Extremal Surfaces And Entanglement Entropy

We have obtained the equation of the extremal hypersurface by considering the Jacobson-Myers functional and computed the entanglement entropy. In this context, we show that the higher derivative corrected extremal surfaces can not penetrate the horizon. Also, we have studied the entanglement temperature and entanglement entropy for low excited states for such higher derivative theories when the entangling region is of the strip type.


Introduction
The study of the entanglement entropy in the AdS/CFT context [1] has attracted a lot of attention due to its potential application in condensed matter systems as well as in quantum information theories. In a seminal work, [2], Ryu and Takayanagi (RT) made a holographic conjecture of the computation of the entanglement entropy. The proposal is given for a static spacetime with a co-dimension two hypersurface, whose area is proposed to be related to the entanglement entropy. A proof of such a proposal is attempted in [3], with further comments in [4]. In another study, a proof is suggested in [5], when the entangling region is of the sphere type. More recently, a suggestive argument was put forward in [6] based on the previous works in 1 + 1 dimensional CFT of [7] and [8] that explains the RT conjecture.
One of the universal result of [2] is that the entanglement entropy becomes divergent at UV, which goes as 2−d , with as the UV regulator, for a d dimensional CFT, and becomes finite at IR. In fact, the approach adopted in [2] is a non-covariant way of doing the calculation, which has been generalized in a covariant way in [9] by Hubeny, Rangamani and Takayanagi (HRT) to derive, in particular, the equation of the extremal hypersurface. To emphasize, this is the generalization of the minimal hypersurface in a covariant way of RT prescription. However, it is interesting to note that the equation of the hypersurface is also derived in [6] starting from the bulk Einstein's equation of motion, which is further studied in [10,11] and [12]. Some other interesting studies on entanglement entropy are reported in [13,14] and [15].
In this paper, we ask several questions and find answers to it. First, how would the equation of the hypersurface look, in a covariant way, upon inclusion of the higher derivative terms to the entanglement entropy functional? We answer this question, by considering the Jacobson-Myers functional (JM) [16] as the starting point for the holographic entanglement entropy functional and then derive the equation of the hypersurface by extremizing it with respect to the embedding fields 1 , X S . It is important to note that such a from of the equation of the hypersurface does not depend on the shape or the size of the entangling region. As an example, with up to the Gauss-Bonnet term in the holographic entanglement entropy functional, the hypersurface reads as where K S ≡ g ab K S ab , whose functional form is given in eq (38). The couplings λ 1 and Λ are undetermined constants and are dimensionfull objects. Let us recall, without turning on, Λ, the finite piece of the the λ 1 dependent part of the entanglement entropy 2 is computed numerically in [17] for a 4 dimensional CFT. We revisit such a computation but for a d dimensional CFT, i.e., for bulk AdS spacetime and the result to the linear order in the coupling λ 1 can be summarized as follows: (a) The divergent term coming from UV goes in the same way as in the absence of the higher derivative term to the entanglement entropy functional, i.e., like 2−d , whereas (b) the finite term, the coefficient of r 2−d , where r is the turning point, depends on the coupling λ 1 very non-linearly but we determine the functional form only to linear order. Explicitly, it reads as where R 0 and R are the radii with and without the higher derivative term to the AdS and λ 1 = aλ a R 2 with a = 2 (d−2)(d− 3) . This value of a follows upon comparing our JM functional with eq(C.24) of [18] and 3 identifying the coupling λ a = λ there . After turning on both the couplings λ 1 and Λ, we find that the entanglement entropy for a d = 6 dimensional CFT follows the same pattern. That is apart from the UV divergent term, 1/ 4 , the finite term coming from IR goes 1/r 4 times some constants. Explicitly, the entanglement entropy to linear order in the couplings reads as In fact, we need to set a = 1/6 and X 2 = −1/24. This follows upon comparing with eq(C.24) of [18] and identifying the couplings as λ a = λ there and Λ a = µ there .
It simply follows from the expression of the entanglement entropy eq(127) or eq(130) that there exists a differential equation relating the entanglement entropy and the size, , as ∂ ∂ ∂ ∂ + (d − 2) S EE = 0, with fixed L and R.
We ask the validity of such a differential equation upon inclusion of the higher derivative term to the holographic entanglement entropy functional? We check that such a form of the differential equation goes through to linear order in the coupling λ 1 and Λ. In fact, such a differential equation can be re-written as In the context of RG flow, in [19], a candidate c-function has been suggested, which shows the necessary monotonicity along the flow from UV to IR as long as the matter fields obeys the null energy condition 4 . In fact, such a c-function 5 is proportional to d−1 ∂ ∂ S EE for d ≥ 3. The vanishing of the derivative of such a quantity with respect to the size, , suggests that we are sitting at the fixed point. However, in [21] another quantity is defined and called as "renormalized entanglement entropy" that gives the rate of flow 6 . Inspired by this, we find the simplest UV finite quantity is ∂ S EE . Using which, we find there exists another quantity, 2 ∂ 2 S EE − ∂ S EE , which is negative but only when d ≥ 5 for non-zero q in the perturbation to the geometry. In a recent study in [22] by Hubeny in the context of the entanglement entropy found that the spatial hypersurfaces does not penetrate the horizon. This motivate us to ask the following question: Is the non-penetrating of the horizon by the probe, spatial hypersurfaces, remain true even after the inclusion of the higher derivative term to the area functional?
The answer to this question is that upon inclusion of the higher derivative term of the type as in eq (36) to the action of the embedding field make the spatial hypersurface not to penetrate the horizon.
It is suggested recently, by studying the low excited states [23], that the entanglement entropy obeys a law like that of the first law of thermodynamics, T ent ∆S = ∆E. The quantity T ent is dubbed as the 'entanglement temperature', which is proportional to the inverse of the length, , i.e., T ent = c −1 . The proportionality constant, c, is a function of d, the spacetime dimensionality of the field theory and depends on the nature of the entangling region [23]. The question that we ask: Do we still expect to see a similar first law like relation even after the inclusion of the higher derivative term to the holographic entanglement entropy functional? If yes, then how does the 'entanglement temperature' go with the length, ? And how does the proportionality constant behave as a function of d? We show that there indeed exists a first law like relation even with the higher derivative term to the holographic entanglement entropy functional and the relation between the 'entanglement temperature' and the size is same as mentioned above. And the proportionality constant is a function of d and the coupling 7 .
The paper is organized as follows: In section 2 and Appendix A, we re-visit the computation of the area functional of the hyperscaling violating geometries and generates both the existing [25,27,28] and new form of the geometry that exhibits the log violation of the entanglement entropy. In section 3, we find the equation of the hypersurface using the technique of [9] but with the higher derivative term to the area functional and explicitly compute the entanglement entropy for AdS spacetime. In section 4, we show following [22] the absence of the penetration of the horizon by the spatial hypersurfaces. In section 5, we find the first law like of thermodynamics by considering fluctuation to the geometry and finally we conclude in section 6. Some of the expression of the solution of the embedding field with the higher derivative term to the area functional has been relegated to Appendix B.
7 Similar type of question is asked in [24] for a 4 and 6 dimensional CFT when the entangling region is of the strip type and sphere type, respectively. Our study is different in the following way. We study the 6 dimensional CFT when the entangling region is of the strip type, which is not studied there. Also, we study the d dimensional CFT by considering the two derivative area functional whose special case, d = 4, is studied in [24].

A differential equation
If we look at the expression of the holographic entanglement entropy, which is proportional to the area, eq(130) for AdS spacetime, then it follows that Note that in deriving such a differential equation we have considered a mild assumption that the UV cutoff, , is independent of the size , i.e., d d = 0. In fact, this is true because the size, depends only on r . The details of the computation of S EE are relegated to Appendix A. This type of differential equation would suggest some form of the 'RG flow' equation for the entanglement entropy. Strictly, such a form of the differential equation involving S EE and follows when the entangling region is of the strip type and holds true irrespective of whether d is even or odd. In fact, a similar equation also follows for r , as is linearly related to it. It reads as It is expected that the limits of integration of the radial direction should be independent of each other. Hence, it is justified to consider that r and are not dependent on each other. The simple looking differential equation eq(7) obeyed by S EE for AdS spacetime gets changed when we change the background spacetime to Lifshitz type as follows This follows from eq(138). From which it follows that In [21] a quantity, S Σ d , is defined with the shape of Σ assumed to be a sphere. This has been used to study the rate of flow of the renormalized entanglement entropy. In our case of a strip type entangling region, we shall assume the existence of the following quantity: In getting such a result, we have used eq (7). Upon evaluating ∂ l S EE explicitly, using eq(130), we find it to be positive for any d. For d = 3, this result is in some sense consistent with [21] because the authors did not find any example that violates the monotonicity of S 3 unlike S 4 .
In what follows, we shall define two more quantities, which are UV finite, ∂ ( ∂ S EE ) and 3 ∂ ( −1 ∂ S EE ) and find whether they can be used to study the rate of flow, i.e., are negative? Given the explicit result of the entanglement entropy for AdS spaetime in eq(130), we find the following equations for any d ≥ 3 In fact there exists the following relation Hence, it is not surprising to see that Hence, it is highly plausible to consider these two quantities, which may give information on the rate of flow. This is further investigated by doing fluctuations to the geometry in subsection 6.3 and there arises interesting restrictions on d.

Log term to S EE
In this section, we shall try to find answer to the following question: Can we generate log term in the entanglement entropy? Let us assume the following form of the metric in 3 + 1 dimensional spacetime ds 2 3+1 = −g tt (r)dt 2 + g xx (r)dx 2 + g yy (r)dy 2 + g rr (r)dr 2 + 2g xy (r)dxdy, (14) we assume that the boundary is at r = 0. Following the proposal of [2], the geometry of the co-dimension two hypersurface becomes ds 2 2 = g xx + g rr r 2 (x) dx 2 + g yy dy 2 + 2g xy dxdy (15) So the area of the hypersurface becomes where we have inverted the function r(x) and written as x(r). Now the hypersurface x(r) that extremizes the area A is dx dr = (g xx (r )g yy (r ) − g 2 xy (r )) g yy (r)g rr (r) [g xx (r)g yy (r) − g 2 xy (r)][g xx (r)g yy (r) − g 2 xy (r) − g xx (r )g yy (r ) + g 2 xy (r )] , where the turning point r is determined when the quantity x (r ) diverges. We consider the entangling region to be like a strip, 0 ≤ x ≤ and −L/2 ≤ y ≤ L/2 and the size /2 = r 0 dr (g xx (r )g yy (r ) − g 2 xy (r )) g yy (r)g rr (r) [g xx (r)g yy (r) − g 2 xy (r)][g xx (r)g yy (r) − g 2 xy (r) − g xx (r )g yy (r ) + g 2 xy (r )] .
In such a case, the extrmized area turns out to be where is the UV-cutoff. From now on wards, we shall be working in the diagonal form of the bulk metric i.e., g xy = 0, for simplicity. From this expression of the area, we can ask: under what condition do we see a log term? The condition to see such a log term, using eq(139) are In which case the area becomes We have put an index k in the area to label it. In order to fix the metric components, let us assume that g yy = r −2w , g xx = r 2(w−1) , g rr = r 2(w−2k−1) , in which case the 3 + 1 dimensional geometry reads as where we have used the indices k and w to label the geometry. Moreover, the entanglement entropy does not fixes the time-time component of the metric tensor. The quantity Note, there exists UV divergence to for k ≥ 1. Once again to regulate it, we have put an UV cutoff. In which case, Explicitly, the area for k = 0 and k = 1 are Note for k = 0 case, the entanglement entropy obeys eq(10) for γ = 1/2. It is interesting to note that the neither the area eq(21) nor the length, , eq(24) depends on w. It means there can be more than one co-dimension two geometry which has the same area and the length . The explicit from of the bulk geometry for k = 0, 1 are For k = 1 case, it is very difficult to find the explicit dependence of the area in terms of and the UV cutoff . Even though the area depends logarithmically on r , but it is not clear it will do so in terms of . Also, equation of the type, eq(10), is difficult to satisfy for γ = 3/2, where γ is defined below in eq (30).
Looking at the geometries for k = 0 and 1, we see the presence of rotational symmetry only for w = 1/2. Hence, let us find out the scaling behavior of such cases.
Case 1: w = 1/2. For this choice of w, there exists a rotational symmetry, i.e., g xx = g yy . The choice k = 0 corresponds to that studied in the previous section and also found in [27]. The other choices of k ≥ 1 are new.
As an example, let us looks at the explicit geometry for k = 1. In which case, we get It means, we can write down the geometry as which falls under the category of the hyperscaling violating geometry provided t has a nice scaling behavior. It is easy to see that for k = 0 the geometry is same as written in eq(131) for d = 3 and γ = 1/2. If we assume that g tt = ρ , then for generic k we can have the following scaling behavior where ρ = r 2k−1 .
Case 2: This corresponds to those solutions which does not respect the rotational symmetry i.e., solutions other than w = 1/2. Let us take different choices of w. In which case, the 3 + 1 dimensional solution becomes For g tt = r −2 with k = 0 and w = 0, it corresponds to AdS 3 × R 1 or AdS 3 × S 1 for non-compact and compact, y, respectively. As we saw earlier, the entanglement entropy does not depend on w and for k = 0 case it goes as S EE ∼ Log( / ). It means for k = 0 and w = 0, we can have a log violation to the entanglement entropy as well.
Subsummary: We obtain the presence of the logarithmic term in the entanglement entropy in 3 + 1 dimensional bulk system with the rotational symmetry along the spatial directions. In fact the only geometry that is found here, corresponding to having k = 0, and is same as that found in [27], which is of the Lifshitz type. So, the only rotationally invariant solution whose entanglement entropy goes logarithmically with the size, , corresponds to the k = 0 case.
In the absence of the rotational symmetry, we have obtained several geometries as written in eq (31). In particular, the direct product of geometries like AdS 3 × R 1 or AdS 3 × S 1 shows the presence of log term in S EE .

Re-visit of S EE as studied in [17]
In [17], the authors considered a geometry that does not have the full rotational symmetry SO(d-1) in a d + 1 dimensional bulk system, while studying the entanglement entropy. We are going to re-investigate this calculation but without the higher derivative correction.
Let us substitute in eq(19) the following form of the metric components as considered in [17] g tt = g xx = g rr = 1/r 2 , g yy = 1/r 2w , g xy = 0.
Now the hypersurface is described by x(r), whose explicit form can be read out from eq (17). On computing the area of the hypersurface of a strip type entangling region and the size Note that the 3 + 1 dimensional geometry as written above is a solution at IR not at UV. Hence, it is expected that such a solution will not show the desired result, −1 , at UV. In such a case, the entanglement entropy obeys the following differential equation

The equation of the hypersurface
In this section, we shall derive the covariant equation of the extremal hypersurface with higher derivative effects. It means, we are including the effects of the finite 't Hooft coupling. The equation motion of the embedding fields, X M (σ a ) essentially gives the form of the hypersurface. The induced metric is given by where G M N denotes the d+1 dimensional geometry of the bulk spacetime, σ a are the coordinates on the codimension-2 hypersurface. Let us assume that the entanglement entropy functional is where we have included higher derivative correction with λ i 's as the coefficients 8 .
A priori there is no good reason to believe the inclusion of higher derivative terms in this particular way. Even though this is purely a guess but we hope, it can be thought of as follows. For the Einstein-Hilbert action, it is suggested in [2] to consider only the first term in eq(36), i.e., setting all the λ i 's to zero. Upon inclusion of the Gauss-Bonnet term to the (bulk) Einstein-Hilbert action, it is suggested in [17] that the entanglement entropy should be given by eq (36) for which λ 2 = 0 = λ 3 = λ 4 . Hence, it follows from these examples that for each power of the 'Ricci scalar' in the bulk theory 9 one should include a 'Ricci scalar' with one less power in the entanglement entropy. Note that the 'Ricci scalar' in the entanglement entropy should be constructed out of the induced metric g ab . Formally, we can write it as where the left hand side is the bulk action and right hand side is the entanglement entropy functional and GB(G) denotes the Gauss-Bonnet term made from the bulk metric G M N . Similarly, if we go for one more higher power of the scalar curvature then it is highly plaussible to consider the terms as written in eq(36) with arbitrary coefficients. In [18], the authors have considered the Jacobson-Myers form of the entropy functional [16] and studied the entanglement entropy when the entangling region is of the sphere and cylinder type. In our study, we do it for the strip type. Before moving onto the calculation of the extremal surface, recently in [14], the author has given a derivation of the Jacobson-Myers entropy functional.
On varying the entanglement entropy functional, eq (36), with respect to the embedding field, X S , gives where the γ c ab and Γ K M N are connections defined with respect to g ab and G M N , respectively. In fact, K S ab obeys an identity K M ab ∂ c X N G M N = 0. The quantity For this form of X ab , one can easily show that, it obeys an identity: ∇ a X ab = 0, where the covariant derivative is defined with respect to g ab . In which case, the equation of motion of the extremal hypersurface becomes Note, the equation of the hypersurface is independent of the shape and size of the entangling region. Now, we shall write down the form of X ab in two different cases.
Gauss-Bonnet combination: Let us consider a very specific combination where λ 2 = Λ, λ 3 = −4Λ, λ 4 = Λ, then X ab takes the following form In which case, the equation of motion of X S can be re-written as where K S ≡ g ab K S ab .
Weyl-square combination: In this case, the λ i 's take the following values: In which case the X ab takes the following form and the equation of motion of X S becomes We have checked that the Weyl-squared term does not contribute to the entanglement entropy till d = 8.

The precise form of the hypersurface: An example for strip
Let us compute the hypersurface eq(40), for the following form of the solution in the bulk which gives rise to the following induced metric with the embeddings as The explicit computation of the components of K S ab gives and the rest of the components are zero. For simplicity, let us set the coefficients λ i 's to zero in which case, the extremal hypersurface, K S = 0, gives 2r g xx g rr − dr 2 g rr ∂ r g xx + r 2 g xx ∂ r g rr − (d − 1)g xx ∂ r g xx = 0.
Upon using the identity, r 3 d 2 x 1 dr 2 = −r , we can re-write the equation of the hypersurface as where x 1 = dx 1 dr . Essentially, we have re-written a second differential equation as a first order differential equation. Now, we can solve the equation of motion and In order to determine the constant of integration, we have used the following boundary condition, x 1 (r ) → ∞. To get a feel of the solution, let us consider a spacetime that exhibits the scale violating behavior along with the trivial and non-trivial scaling of the spatial direction [29], namely, In which case the differential equation can be exactly solved where c 1 is a constant of integration and 2 F 1 [a, b, c, x] is the hypergeometric function. The precise form of c 1 is determined by imposing the boundary condition that x 1 (r = r ) = 0 [22], which gives We can relate with the constant of integration, c 1 , as /2 = −c 1 . It is easy to see that in the δ = 1, γ = 0 limit, it re-produces the result of [2]. However, in the δ = 0 and γ = 0 limit, the solution becomes .
The entanglement entropy for a generic diagonal and rotationally invariant metric with the entangling region as a strip, 0 ≤ x 1 ≤ , − L/2 ≤ (x 2 , · · · , x d−1 ) ≤ L/2, takes the following form [29] 2G N S EE = L d−2 r dr g d−2 xx (r)g rr (r) where in the second equality we have substituted the geometry as written in eq(51). Now, we shall give results to this integral in two different cases i.e., δ = 0, 1 and γ = δ.
For δ = 0, γ = 0 case : In this case the entanglement entropy gives the following result It looks from this expression as if the area is a complex quantity but it is not because γ is negative. The entanglement entropy is completely divergent and the divergence goes as −|γ|(d−1) .
For δ = 1, γ = 0 case : In this case the entanglement entropy gives the following result . The dependence on the goes as 2−d+γ(d−1) . It is easy to notice that in the γ → 0 limit, it reduces to that written in eq (127) and gives the precise entanglement entropy as obtained in [2] and the 2−d behavior.
For γ = d−2 d−1 , it is easy to see from eq(55) using eq(139) that there arises a logarithmic dependence of the entanglement entropy as obtained in [27]. For completeness, it comes out as Hypersurface at finite coupling but for λ 2 = λ 3 = λ 4 = 0: Now, let us include the effect of the finite coupling, λ 1 , for the diagonal d + 1 dimensional bulk spacetime, which is AdS. Upon doing a tedious but straight forward calculation, we find the following expressions 2r 2 g xx g 2 xx + r 4 g xx g xx g rr − 2r g 2 xx g xx − 2r 2 g 2 xx g xx − 2r 4 g xx g rr g xx + r 4 g rr g 2 where g ij = ∂ r g ij and r = dr dx 1 . In which case, the equation of motion becomes where we have included the contribution only from the first two terms of eq(36), i.e., have set λ 2 = λ 3 = λ 4 = 0. It is easy to notice that for d = 2 the contribution from the induced Ricci scalar vanishes identically. The above equation of motion can be re-written as This gives a cubic equation in x 2 1 and all the solution of it are not real. In fact, the real solution for x 1 is a huge expression and finding the exact analytical solution of the hypersurface, x 1 (r), is a daunting task. However, the derivative of the function, x 1 (r), which is real given in Appendix B.
On computation of the holographic entanglement entropy functional, we find the entanglement entropy takes the following form Substituting the solution of x 1 (r) to quadratic order in λ 1 from Appendix B into the above expression of the entanglement entropy, gives dr 2c 2 g 2 xx (r)g rr (r)g xx (r) − 2g d+1 xx (r)g rr (r)g xx (r) + 6c 2 g xx (r)g rr (r)g 2 xx (r) + (d − 5)g rr (r)g d xx (r)g 2 xx (r) + 4g rr (r)g d+1 xx (r)g xx (r) − 4c 2 g rr (r)g 2 xx (r)g xx (r)

Entanglement entropy
We can get the exact expression of the entanglement entropy upon substituting the solution from Appendix B into eq(62) for the geometry eq(51). Since, the form of x 1 (r) is messy, let us use the leading order (in λ 1 ) form of it and perform the computation of the entanglement entropy. In which case where the ellipses stands for higher order terms in λ 1 . On performing the integrals results in It is worth mentioning that till this order in λ 1 do not give any logarithmic violation of the entanglement entropy for any choice of γ with δ either 0 or 1 except γ = d−2 d−1 with δ = 1. To get a feel of the entanglement entropy for AdS solution, which means setting δ = 1 and γ = 0, restoring the AdS radius R 0 where we have taken the → 0 limit and kept both the divergent and finite terms. In the absence of the higher derivative correction, the UV divergence was found to be 2−d in d + 1 dimensional bulk spacetime [2]. And, upon inclusion of the next higher derivative term to the holographic entanglement entropy functional gives the same power of the UV divergent, 2−d . This is observed, however only, for d = 4 in [17].
We can re-express the above expression of the entanglement entropy in terms of the size and R. In which case, it reads as Note that R 0 and R are the radii of AdS spacetime with and without the higher derivative correction, whose precise relation is given in section 6 and λ 1 = aλ a R 2 . Upon comparing eq(36) with eq(C.24) of [18], we find a = 2 (d−2)(d−3) , after identifying the couplings λ a = λ there . It is easy to see that up to linear in the coupling, λ a , the the finite part and the singular part of the entanglement entropy obeys following differential equations where S f p EE and S sp EE stands for the finite part and singular part of the entanglement entropy, respectively. However, there exists another differential equation for the full S EE Let us re-write the expression of the entanglement entropy as We can check that the quantities defined in eq(12) holds good for d = 4 when we take the coupling to stay within the following window − 7 36 ≤ λ a ≤ 9 100 . This bound follows from the study of causality and the positivity of the energy flux in [30,31,32].

Gauss-Bonnet combination
In this section, we shall find the form of the extremal hypersurface as well as the entanglement entropy to leading order in the coupling λ i for a very special combination of the λ 2 , λ 3 and λ 4 . In which case the holographic entanglement entropy functional takes the following form as in [18] 4G N S EE = d d−1 σ det(g ab ) 1 + λ 1 R(g) + Λ R 2 (g) − 4R ab (g)R ab (g) + R abcd (g)R abcd (g) .

(71)
In what follows, we shall be interested to calculate the entanglement entropy when the entangling region is of the strip type. In which case, the induced geometry is as written in eq(46). For simplicity of doing the computation, we shall fix the dimensionality of the bulk spacetime.
For d = 5: The bulk spacetime is a 5 + 1 dimensional system whereas the induced metric is a 4 dimensional spatial metric. In this case, the Gauss-Bonnet combination, R 2 (g) − 4R ab (g)R ab (g)+R abcd (g)R abcd (g), is non-zero but topological. It gives a non-zero contribution to the action but not to the equation of motion of the embedding field. This also agrees with the computation of the equation of motion of the embedding field, X M , following from eq(42). In this case, the equation of motion can also be obtained from eq(60) by considering d = 5. Hence, it is easy to conclude that the form of the hypersurface is same as in the previous case. Moreover, the holographic entanglement entropy becomes Substituting the solution, x 1 , from Appendix B, for the AdS geometry and doing the r integral resulting in the entanglement entropy to linear order in λ 1 and Λ where and r are the UV regulator and the point of maximum extension along the r direction, respectively. It is interesting to observe that to the linear order in Λ, the Gauss-Bonnet coefficient does not enter in the finite term of S EE whereas it enters in the divergent piece.
For d = 6: For this case, with the induced metric as written in eq(46) gives the following holographic entanglement entropy functional The equation of motion that follows takes the following form This equation of motion can be re-written as d dr Upon solving the equation of motion, we find to linear order in λ 1 and Λ as where c is a constant of integration and is determined by demanding that as r → r , x 1 (r ) diverges. Substituting this form of the solution into the action, results in Let us use the geometry of AdS spacetime and do the r integral from , the UV cutoff, to the maximum extension in IR, r . This gives where we have set the constant c = R 5 0 /r 5 . In our previous studies, we found that the divergent term to the entanglement entropy goes as 2−d and the finite term has the following dependence, r 2−d . For the Gauss-Bonnet term in the holographic entanglement entropy functional, we found this behavior again. Now computing the size from eq(77) to linear order in the couplings where the ellipses stands for the terms higher order in the couplings. Re-expressing the entanglement entropy in terms of the size Using the following relation R 0 = R (1 − 3aλ a − 12X 2 Λ a ), we can re-write where we have set λ 1 = aλ a R 2 and Λ = 3X 2 R 4 Λ a . Upon comparing eq(36) with eq(C.24) of [18], we find a = 1/6 and X 2 = −1/24 for d = 6, after identifying the coupling λ a = λ there and Λ = µ there . From this expression of the entanglement entropy, it is easy to notice that S EE obeys the following differential equation for d = 6 4 Higher derivative corrected extremal surfaces can't penetrate the horizon The higher derivative corrected equation of motion with λ 2 = λ 3 = λ 4 = 0 in any arbitrary dimension can be re-written as Now we shall set up our logic following [22] and show that after the inclusion of the higher derivative terms, i.e., with λ 1 = 0, the extremal surfaces can't penetrate the horizon, r h . The argument goes as follows. Eq(84) essentially describes the equation of the hypersurface that ends on the boundary and goes deep into the bulk. Let us assume that it goes till r = r , which is the deepest that it can go and then turns around and ends at the boundary. So at this point, r , the derivative of the function r with respect to x 1 vanishes, i.e., dr dx 1 r = 0. Putting this piece of information into eq(84) gives Before moving onto discuss the λ 1 = 0 case, let us first discuss the λ 1 = 0 case. Also, we want to make few assumptions on the metric components. Let there be a horizon at r = r h , if there exists more than one then this is the outermost horizon. The quantity g rr changes sign as we go beyond r h , i.e., g rr < 0 for r > r h 10 and assume ∂ r g xx is always negative 11 . On summarizing the assumptions: g rr (r) = +ve for r < r h (Outside the horizon) −ve for r > r h (Inside the horizon) and g xx (r) is + ve for any r; whereas g xx (r) is − ve for any r (86) λ 1 = 0: Let us demand that the extremal surface goes deep into the bulk and has the maximum extension. It means we need to impose the above mentioned condition along with the further condition that the quantity d 2 r dx 2 1 r < 0. For vanishing λ 1 , there follows from eq(85) at r = r that [2r g rr − (d − 1)g xx ] r = 0.
For r < r h , i.e., the point r is outside the horizon. In which case, we can easily satisfy the above equation. To make things clear (r g rr ) r is -ve whereas g xx (r ) is -ve.
For r > r h , i.e., the point r is inside the horizon. In which case, we can't satisfy the above equation. It means that the extremal surfaces cannot penetrate the horizon because the sum of two positive quantity can't give zero. This is the argument put forward in [22].
Once again imposing the condition that the extremal surface goes deep into the bulk and has the maximum extension means we need to put the conditions as mentioned above along with d 2 r dx 2 1 r < 0. The corresponding equation for the hypersurface at r = r is given by As the hypersurface goes inside the horizon, r > r h , we can satisfy the above equation provided λ 1 > 0. It means for positive values of λ 1 , the spacelike hypersurfaces can cross the horizon. However, as we shall check explicitly for AdS black hole spacetime the condition  δg ab = ∂ a X M ∂ b X N δG M N . If we set λ i+1 = 0 for i ≥ 1, for simplicity, then the change in the entanglement entropy can be calculated from where h ab = δg ab . The indices are raised and lowered using g ab and its inverse. ∇ a is defined with respect to g ab . In what follows, we are going to use a particular kind of metric fluctuation, namely that is used in the paper [23], where the fluctuating metric is diagonal and asymptotes to the AdS geometry. Since, we did the computation of the entanglement entropy in full generality for the diagonal form of the metric as in (63), which suggests we can now do the computation for the fluctuating geometry as well.

One parameter fluctuation with Λ = 0
For completeness, let us write down the complete form of the metric with fluctuations and the bulk cosmological constant, Λ c where we have restored AdS radius. m is a constant and assumed to be very small, in the sense of [23]. In which case where we have set λ 1 = aR 2 λ a with a a constant. Now, we shall move onto the computation of the energy (or mass) of the excited state using the AdS/CFT dictionary as worked out in [34] which in our case using N = √ g tt , u t = 1/ √ g tt and det(σ) ij = g d−1 2 The t − t component of the energy momentum tensor can be calculated from [33] where √ grr . The quantityR is defined in such a way that as λ → 0, it approaches the size of the AdS spacetime 12 , i.e., lim λ→0R → R. It means we can writeR = R + λR 1 for small λ and we shall determine R 1 by demanding that T tt is not diverging as we approach the boundary, r → 0. The quantity λ is same as α 2 in the notation of [33].
Substituting all these ingredients into T tt gives where . The mass comes as On comparing with the bulk action and the holographic entanglement entropy functional of [17] and [18] with ours, we find that 2λ = λ 1 . So the mass of the excited state becomes (100) 12 We use R to denote the size of AdS spacetime without the higher derivative term, whereas R 0 with higher derivative term. The relation between them can be read out from [35], The AdS radius R 0 is related to R as To leading order in λ a , we take R 0 = R−(a/4)(d−2)(d−3)Rλ a . Finally, taking the ratio of the change in the entanglement with the mass gives If we assume that there exists the following first law like of thermodynamics T ent ∆S EE = ∆M , then The constant a can be fixed by comparing eq (36) with eq(C.24) of [18] and there follows At least for d = 4, we find that the quantity c is positive, which is expected. Recall, that the minimum value of λ a = −7/36, follows from [30,31,32]. In fact, for any d with positive T ent requires us to set 13 λ a ≥ −1/4. By doing the fluctuation to other parts of the metric component, the authors of [36] found a modified first law like relation which involve both the 'entanglement temperature' and 'entanglement pressure'. For non-conformal theories the T ent is obtained in [38] and [39].

Fluctuation for non-zero λ 1 and Λ
Let us calculate the change in the entanglement entropy and the mass due to the fluctuation in the geometry for d = 6. Doing the one parameter fluctuation, m, as done before in eq(93), we find that the change in entropy comes out as To this order, we can again read out the t − t component of the energy-momentum tensor from [33] We shall expandR = R + λR 1 +ΛR 2 to linear order such that as we take the couplings to zero, we do get back the size of the AdS spacetime, R. The sizes R 1 and R 2 will be determined by demanding that T tt becomes finite as we approach the boundary. Or in the limit of m → 0, the T tt component should vanish as well [34]. It gives R 1 = 8/R 0 and R 2 = −72/(5R 3 0 ). Using all these ingredients into eq(96), we find the mass becomes Now let us set the following relation between the bulk couplings λ andΛ with that appears on the holographic entanglement entropy functional λ 1 and Λ following [18] with a and X 1 are real numbers. The size of the AdS radii are related as Note that while writing down such an equation, we have already used the relation between λ,Λ and λ 1 , Λ as written above. To linear order in the coupling we take R 0 = R 1 − 1 2 λ a + 1 2 Λ a . Finally, taking the ratio The first law like of thermodynamics follows, T ent ∆S EE = ∆M , if we identify Let us fix the constants a and X 1 by comparing eq(36) with eq(C.24) of [18]. It follows that a = 1/6 and X 1 = −1/8 for d = 6, after identifying the couplings as λ a = λ there and Λ a = µ there . In which case Positivity of T ent along with λ a ≥ −1/4 requires us to set Λ ≥ 0.

Two parameters fluctuation
Now, we include the second parameter and study the change in the entanglement entropy as a function of these two parameters, m and q 2 , to the AdS geometry. Let us, write down the geometry with fluctuation as follows The original motivation to take such a form of the geometry is to compute the entanglement entropy with electric charges for RN-AdS black hole. But, it is difficult, in practice, to carry out the radial integration involved, analytically, in the calculation of the entanglement entropy. Hence, we shall treat m and q 2 as small parameters With this kind of fluctuation, we shall compute the entanglement entropy. In fact, this computation is very easy to do, in the limit of vanishing of all the λ i 's in eq(63). The radial integral will be performed from the UV cutoff, , to the turning point, r . Moreover, the size is related to the turning point, r . We obtain the entanglement entropy in terms of as For q = 0, it is easy to see that for any d ≥ 3. Whereas 2 ∂ 2 S EE + ∂ S EE is not necessarily negative. So, also the quantity 2 ∂ 2 S EE = ∂S Σ 3 ∂ . For q = 0, the following quantity but only for d ≥ 5.

Conclusion and Open question
The entanglement entropy is supposed to provide us the amount of classical/quantum information stored in a given region. The beautiful idea of [2] has led us a new way to quantify it, using the celebrated AdS/CFT correspondence. In this paper, we used the Jacobson-Myers functional [16] along with the prescription of [2] to compute the entanglement entropy of different kind of systems. Such systems are described by having different amount of symmetries and are called as Lifshitz solutions.
As per [2], one of the important ingredient require to compute the entanglement entropy is the hypersurface whose boundary coincides with the boundary of the given region under study. The explicit form of the hypersurface is found using the prescription of [9] and because of its covariant nature, the hypersuface is independent of the nature of the entangling region but depends on the bulk couplings. The form of the hypersurface is obtained, essentially, by extremizing the Jacobson-Myers functional. Apparently, it is not clear whether this form of the hypersurface holds good even for time dependent geometries as well.
Upon computing the value of the action over the hypersurface which is considered to be of the shape of a strip gives us the desired result of the holographic entanglement entropy, S EE . For a given size, , the entanglement entropy obeys the following differential equation for AdS spacetime in d ≥ 3 We have checked that even in the presence of the higher derivative terms to the holographic entanglement entropy functional, the entanglement entropy obeys the above mentioned, simple looking, differential equation for the AdS spacetime with the following caveat.
Since, the analytic computation of S EE to the higher orders in the couplings are very cumbersome. So, we have computed S EE only to linear order in the couplings and checked the above mentioned differential equation.
In [21], a useful quantity, "renormalized entanglement entropy" S Σ d is introduced, with which the authors have suggested to study the rate of flow for a sphere type entangling region. In our case, we have consider the scale R in [21] as and the surface, Σ, as the strip and define to study the flow as ∂S Σ d ∂ < 0. We have checked with the help of the differential equation obeyed by S EE and the exact form of S EE that such a quantity, ∂S Σ 3 ∂ < 0, holds true only for d = 3.
By going through an example of one parameter fluctuation, m, to the geometry, we have computed the entanglement entropy. From the result, it is highly suggestive to consider ∂ ( −1 ∂ S EE ) as the quantity that should give the rate of flow. As the quantity ∂S Σ 3 ∂ is not necessarily negative. However, by studying two parameter fluctuations, m and q, we find that such a quantity becomes negative only for d ≥ 5.
It is a priori not completely clear whether this gives the (complete) RG flow. Presumably, it is interesting to include the quantum corrections to the entanglement entropy of RT along the lines of [40,41], and find the full RG flow structure, which we leave for future research.
We have, also, studied the first law like of thermodynamics for low excited states with higher derivative term. In which case, the entanglement temperature T ent goes inversely with the size, . In fact, the proportionality constant is a function of the dimension, d, and the couplings.
sional spacetime with dynamical exponent z as where the boundary is at r = 0 and R is the size of the bulk Lifshitz spacetime. It is easy to see that it has the following scaling symmetry r → λ r, t → λ z t, x i → λ x i .
We can re-write the geometry by doing the following change of coordinates r z = ρ, t = t/z, x i =x i /z, R/z =R as which has the following scaling symmetry The rationale behind taking such a non-linear scaling of the radial coordinate is that, we do not want to change the fact that we started out with a spacetime which has a dynamical exponent z. Let us compute the entanglement entropy of these two spacetimes using the proposal of RT. According to the proposal the entanglement entropy is computed for a fixed time for which the d−1 dimensional spacelike hypersurface, γ, extrmizes the area of this hypersurface. Finally, the entanglement entropy is conjectured to take the following form, S γ = Area(γ) 4G N . Let us assume that the precise form of the hypersurface is determined by the function r(x 1 ) and ρ(x 1 ), in which case, the induced metric for these two cases are Let us consider the entangling region is of the strip type. It means 0 ≤ x 1 ≤ and −L/2 ≤ (x 2 , · · · , x d−1 ) ≤ L/2. In which case, the area becomes Essentially, we are trying to find the cases, where there appears a log term in the area. For this choice of γ = d−2 d−1 , the area up to a divergent term becomes A(γ I ) = 2R d−1 L d−2 r dr r which is the result found recently in [27].