely equal to each other (as it is the case of the Schwarzschild solution), containing a tidal charge Q 1 = 4M 2 reproducing a solution that is tidally charged by the Weyl fluid [45]: It is worth to emphasize that the metric of Eq. (14) has a degenerate event horizon at r h = 2M = M 1 . Since the degenerate horizon lies behind the Schwarzschild event horizon, r h = M 1 < rs = 2M 1 , bulk effects are then responsible for decreasing the gravitational field strength on the brane. Now the exterior solution for k = 2 can be constructed, making Eq. (12) to yield where Q 2 = 12M 2 and M 2 = 3M . The radial component, on the other hand, reads where the coefficients cm ≡ cm(M 2 , Q 2 , s) are The asymptotic Schwarzschild behavior is then assured when s = −M 2 /96. In this case, the degenerate event horizon is at re ≈ 1.12M 2 [5]. Hence, the bulk Weyl fluid weakens gravitational field effects. The classical tests of GR applied to the EMGD metric provide the following constraints on the value of the deformation parameter, k 4.2 for the gravitational redshift of light. The standard MGD corresponds to k = 0, whereas the Reissner-Nordström solution represents the k = 1 case with the ADM mass M 1 , instead.

HEE in MGD spacetimes
The EE S A in QFTs represents the von Neumann entropy of the reduced density matrix, when one spreads degrees of freedom inside a 3D spacelike submanifold B in a given 4D QFT, which is a complement of a manifold A. S A ia responsible to quantify the correlation between A and B, seen as two physical subsystems. In other words, S A corresponds to the entropy observed in A, by an observer that has no access to B. The EE does not vanish at the zero temperature limit (Fig. 1). As the amount of information in the subsystem B can be computed by the EE S A , one may argue which component of the AdS 5 bulk is in charge for computing S A in the dual gravity.
The definition of EE can be implemented, once one considers QFTs [36]. At zero temperature, the quantum system is described by the pure ground state |Ψ . Then, the density matrix is that of the pure state ρ tot = |Ψ Ψ |. The von Neumann entropy of the total system is clearly zero S tot = −tr ρ tot log ρ tot = 0. Splitting the total system into two subsystems A and B, the observer that has access only to the subsystem A will feel as if the total system is described by the reduced density matrix ρ A = tr B ρ tot . Now one defines the EE of the subsystem A as the von Neumann entropy of the reduced density matrix ρ A , namely, S A = −tr A ρ A log ρ A . If the density matrix ρ tot is pure, then as B is the complement of A, it follows that S A = S B . This equality is violated at finite temperature. One can find the subadditivity relation, More precisely, considering a QFT on a 4D spacetime splitting, R × Σ 3 , into timelike vector field and a 3D spacelike manifold, Σ 3 . Define a 3D submanifold B ⊂ Σ 3 at fixed time t = t 0 as the complement of A with respect to Σ 3 . The boundary ∂A of A, divides the manifold Σ 3 into two complementary submanifolds A and B. As the EE diverges in the continuum limit, an UV cutoff a is needed. Then the coefficient of the divergence is proportional to the area of the boundary ∂A, where α is a constant. Employing the Poincaré metric of AdS 5 with radius R, the dual CFT 4 is supposed to live on the boundary of AdS 5 which is R 1,3 at z → 0 spanned by the coordinates (x 0 , x i ). The bulk conformal coordinate z in AdS 5 is interpreted as the length scale of the dual CFT 4 . Since the metric diverges in the limit z → 0, we put a cutoff by imposing z ≥ a. Then the boundary is situated at z = a. Although AdS/CFT is based on an AdS spacetime (19), it can be also used to any asymptotically AdS 5 spacetime, encompassing AdS black branes. Now we are in a position to present how to calculate the entanglement entropy in CFT 4 from the gravity on AdS 5 . In the setup (19), one extends ∂A to a surface γ A , such that ∂γ A = ∂A. One has to choose the minimal area surface among them. In this setup the EE S A in CFT 4 can be computed [36,37,38].
To choose the minimal surface as in (20) means that one defines the severest entropy bound [46] so that it has a chance to saturate the bound.
There is an identification of the 4D entanglement entropy QFT with a certain geometrical quantity in 5D gravity, then generalizing the black hole entropy. In the particular case of the membrane paradigm, this identification implements the relationship between black hole entropy and entanglement entropy in the induced gravity setup [46].
We will study the HEE from two perspectives: the MGD, in this section, and the EMGD solutions, in the next one. For both of them, one needs to understand how the first law of HEE holds in the context of the membrane paradigm. The dual theory can be defined on a boundary located at two kind of distance ranges: (i): far from the horizon -a finite large radial coordinate denoted by r∞, and (ii): almost on the horizon -a small displacement from the horizon, named δr ≡ r −r, wherer is the horizon situs on spacetime. The MGD HEE will be implemented under these perspectives and scrutinized in what follows. The metric in Eq. (9) is employed, where the temporal and radial components are respectively set by Eqs. (10a) and (10b).

Far from the horizon
In the region far from the horizon, the boundary manifold is placed at r = r∞ that is far away from the event horizon. Let one considers a circle, in spherical coordinates defined by the azimuthal angle θ = θ 0 , responsible to enclose the entangling surface. The radial coordinate function, r = r(θ), describes the minimal surface whose boundary is the entanglement surface. In addition, the minimization of the area function, with boundary condition r(θ 0 ) = r∞, plays a prominent role in computing the minimal surface.
Obtaining the global minimum of the area yields the HEE, by employing Eq. (20). Eq. (21) reads where L MGD = 2πr (1 − y 2 )Fṙ 2 + r 2 1/2 , y = cos θ and y 0 = cos θ 0 . The dot designates the derivative with respect to y and F = F (r(y)) ≡ e λ(r(y)) . Applying the variational method, one varies Eq. (22) with respect to r(y), yielding the following ODE: Eq. (23) is strongly nonlinear. Therefore, a way to attenuate it is to attribute F ≡ F (r(y)) = 1, to yield r = w 0 /y as the simplest solution to be achieved. In addition, according to Ref. [47], one can derive nontrivial solutions of Eq. (23), working with series expansions, respectively for F (r(y)) and r(y): Here ε denotes a small dimensionless parameter, relating the black hole mass M to r∞ by ε = M r∞ .
The O(ε) terms in the expansions (24a) and (24b) may indicate corrections regarding the black hole collapse itself. It is worth to emphasize that the 0 th -order term, r(y) = w 0 /y, in (24b) is the solution corresponding to F = 1. Now, considering the F function for the MGD spacetime, encoded in Eq. (10b), one finds, up to the 2 nd -order in the g j (y) functions in the series (24a), where, due to dimensional analysis, the MGD parameter related to the expansion parameter can be written as = ξM. Higher order terms in Eq. (24a) can be forthwith derived. The set of auxiliary functions {g 1 (y), g 2 (y), . . .} in Eq. (25) is important to solve Eq. (23) order by order [47,48]. We intend here to pursuit the possible modifications to the HEE up to the 2 nd -order. Hence, the calculation of the r-functions immediately follows, which are necessary to provide the HEE corrections up to 2 nd -order. The 1 st -order ODE, taking 1 st -order terms in ε, reads Eq. (26) carries D 1 and D 2 as constants of integration, whose values are determined by the finiteness condition. Hence, to avoid divergences at y = 1, as y = cos θ ∈ [cos θ 0 , 1], one needs to set D 2 = (2− ξ)r∞. Besides, using the boundary condition r 1 (y) = 0 yields D 1 = (ξ−2)r∞{y 0 +2 log[y 0 /(1+y 0 )]}. Therefore, the first r-function reads Importantly, there is a subtle restriction due to limitations in the perturbative expansion, as aforementioned in Ref. [47]. In fact, the y = 0 point is never reached. Hence, the validity of the solution r 1 (y) is contained in the interval θ 0 < π/2 or, equivalently, y ∈ (0, 1). Going to the 2 nd -order in ε, and employing the r 1 (y) solution in Eq. (27), yields r 2 (y) + 5y 2 − 3 y (y 2 − 1)ṙ 2 (y) + 3y 2 − 1 y 2 (y 2 − 1) r 2 (y) = P(y) . with Proceeding analogously as in the solution of Eq. (26) implies that where Once more, computing of D 4 and D 3 requires the preclusion of divergences at y = 1 and the boundary condition r 2 (y 0 ) = 0, respectively. With this setup, they read with J(ξ) = 32(ξ − 2) 2 . Hence, the complete form of the second r-function is given by As the last step, we proceed to the expansion L MGD = L MGD 0 +εL MGD 1 +ε 2 L MGD 2 +· · · , within the formula for the area shown in Eq. (22). From now, as formely mentioned, the r-functions are employed to compute each order of the contribution for the HEE, S MGD = S 0 + S MGD 1 + S MGD 2 + · · · . Besides this expansion will be considered, including terms of 2 nd -order. Next, the detailed computation of each order is provided.
For the 0 th -order, one has the following expression: whereas the 1 st -order reads Compared with the results obtained in Ref. [47], our results show an interesting novelty. Although the 0 th -order term of the entanglement entropy remains the same, the 1 st -order corrections for the HEE display the MGD parameter, ξ, which carries the signature of the finite brane tension, within this order of correction, into the HEE. The general relativistic limit, σ → ∞, yields ξ → 0, recovering the 1 st -order correction to the HEE in Schwarzschild spacetime. Besides, the 0 th -order of the entropy is proportional to r 2 ∞ , since w 0 = r∞ cos θ 0 , whereas the 1 st -order one is proportional to r∞, with the MGD parameter increasing the numerical factor. This indicates a small contribution of the 1 st -order, compared to the 0 th -order -as pointed out in [47] -even in the presence of the MGD parameter ξ.
To analyze the signature of the MGD parameter on the correction, at a given order, in the HEE, a new quantifier can be introduced. We define the n th -order corrections ratio as Now, the next order reads One can notice the contribution of the MGD parameter, encoding the finite brane tension, as one compares with the HEE for the Schwarzschild spacetime, corresponding to → 0 and, hence, ξ → 0. Henceforth, in the general relativistic case of a rigid brane, σ → ∞, one recovers the 2 nd -order correction for Schwarzschild spacetimes. On the other hand, the 2 nd -order corrections ratio are given by Both corrections, the 1 st -and the 2 nd -order ones, have the MGD parameter as a dominant variable, when considering the minimal surface in large range, correspondly, the lower limit very close to zero. The 1 st -order ratio does not depend on such range. However, the 2 nd -order ratio has the limit As ξ < 0, it is observed an increment of the value of this order of correction to the HEE. Irrespectively of the limit taken, the limit ξ → 0 recovers the 2 nd -order correction for the HEE in a Schwarzschild spacetime. In general, the ratio depends on the finite brane tension and the lower limit of the minimal area. Fig. 2 displays such behavior. It is particularly important to notice that, since ξ < 0, a decrement of such contribution is observed, providing another relevant signature of the MGD parameter.
Here, lower values of the brane tension contribute to diminish the HEE in MGD black holes. By completeness, let us examine a restriction on ξ to obtain the 2 nd -order contribution to the HEE in both MGD and Schwarzschild spacetimes. In such situation, the equality Φ MGD 2 = 1 holds, whenever the terms on the rhs of Eq. (37) equal to 1. Let us denote the values of ξ (eventually dependent on y 0 ) that satisfy this condition by ξ 0 . Looking at Eq. (37), there are two solutions: the trivial one, ξ 0 = 0, and ξ 0 (y 0 ) = 6 + 16 This result is quite relevant. In fact, the MGD parameter could produce an equal correction ratio, depending on the lower limit of integration to compute the minimal area. However, as ξ < 0, such an exclusive value is not allowed, due to the fact that ξ 0 (y 0 ) > 0, for any value of y 0 in (0, 1). Besides, Fig. 3 displays the behavior of the 2 nd -order correction to the HEE in MGD spacetimes. It shows that the order of the correction in MGD spacetime is always negative and more intense than the same order of correction in Schwarzschild spacetime. This fact could be noticed by realizing the positivity of the ratio between both of them. Fig. 4 considers the 2 nd -order correction for the MGD spacetime, by fixing ξ and y 0 to different values of the black hole mass. For comparison, Fig. 5 displays the increment of the 2 nd -order correction in a Schwarzschild black hole, as a function of the mass.
One can notice the increment of this order of correction as the black hole mass increases and, simultaneously, the decrement of y 0 , which contributes with the extension of the minimal area. The smaller the brane tension, the greater the magnitude of correction in this order is, even with a minimal surface of small size. Besides, Fig. 6 illustrates the behavior of the HEE 2 nd -order corrections in both MGD and Schwarzschild spacetimes, whereas the minimal surface size is a function of the black hole mass, M .   A small value of the brane tension contributes to the increment of the HEE 2 nd -order correction in a MGD spacetime more intensely than the same correction in Schwarzschild spacetimes. The surface representing the HEE 2 nd -order correction in Schwarzschild spacetimes has an almost steady declination, when compared to the declination to the HEE 2 nd -order correction in a MGD spacetime.
Finally, one can notice the first law of HEE, as δS = S − S 0 ∝ M , regarding a vast range of the brane tension, within precise phenomenological bounds [20,21].
With the ρ-functions, we can compute and analyze the area of the entangling surface. First, the expansion of the integrand in Eq. (42) is adopted after the appropriate expansion in ε, Inserting Eq. (47) into Eq. (42) and executing the expansion of A, which reads A = A 0 + A 1 + A 2 + . . . , that is, the expansion ofL MGD , implying that corresponding HEE corrections yield The calculation of S MGD 2 is awkward enough to handle analytically. For solving it numerically, we plot the S MGD 2 function in Fig. 7, for different values of α. With the MGD parameter = 0, meaning α = 0, one recovers the HEE 2 nd -order correction for a Schwarzschild black hole. As the MGD parameter increases, one can observe the displacement -upwards and to the left -of the maximum of this order of correction looking at Fig. 7, as y 0 decreases. This means that the MGD HEE 2 nd -order correction increases simultaneously to the requirement of the extension of the range of integration, that is, the size of the dual quantum subsystem.

HEE in EMGD spacetimes
As the HEE was already scrutinized in the last section for the MGD solution, the next step is to analyze the HEE for the EMGD metrics, where the notations EMGD 1 and EMGD 2 are adopted for the k = 1 and k = 2 cases, respectively.

EMGD k = 1 case
The EMGD k = 1 case, represented by the solution in Eq. (14), deals with the ADM mass M 1 and the tidal charge Q 1 , being a Reissner-Nordström-like metric.

Far from the horizon
Considering such boundaries far away from the horizon, the outcomes for the HEE corrections are similar to those ones found in Ref. [47], once the direct replacements M → M 1 and Q 2 → Q 1 -up to the 2 nd -order correction of HEE-emulate the results presented in [47]. Therefore, the 1 st and 2 nd -order corrections read We opt not to display the 0 th -order, as it is the same as the one presented in Ref. in [47], being independent of the ADM mass M 1 , for this case. Assigning the ADM mass M 1 and tidal charge Q 1 to the mass parameter M , which is the black hole Misner-Sharp mass function in the Reissner-Nordström metric, the contribution from the MGD can be then closer investigated. Hence, after those respective identifications, one gets Such factor varies independently of the mass parameter, M , and lim y0→0 Φ EMGD1 Fig. 8 shows the global profile of this factor. The Φ EMGD1 2 function is not monotonic, presenting an inflection point. Looking closely to values of y 0 , one may observe the transitions from an initial increment to an intermediate lowering, and next, increases again. Fig. 9, which magnifies Fig. 8 for y 0 near the origin, displays this behavior. In addition, there are two brief and important features to emphasize. Firstly, at the first sight, inspecting Eq. (50) and setting Q 1 → 0, one promptly verifies that the 2 nd -order correction, considering the ADM mass related to the mass parameter M , is four times the same order correction to the Schwarzschild spacetime. Second, the HEE 2 nd -order correction in the EMGD 1 case, related to the mass parameter M , is always negative. It can be interpreted as an increment of attenuation in the entropy function, as the HEE 2 nd -order correction in the Schwarzschild spacetimes is also negative.
Hereon, let us take a look at the mass parameter after choosing a specific size of the entangling surface, which means to delimitate the minimal area. For y 0 values close to zero, the increment of Fig. 8 Global profile of the factor between the HEE 2 nd -order corrections with respect to y 0 . Fig. 9 Profile of the factor between the HEE 2 nd -order corrections when y 0 is close to 0. the mass parameter M , accentuates the 2 nd -order contribution for EMGD 1 , when one works with an entangling surface with a specific size. On the other hand, there is no such accentuation when the y 0 integration limit equals 1, even when the black hole mass increases. It is very illustrative to display the profile of such correction in Fig. 10, to compare with the same order of correction of the Schwarzschild black hole displayed in Fig. 5. As the black hole mass increases, the attenuation becomes greater. In addition, the attenuation increases faster for small values of y 0 . Otherwise, the attenuation continues to increase in a slower rate. Let us implement the same procedure for the 2 nd -order correction in EMGD 1 related only to M .
One can notice that the same analysis can be accomplished to the EMGD 1 related only to M . Moreover, the attenuation is more intense in the EMGD 1 case, when compared to the Schwarzschild one. It is also worth to emphasize that such analysis considered the tidal charge and the ADM mass as functions of the mass parameter M . To clarify this point, we take two values for y 0 , one of them close to 0 and another one close to 1, displaying both corrections in Fig. 11. Finally, both corrections can be plotted making M and y 0 to run in their specific ranges, as shown in Fig. 12. As one can observe, a more restrictive interval for y 0 is considered, to realize the profile of each minimal surface. It is straightforward to observe how the range of integration characterized by y 0 establishes a major difference between both 2 nd -order corrections, as the black hole mass increases. On the other hand, the difference is insignificant when the size of the minimal surface is reduced as y 0 increases.

Almost on the horizon
From now on, we initiate the analysis of the EMGD 1 black hole entropy, concerning the boundary almost on the horizon. The solution for this case is based on the metric in Eq. (14). According to Ref. [5], this metric corresponds to an extremal black hole, which has degenerate horizons represented byr = M 1 . In this sense, the functions describe the constant t-fold induced metric as which is built with the variable change q(ρ) = ρ 2 +r. Above, one also denotes p(ρ) = 4(ρ 2 +r) 2 ρ 2 . Proceeding to the computation of the area functional and its minimization yields the highly nonlinear ODE,ρ Next, similar steps implemented from Eq. (42) to Eq. (44) will be employed. In fact, it consists of a perturbation procedure to obtain an approximated solution up to 2 nd -order of Eq. (55). The expanded ODE is awkward and difficult to solve through analytical methods. On the other hand, one can look at the 0 th -order in ε, which is We employ the boundary conditions, constraining Eq. (44), to filter the infinite possible analytical solutions to Eq. (56), implying that ρ(y) = ρ 0 .
In full agreement with [47], such constant solution is the only one that attends strictly the boundary condition. It disposes quite differently of the Schwarzschild or MGD spacetimes looking for a minimal surface almost on the horizon. Such so restrictive solution only could emphasize that Eq. (55) needs to be investigated at higher orders, once the constant solution shown by Eq. (57) is not a solution of the full Eq. (56). Finally, we reinforce the solution Eq. (57) as a completely safe one, up to 2 nd -order. Thus, with the solution (57), we are able to estimate the entropy as follows: and Eq. (57) yields which has R EMGD1 Bound = ρ 2 0 +r representing the boundary surface radius. Sincer = M 1 = 2M , the entropy is increased, compared to that one established for the extremal RN black hole in Ref. [47]. Such an entropy increment is explicit through the ratio standing S extRN = π/2 (1 − y 0 ) ρ 2 0 + M 2 as the entropy of an extremal RN black hole, where the horizon isrextRN = M . With that, we obtain the entropy gain without any mention to the range of the minimal surface. Importantly, the ratio is positive, indicating the increment of the entropy in the EMGD 1 scenario for extremal black holes. Fig. 13 points out such profile. Fixing ρ 2 0 provides a first range with a fast-growing entropy until M = 10ρ 2 0 . After this, there is a very slow-growing, stabilizing at a ratio equal to 4. On the one hand, it does not matter how large the black hole is, the ratio stabilizes at 4, even with the displacement of the extremal horizon in the EMGD 1 case. On the other hand, entropies of black holes with 10 −2 ρ 2 0 M 20ρ 2 0 have meaningful increments, which shows simply and directly the contribution from the EMGD 1 approach.

EMGD k = 2 case
We settle here an analogue construction to the one in Sect. 3.1, using the EMGD metric with the temporal and radial components respectively given in Eqs. (15) and (16), for k = 2.

Far from the horizon
Let us consider the steps in Eqs. (21) and (22). The replacement L MGD → L EMGD2 is then necessary, as a distinct F must be taken into account. In fact, using the EMGD 2 metric, one gets a quite similar ODE shown in Eq. (23), which is the metric radial component. Once again, that similar Eq. (23) with the current F must be solved perturbatively. Before it, one establishes the parameter of expansion ε = M 2 /r∞ and the corresponding parameters Q 2 = κ 2 M 2 2 and s = ωM 2 , that follow a similar reasoning of the previous cases 1 .
Applying a similar procedure realized in Sect. 3.1, we need to compute the auxiliary g-functions for the series expansions necessary to find the r-functions, which are crucial to calculate the HEE corrections up to 2 nd -order. Hence, applying the expansions (24a, 24b), we find the g-functions: that are necessary functions to find the respective ODEs that lead us to determine the r 1 (y) and r 2 (y) functions. Each one of them is solved strictly as engaged in Sect. 3.1, using the boundary conditions to compute the constants of integration for each function. Hence, at 1 st and 2 nd orders, as follows, it implies respectively thaẗ whose solution is which has solution given by Once again, we use the r-functions to proceed with the expansion of L EMGD2 towards the computation of the area and, consequently, the HEE expression up to 2 nd -order. Thereupon, the 0 thand 1 st -order of the HEE corrections are, respectively, It is worth to emphasize that Eq. (69) has the presence of the EMGD 2 parameter. It is quite different, compared with the k = 1 case, where there is no EMGD 2 parameter in such order of correction. One can notice a growth like the 1 st -order correction from [47] as well as succeeded in the EMGD 1 . This occurs due to the ADM mass, which corresponds to 3M in this k = 2 case. Hence, S EMGD2 1 = S MGD 1 . Again, there is no contribution from the charge as well as one noticed in Ref. [47] to RN spacetimes.
Carrying on, the 2 nd -order of the HEE correction reads S EMGD2 2 = π 288 4M 2 2 (y 0 − 1) (3y 0 + 11) − 6 log(y 0 ) + 16 log Looking at the previous cases, the MGD and EMGD 1 , there is a leading difference here. Even in the s → 0 regime, there is a numerical difference, when compared to the EMGD 1 . It would be nice to plot some comparison with the Schwarzschild black hole or, strictly, with the RN without the s parameter, to scale the numerical contribution.
Following an analogue procedure established in EMGD 1 case, let us put the ADM mass and the tidal charge in terms of Schwarzschild mass parameter, which are M 2 = 3M and Q 2 = 12M 2 , respectively. Besides, we use ζ = s/M as well as it has been done in the MGD case. Over again, the main purpose here is also fixing M to analyze the influence of a finite brane tension at this order of HEE correction. Continuing, the expression below carries only the lower-limit of the integration in the area functional and the parameter ζ. In this sense, we clearly could investigate the ratio related with the 2 nd -order correction for Schwarzschild spacetimes, that is, where Next, Fig. 14 illustrates, for two values of ζ -the first one representing a high brane tension and another one depicting a low brane tension -how the size of the minimal 2 surface affects the 2 nd -order correction. To compare the HEE 2 nd -order correction in EMGD 2 to the one of a Schwarzschild one, for different M values, we set ζ = −0.1, in Fig. 15, and ζ = −100 in Fig. 16, specifying two kinds of ranges: a first one close to 0, and another one close to 1.  To analyze a wide range scenario to ζ and y 0 , we plot Fig. 17. Besides, the ratio to this order Fig. 17 The 2 nd -order corrections to EMGD 2 and Schwarzschild spacetimes pondering light and heavy tension on the brane as well as the full range of y 0 .
which graphically is presented in Fig. 18. In a general framework, leaving ζ and y 0 free to run within their valid interval of values, Fig. 19 shows the 2 nd -order ratio. For completeness, we establish Note that both ratios above are identical to those ones obtained in the MGD case. Some features can be extracted out of Eq. (75) and Fig. 19: (i) when the size of the minimal area is reduced, which is implemented with y 0 0.9, a low brane tension hugely contributes to the increment of the ratio; (ii) when y 0 → 0, the parameter related to the brane tension is dominant.

Almost on the horizon
Specifically, we now deal with the metric (9), which carries the time component (15) and the radial one (16), with coefficients cm's displayed in (17), as wherer = re = 1.12M 2 stands for the degenerate event horizon determined in [5] and µ ≈ 0.4533. We must implement the subtle displacement of the event horizon, that is, r = ρ +r, ρ > 0, and fix the boundary on the horizon with ρ 0 = εr with ε 1. Once again, the θ = θ 0 circumference maps the entangling surface. Hence, the resulting induced metric on the t-constant manifold is where r → q(ρ) and p(ρ) = (ρ + µr) (ρ +r) Finding ρ ≡ ρ(θ) means to minimize the surface area whereL EMGD2 = 2πq(ρ) p(ρ)(1 − y 2 )ρ 2 + q(ρ) 2 1/2 , y = cos θ is employed to attain ρ ≡ ρ(y). The variation of Eq. (42) with respect to y, and taking δA = 0, gives where q = q(ρ), p = q(ρ). Now, the perturbation procedure previously used in Sect. 3.2 is also applied here to build two ODEs up to 2 nd -order in ε by the expansion ρ = ερ 1 + ε 2 ρ 2 into the Eq. (82). Such ρ-functions are important to execute the series expansion of the integrand in Eq. (81) up to 2 nd -order, which will be substantial to determine the HEE corrections in the present case. Thus, the first one of them, that is, the 1 st -order in ε ODE is Eq. (83) has the general solution with Pη(y) and Qη(y) as Legendre polynomials of the first and second kind, respectively, and η = 1/2 (−1 + √ 1 − 4γ). Requiring regularity at y = ±1, one needs to set A 2 = 0 since Qη(y) is not regular in such points. The boundary condition ρ 0 = ερ 1 (y 0 ) determines A 1 and leaves us with The 2 nd -order ODE reads where Ω(y, γ, β) = 1 and Eq. (87) is a linear non-homogeneous ODE. The presence of the Ω(y, γ, β) permits a variety of solutions conditioned to the parameters β and γ, which by themselves are constrained to the physical parameters of EMGD 2 case, i.e., the ADM mass M 2 , the tidal charge Q 2 and the EMGD 2 parameter s within c-coefficients explicitly detailed in (17). Therefore, the general analytical solution for Eq. (87) is written as . Therefore, we may pursuit a wide family of solutions to Eq. (87) depending on the aforementioned parameters, which are crucial to estimate the final shape of the ρ 2 (y) in Eq. (90). The constants of integration B 1 and B 2 depend on the computation of the integral carrying the Ω-function.
Hereon we opt to work with two main scenarios. The first one consists to regard only the 1 storder at ε, considering ε 2 ρ 2 (y) insignificant, compared to ερ 1 (y). In fact, it is also consistent with the MGD and EMGD 1 scenarios, where ρ 2 = 0. The second one goes to the 2 nd -order with some kind of simplifications to the Ω(y, γ, β) through free choice of values for the γ and β parameters to fit consistent solutions.
First scenario: cutting off ε 2 ρ 2 . In this case, only the ε-order for ρ-function is imperative, leading us to deal with a simplified solution.
Next, it is important to expand the integrand in the Eq. (81), which yields and calculate the perturbative entropy function S = S 0 + S 1 + S 2 + · · · . It is crucial to keep in mind that we stand for up to second order. Now, using only ρ 1 (y), we determine the contributions to the entropy, order by order, up to the second one. Thus, the 0 th , 1 st and 2 nd -orders are, respectively, A first novelty concerns about a non-vanishing 1 st -order correction for the HEE, which did not happened either in the MGD or in the EMGD 1 cases. The computation of a numerical value depends on the parameters γ and y 0 . Then we must plot Eqs. (93) and (94) considering some values for those parameters. Fig. 20 shows three values for γ -the parameter gathering the ccoefficients with information about the ADM mass and the tidal charge as well. Meanwhile, the range −1 < y 0 < 1 is imposed, regarding the lower limit of integration that determines the size of the boundary. On the one hand, there is a change of sign of the HEE 1 st -order correction between the asymptotes, for each value of γ. It indicates a substantial contribution from the EMGD parameters. On the other hand, we see only negative corrections at 2 nd -order correction.
It is worth to emphasize that chosen values for γ generate the simplest polynomials as a manner to investigate a particular behavior of such order of correction. In a more realistic scenario, we will need precisely the physical values for both the ADM mass and the tidal charge, to fully understand the contribution at this order.
Second scenario: samples for the Ω(y, γ, β) function. At this point, first, we choose two pair of values for γ and β to determine Ω(y, γ, β), permitting us to determine the HEE corrections. Second and last, we attribute a value for γ to find the corresponding numerical value for the EMGD 2 parameter dealing with a unit value for the mass parameter M .
The next step comprise to calculate the HEE corrections up to 2 nd -order, as implemented, employing Eq. (96). Also, it is necessary to remember that ε = ρ 0 /r. Therefore, it implies that 1 st -order correction, which is not present in cases like the MGD or the EMGD 1 . In addition, there is a sign change of such correction as well as can be observed in the case where ρ 2 is insignificant. As a second example, let us take γ = −6 and β = 8. Similarly proceeding as in the previous example, the ρ-functions can be derived, as Hence, one obtains Ω(y, −6, 8) = 42r 1 − 3y 2 2 / 1 − 3y 2 0 2 , yielding ρ 2 (y) = − 9r y 2 − y 2 0 5 (3y 2 0 − 1) One more time, with these ρ-functions, we expand the integrand in Eq. (81), which leave us with Finally, after employing Eq. (81), the calculations of the HEE corrections, order by order up to the second one, read  The appearance of the 1 st -order correction happens again, with the sign-changing noticed before in the first example.
With the integrand in hands, we can compute the HEE corrections, order by order, up to second one, as follows where the numerical coefficients K i are listed in Appendix B. Fig. 23 shows the shape of the last two entropy functions above. The profile of the 1 st -order correction has a sign-changing noticed

Conclusions
About the MGD case, we calculated the HEE of the MGD solution to investigate the influence of high energy effects caused by the MGD parameter , encoded in the ξ parameter, from the AdS/CFT membrane paradigm. There are two perspectives, namely, the almost on the horizon and far from the horizon regimes. Far from the horizon, the HEE 0 th -order is not affected by ξ, which is a good feature of the deformation, as Eq. (33) exactly matches the HEE for Schwarzschild spacetimes, as pointed out in Ref. [47]. The novelty clearly appears when one reaches the HEE 1 st -order correction, since the ξ parameter is present in Eq. (34) as well as in the ratio casted by Eq. (36). The fact that ξ < 0, due to the same sign of , contributes to an increment of the correction term, however without any modification of its sign, which is made explicit by Eq. (36). Once more, the MGD parameter carrying on brane effects is featured in the HEE 2 ndorder correction, as revealed by Eqs. (37). Computations in this direction shed new light about holography in asymptotically flat spaces.
Comparatively with the HEE for Schwarzschild black hole, one notices the exponential rise of such order of correction, when the brane tension is lowered, as illustrated by Fig. 2. Therefore, lower brane tension values have profound influence in the increment of this order of correction, as one can see in Fig. 3. Another feature is the agreement with the first law for the HEE, evinced by Eqs. (34) and (37). Fig. 4 shows that the more the MGD black hole mass increases, the higher the magnitude of the 2 nd -order correction is, concomitantly to the rise of the size of the subsystem, which is characterized by y 0 . Fig. 5 permits us to obtain a better comparison of this feature, while one looks at the HEE 2 nd -order correction for a Schwarzschild black hole. As expected, when ξ = 0, the HEE corrections for Schwarzschild spacetimes are recovered at 1 st -and 2 nd -order, accordingly. Even when one considers the boundary far away from the event horizon of the MGD black hole, it is observed substantial differences when confronted to the HEE of a typical Schwarzschild black hole.
Regarding the entangling surface almost on the horizon, the MGD parameter, , that encodes the finite brane tension, demonstrated its strength to modify the HEE 2 nd -order correction, as one can notice in Fig. 7. The MGD influence is codified by the parameter α, which is correspondent, in a brief mode, to . The 0 th -order and 1 st -order are not susceptible to such parameter and both of them match to those ones established in Ref. [47]. On the other hand, the low brane tension weighs significantly to lift the maximum value of the HEE 2 nd -order correction, according to Fig. 7. Here, so close to the event horizon of a MGD black hole, the correction at 2 nd -order is more sensitive to the MGD parameter.
Extending the analysis from the Schwarzschild black holes and illustrated by Fig. 7, an important novelty consists of the maximum value arising in the MGD HEE 2 nd -order correction. This is associated with lowering the brane tension and, concomitantly, requires a large size of the dual quantum subsystem.
In the EMGD 1 case, a similar scenario to the Reissner-Nordström spacetime occurs. Far from the horizon, similarly to the HEE for the Schwarzschild spacetime, a subtle numerical shift of the HEE 1 st -order correction is verified with Φ EMGD1 1 = 2. It happens due to the presence of the ADM mass in Eq. (49). Meanwhile, the 0 th -order is not altered. It is worth to emphasize that the correspondence between the tidal charge, Q 1 , and the ADM mass, M 1 , with the Schwarzschild mass M , is mostly necessary to analyze the relative behavior of the HEE corrections for the EMGD 1 spacetime. The ratio (52) shows the peculiarity of such correspondence, which is sustained by Fig. 8. The influence of the black hole mass is notorious with the large size of the entangling surface characterized by y 0 , as shown by Fig. 12. The increments in the HEE 2 nd -order correction are accentuated accordingly with the mass increment and the enlargement of the minimal area. Figs. 11 and 12 make us to comprehend that the greater the mass, the greater the deviation of the HEE 2 nd -order correction for the EMGD 1 is, related to that one for the Schwarzschild spacetime.
Considering the entangling surface almost on the horizon for the k = 1 scenario, we have only an extremal black hole with the degenerate horizonr = M 1 . The HEE for this case is very close to the HEE for the Reissner-Nordström spacetime. The crucial distinction relies on the numerical value of the full entropy displayed by Eq. (59) in consequence of the weakening of the gravitational field carried by the position of the event horizon in an EMGD 1 spacetime, that is,r = M 1 = 2M . The relationship between those entropies displays a limit equal to 4 and it is sustained by Eq. (60) and exhibited by Fig. 13, where it is possible to notice a fast-growing ratio as the mass of the black hole increases.
The EMGD 2 case brings on the possibility to settle additional HEE corrections to a certain class of black holes beyond Reissner-Nordström spacetimes. Far from the horizon, the HEE 0 th -order is not affected, behaving like a constant, as the HEE for all cases are confronted. As occurred in the MGD case, the HEE 1 st -order correction displays already the specific quantifier related to the brane tension, i.e., the parameter s, as shown by Eq. (69). Besides, the HEE 2 nd -order correction is richer, despite its structural similarity when faced up to the same order in either the MGD or the EMGD 1 cases. The mass terms are preserved, which is a welcome feature to hold the first law of HEE. The new establishment has tuned with the quadratic term in s and the mixed one with M 2 and s, as supported by Eq. (70).
The Φ-ratios were also computed, scaling with Schwarzschild mass M . In Fig. 14, we observe two simple scenarios fixing the brane tension parameter. It unveils the fast-growing of the HEE 2 nd -order correction according to the mass parameter and the size of the minimal area. Fig. 18 exposes how the brane tension affects, relatively, the HEE 2 nd -order correction, where it is clear that lower tension branes have exponential gains, consonantly with the size of the minimal area, that is, the range of the dual subsystem that entanglement entropy stands for. In addition, looking at Fig. 15 and Fig. 16, one can notice the significant deviation between the HEE for a Schwarzschild black hole and the HEE for the EMGD 2 spacetime. For completeness, Fig. 19 shows how the 2 ndorder ratio behaves under the simultaneous variation of the brane tension and the size of the dual subsystem.
With the entangling surface almost on the horizon, we employ the expansion of an auxiliary function characterizing the proximity to the horizon which head us to general analytical solutions depending on parameters related to the ADM mass, tidal charge, and brane tension. Therefore, we expend efforts to analyze some possible scenarios towards the profile of HEE corrections in this present case. Firstly, based on a meaningless ρ 2 , we determine HEE corrections very similar to the previous cases, i.e., MGD and EMGD 1 , as shown by Eqs. (92) and (94). The dependence of the starting point at horizon ρ 0 is sustained at 1 st -order and 2 nd -order corrections. In addition, this approximation requires using values to γ and the plots in Fig. 20, even dealing with simple Legendre polynomials, displaying the sign-changing demeanor of the two orders of corrections for the HEE. Of course, if we limit ourselves to a certain region into the boundaries, which means to limit the size of the dual subsystem, we get away from the asymptotic regions. Besides, among the asymptotes we observe the similar behavior of both HEE orders of corrections. Secondly, we scrutinize three examples, each one of them demonstrates sign-changing behavior of the HEE 1 stand 2 nd -order corrections. The 0 th -order, as usual, remains immutable. According to the rank of the Legendre polynomials corresponding to the choices for β and γ, we handled with one to two asymptotes marking the regions where the change of sign of that order of correction occurs, as one can realize in Figs. 21 and 22. The last example was built attributing a mass reference and, subsequently, fixing the brane tension parameter, s, with determined value for γ, which is purposely attached to the order of a rank-4 Legendre polynomial. Its functionality as a toy model reveals the same sign-changing aspect of the orders of corrections for the respective HEE. The new aspect noticed here was in virtue to the local maxima and minima presented at HEE 2 nd -order correction as showed by Fig. 23. Such presence of extremal points reveals a real constraint to the corrections for the HEE. Finally, specific values for the physical parameters bring to us the most realistic results for the HEE in EMGD 2 spacetimes. Without lose of clarity the constructions of the toy models aforementioned was essential to the simplest landscapes.