Weak Minimal Area In Entanglement Entropy

We re-visit the minimal area condition of Ryu-Takayanagi in the holographic calculation of the entanglement entropy. In particular, the Legendre test and the Jacobi test. The necessary condition for the weak minimality is checked via Legendre test and its sufficient nature via Jacobi test. We show for AdS black hole with a strip type entangling region that it is this minimality condition that makes the hypersurface not to cross the horizon, which is in agreement with that studied earlier by {\it Engelhardt et al.} and {\it Hubeny} using a different approach. Moreover, demanding the weak minimality condition on the entanglement entropy functional with the higher derivative term puts a constraint on the Gauss-Bonnet coupling: that is there should be an upper bound on the value of the coupling, $\lambda_a<\f{(d-3)}{4(d-1)}$.


Introduction and summary
The recent conjecture on the holographic formulation of the entanglement entropy by Ryu-Takayanagi (RT) [1] has given a new direction to do explicit calculations in the field theory provided it admits a dual gravitational description 1 [2]. In order to compute the entanglement entropy of a given region, A, with its complement in the field theory, it proposes to consider a co-dimension two hypersurface, Σ, in the bulk in such a way that its boundary coincides with that of the region under study, i.e., ∂A = ∂Σ. Moreover, we need to consider the hypersurface that minimizes the area. In which case, the entanglement entropy is simply given by the area of the hypersurface divided by 4G N , where G N is the Newton's constant and it reads as Recall that the area of a co-dimension two hypersurface is given by where X M and G M N are the embedding functions and the bulk geometry, respectively. Setting the first variation of such an area functional to zero gives the following equation, which is essentially the equation of the hypersurface [4], and is further studied 2 in [5,6,7,8] g ab K S ab = 0, and where g ab is the inverse of the induced metric, g ab . γ c ab and Γ S M N are the connections defined with respect to the induced metric on the hypersurface and the bulk geometry, respectively.
In order to find the entanglement entropy, we can solve for X M 's in eq(3) for a given bulk geometry and substitute that into the area integral. However, it is not apriori clear that the solution of eq(3) will necessarily give us a minimum area. It can give maximum, minimum or a point of inflection/saddle point 3 . It is suggested in [10] that by working with the Euclidean signature, the extremization of the area functional will automatically give a global minimum of the area functional. However, with the Minkowski signature, the extremization gives saddle points and one need to opt for the solution that gives a minimum area.
In this paper we want to find the (weak) minimal area condition with the Minkowski signature for generic Σ that follows from eq (2) and study the consequences through some examples.
In order to check the minimality condition on the area or equivalently on the entanglement entropy functional, let us find the second variation of the area functional eq (2), which gives where the column vector V = ∂ δX δX and we have dropped the indices, for simplicity.
Note that in getting the result, we have dropped a total derivative term, which essentially will give a boundary term and we assume that it is not going to contribute at the boundary. Also, a term proportional to the equation of motion term. If we want the area to be a minimum then the determinant of the matrix M should be positive. The Jacobi test says about the positivity of the matrix M and it corresponds to the sufficient condition for the weak minimum.
In calculus the Legendre test says that and it gives a weak condition on the minimality of the function, in this case the area. Generically, it is very difficult to combine eq(3) and eq(5) so as to draw any useful conclusion 4 .
Instead, in what follows, we shall calculate the quantity, eq(5), in different examples and check whether the area is (weak) minimum or not. In this paper we study the consequence of using such weak minimality condition in different spacetime, such as AdS spacetime with and without the black holes, hyperscale violating geometries and geometries with higher derivative terms. In the case of the black hole geometry the Legendre test along with the RT conjecture gives us a very interesting result that is the spacelike hypersurfaces do not cross the horizon. This conclusion matches precisely as studied in [12], where the author did not find any solution to the embedding field, X M , of eq(3) inside the horizon and further studied in [6] at finite 't Hooft coupling and more generally in [11].
By studying different examples, we find that the second variation of the area functional can be written as where r = dr dx and x is one of the coordinate on the hypersurface. In getting such a form, we have used the boundary condition: δr| boundary = 0 and the integral is over the world volume coordinates of the co-dimension two hypersurface. With this choice of boundary condition, we can introduce another function W as In which case, we can re-write the second variation of the area functional as Let us demand that the function, W , satisfies the following equation In which case, the second variation of the area functional becomes minimal only when the following condition is satisfied In this paper, we shall be checking this particular condition by studying some examples. It is also very interesting to ask the minimal nature of the entanglement entropy functional even in the finite 'tHooft coupling limit 5 . In this context, it is argued in [13] and [14] based on the strong subadditivity property that the first higher derivative correction to the entanglement entropy functional indeed obeys the minimality condition. In our case, we assume that the entanglement entropy functional do admit such a minimal hypersuface. For our purpose, we consider the following entanglement entropy functional, as also studied in [22,16] and [17] where R(g) denotes the Ricci scalar made out of the induced metric. We found the following constraint upon demanding the minimality of the entanglement entropy functional Note that we have used R A to denote the radius of the AdS spacetime.

Example: Strip type
In this section, we shall check the minimality of the area functional by doing some explicit calculation for the strip type entangling region. This will be performed by finding the embedding field that follows from eq(3). The strip on the field theory is defined as 0 ≤ x 1 ≤ and −L/2 ≤ (x 2 , · · · , x d−1 ) ≤ L/2. Moreover, the bulk geometry is assumed to take the following form With the following embedding fields X t = 0, X a = x a = σ a , X r = r(x 1 ), the induced metric is In this case, the area takes the following form: xx √ g xx + g rr r 2 , whose second variation gives the following column vector, V , and the matrix, M This means δrδr.
Generically, it is very difficult to draw any conclusion on the determinant of matrix M . However, it is easy to show that the quantity A is positive. This follows by considering the solution that follows, in fact as constructed in [6] , in which case and the expression for det(M ) are very cumbersome to write down explicitly. The quantity, r , is determined by requiring that r vanishes there.
Let us re-write the second variation of the area functional as follows where we have dropped a total derivative term. This is because we have set the following boundary condition: δr( ) = 0 and δr(0) = 0. Note, 1 In order to check the minimality condition on the area functional, we need to look at the condition A > 0 and C − 1 2 d dx 1 B > 0. Generically, it is very difficult to draw any conclusion on the second condition C − 1 2 d dx 1 B > 0 for eq (17). So, we shall check it on case-by-case basis.
AdS: To begin with, let us consider the AdS spacetime with radius R and the boundary is at r = 0, in which case where we have considered g xx = R 2 /r 2 = g rr . The quantity HSV: For Hyperscale violating (HSV) solution in the convention of [18] with g xx = R 2 /r 2−2γ = g rr and γ < 0, gives as γ < 0.
Black hole: Let us consider a black hole, for simplicity, we assume it asymptotes to AdS spacetime with the boundary to be at r = 0. In this coordinate system the horizon is located at r = r h > 0. Moreover, g xx (r) is positive for all values of r and it takes the following form g rr (r) = +ve for r < r h (Outside the horizon) −ve for r > r h (Inside the horizon).
This means from eq (18), it follows that as we go inside the black hole the quantity, A, becomes negative whereas outside the horizon, it stays positive. So, we see that it is precisely the (weak) minimality condition on the area functional that says we better stay outside the horizon. This can be interpreted as if the minimality condition makes the hypersurfaces not to cross the horizon. This is one part of the minimal condition on the second variation of the area functional. Let us check the other part, namely, C − 1 2 dB dx 1 , in eq (19). Let us take the following choice of the metric components In which case we get where r is the turning point of the solution, which is the maximum reach of the hypersurface in the bulk. Let us re-scale: r = ur and r h = nr , so that u and n are dimensionless. For simplicity, we take d = 4, in which case Generically, r 5 is a function of two variables n and u. To make the function dB dx 1 > 0 means we need to set 3n 8 + u 8 + 18n 4 u 10 > 2u 4 (5u 10 + 6n 4 ). Before moving onto find the condition of that happening, let us recall that r = u n r h . It means when u > n we are inside the horizon and for u < n outside the horizon. If the turning point r is inside the horizon then r = r > r h . This means n < 1. Similarly, for r outside the horizon then r = r < r h , which means n > 1. In summary u < n, n > 1 (Outside the horizon) u > n, n < 1 (Inside the horizon) dB dx 1 is plotted for AdS 5 black hole inside the horizon for which 0 ≤ n < 1 and 1 < u ≤ 2.
For simplicity, we shall restrict n to stay from 1 < n ≤ 2 for outside the horizon which means 0 ≤ u < 1. Whereas for inside the horizon, we shall take 0 ≤ n < 1 means 1 < u ≤ 2.
The quantity r 5 is plotted inside the horizon for AdS black hole in 4 + 1 dimensional spacetime in fig(1). It is easy to notice that both the quantities A andÃ are negative inside the horizon of the AdS black hole background.
The second variation of the area functional inside the horizon becomes negative means the hypersurface that follows from eq(3) is a maximum instead of a minimum. This simply means there does not exists any hypersurface inside the horizon that minimizes the area functional. Recall, according to Ryu and Takayanagi conjecture, we need to find the area of the hypersurface that minimizes the area functional. So, we can interpret the absence of the minimal area hypersurface inside the horizon as the non-penetration of such hypersurface into the horizon. This conclusion is in perfect agreement with that reached by Hubney in [12].
Outside the horizon: Let us look at the second variation of the area functional outside the horizon. We shall see that in this case it becomes positive except for some small values of u and n. A priori, it is not completely clear why does it fails to show the positivity of the second variation of the area functional for the allowed values of u and n outside the horizon.
It is easy to see that the quantity A is always positive outside the horizon, which follows simply from eq (18). The information about the other quantity, r 5 R 3 C − 1 2 dB dx 1 , can be obtained numerically, which is plotted in fig(2).
The second figure is plotted for 3n 8 + u 8 + 18n 4 u 10 > 2u 4 (5u 10 + 6n 4 ) in fig(2), it is clear that for some restricted but small values of u and n, it does not obey this restriction. In Figure 2: The first figure is for r 5 R 3 C − 1 2 dB dx 1 and is plotted for AdS 5 black hole outside the horizon for which 0 ≤ u < 1 and 1 < n ≤ 2. The second figure is for 3n 8 + u 8 + 18n 4 u 10 > 2u 4 (5u 10 + 6n 4 ).
fact, the non-positivity of the second variation of the area functional happens close to the horizon.
The negativity of r 5 for small values of u and n says that not all regions outside the horizon can give a minimal area functional 6 . It means we need to resolve this apparent contradiction.
Resolution of the puzzle: Let us rewrite the second variation of the area functional as We can write this because of the boundary condition δr( ) = 0 and δr(0) = 0 for some function W , whose form will be determined, shortly. Now, we can re-write the second variation of the area functional as Let us demand that the function W satisfies the following equation This means in order to have a minimum area functional the quantity A should be positive, i.e., A > 0, which follows very easily.
Confining Solution: Let us study the minimality condition on the area functional in the case for which the background solution shows confining behavior. To generate such a confining background, the easiest method is to start with the uncharged black hole solution and perform a double Wick rotation. In the end the solution that asymptotes to AdS d+1 with unit AdS radius reads as The coordinate x 1 is now periodic with periodicity 2πβ, whose explicit form is not important for us. The IR is at r = r 0 and the UV is at r = 0. We can proceed further by studying two cases, depending on the fields that we are exciting.

Case 1:
The induced metric on the co-dimension two hypersurface takes the following form In which case, the area of the induced geometry for the strip times a a shrinking circle type entangling region, 0 ≤ The solution to the equation of motion takes the following form where the constant of integration c 0 is determined as follows: dr dx 2 r → 0. This means The second variation of the area functional can be written as follows where we have dropped a boundary term using the boundary condition δr(0) = 0 and δr( ) = 0. The quantities are Once again we can introduce the function W as is done in the introduction and finally we are interested in the quantity A. Using the solution for r results in It is easy to see the positivity of, A, because the radial coordinate stays from 0 ≤ r ≤ r 0 . Hence, the minimality of the area functional for this case is checked.

Case 2:
In this case, we consider the embedding field as studied in [29], i.e., the field, r, that is excited is a function of the compact coordinate x 1 . In the (r, x 1 ) plane it will be a cigar. In which case the induced geometry reads as The area functional reads as The equation of motion that follows gives the following solution where the constant of integration, c 0 , is found by demanding that the quantity, r , vanishes in the limit r → r . It sets c 0 = √ f (r ) r d−1 . On finding the second variational of the area functional using the boundary condition δr(0) = 0 and δr(2πβ) = 0 gives For our purpose, the precise form of the quantity A is not important as we are interested to find only the form of A and its sign. In the present case, it reads as Using the solution as written above, it is easy to conclude that It is interesting to note that the quantity, A, in both the cases, r(x 1 ) and r(x 2 ) gives a minimum to the area functional.
As an aside, the existence of two valid configurations means that there can be a phase transition induced quantum mechanically depending on the energy of these two configurations, which is studied in detail in [29] and [30]. But for our purposes we see that both are becoming minima to the area functional, which we set out to find.

Sphere
Let us consider another example, where the entangling region, Σ, is of the sphere type. In this context, we assume that the bulk geometry is +g rr dr 2 (42) Using the RT prescription, the geometry of the co-dimension two hypersurface takes the following form where r = dr dρ . The area functional reads as where ω d−2 is the volume form associated to the unit d − 2 dimensional sphere, S d−2 . The equation of motion that follows takes the following form where ρ = dρ dr . Upon considering the background geometry as AdS spacetime with radius R, g xx = g rr = R 2 /r 2 , the solution that follows takes the following form: ρ = √ c 2 − r 2 , where c is a constant of integration.
Let us find the second variation of the area functional as written in eq(44) for AdS soacetime where In getting the above mentioned second variation of the area functional, we have used the equation of motion obeyed by r = r(ρ). Once again we can re-write the second variation as where we have used the following boundary condition δr(0) = 0 and δr(∞) = 0. On computing the following quantity In getting the second equality, we have used the geometry of AdS spacetime. Note that both r and ρ are real and positive, hence the above quantity is positive. It is easy to find that So, we have computed the integrand of the second variation of the area functional and it comes out be positive, suggesting the minimality of the area functional for the following hypersurface r = √ c 2 − ρ 2 .

With higher derivative
In the presence of the higher derivative term in the entanglement entropy functional, it is not a priori clear that the entanglement entropy functional will be a minimum, automatically. Moreover, we cannot apply eq(5) in the determination of the Legendre test. However, it is suggested in [13] and [14] that for a very specific type of entanglement entropy functional one can get a minimal entanglement entropy functional. In the present case, we shall determine the consequence of the imposition of the minimal nature of the entanglement entropy functional for the AdS spacetime only, which depends crucially on the value of the coupling, λ 1 , as defined latter. The precise form of the entanglement entropy functional with the higher derivative term can be considered as described by the Jacobson-Myers functional [28]. In fact, for our purpose, we shall consider the structure as studied in [22,16] and [17] where λ 1 is the coupling constant. Let us evaluate the entanglement entropy for the strip type entangling region as discussed earlier. Using the structure of the induced metric g ab as written down in eq (14) gives [6] 2G where x 1 = dx 1 dr . This for the AdS spacetime 7 with the boundary at r = 0 and with the AdS radius R 0 becomes The equation of motion that follows takes the following form The second variation of the entanglement entropy functional can be expressed as In getting this expression we have performed an integration by parts and dropped the total derivative term using the boundary condition δr(0) = 0, δr( ) = 0 also used the equation of motion. The quantities defined above are Now, we can employ the same technique as stated earlier in eq (26) and eq (27). So at the end, we are interested to know whether the quantity A is positive or not. Using the solution that follows from eq(54) to the leading order in the coupling λ 1 gives where we have used the relationship between the sizes of the AdS spacetime, R 0 , and R A in the finite 'tHooft coupling limit as where f ∞ obeys the following relation: 1 − f ∞ + λ a f 2 ∞ = 0, see e.g., [19,22]. Demanding that the quantity A is positive gives the following restriction on the coupling Using the couplings used in [22], we can rewrite 8 the coupling .
The inclusion of the finite 'tHooft coupling correction to the entanglement entropy functional does not make automatically the entanglement entropy functional a minimum. Moreover, it is suggested in [14] that when the extra piece other than the area of the co-dimension two surface term in the entanglement entropy functional has the form of f (R), where R is the induced scalar curvature of the co-dimension two surface then one expects to have a minimum in the entanglement entropy functional.
Upon demanding the minimal condition on the entanglement entropy functional puts a restriction on the coupling as written in eq(59) and eq(60). Hence, we can interpret that the minimality condition essentially says that the coupling has an upper bound which is positive.
Discussion: It is suggested in [13] that the strong subadditivity property 9 should be obeyed by the entanglement entropy functional eq(51), and the integration is done over a hypersurface which minimizes the entanglement entropy functional. We noticed that such minimality of the entanglement entropy functional does not happen for all values of the couplings, λ a , however, it does happen only when we put a serious restriction on the coupling λ a as in eq(60). Hence, the imposition of the minimization condition on the entanglement entropy 8 where λ a here is same as λ in [22]. 9 The strong subadditivity property, S(A)+S(B) ≥ S(A∪B)+S(A∩B), is proven in the holographic case but without the higher derivative term in [24]. It is certainly interesting to ask whether the entanglement entropy functional as suggested, generically, in [14,15] do automatically respect the strong subadditivity property and the hypersurfce under study becomes a minimal surface. Moreover, we need to find the precise connection between the strong subadditivity and the minimal hypersurface.
functional with the higher derivative term as suggested in [13] puts a serious restriction on the coupling λ a .
In 4+1 dimensional AdS spacetime, it is suggested in [23] using the positivity of the energy fluxes and the causality that the coupling stays in a small window and can become a small negative number to a small positive number, which in our notation becomes − 7 36 ≤ λ a ≤ 9 100 . From the study of the minimality of the entanglement entropy functional, we find for d = 4, that the coupling should have an upper bound, i.e., λ a < 1 12 . Generalizing it to arbitrary d + 1 dimensional spacetime, it is found in [25,26,27] that the coupling, in our notation, should stay in the following range It is interesting to note that in the large d limit, d → ∞, both eq(60) and eq(61) gives the same upper bound, namely, 1/4.

Conclusion
In this paper, we have studied the consequences of imposing the minimality condition on the entanglement entropy functional. When such a functional is described by the area of a co-dimension two hypersurface, the RT conjecture states that the co-dimension two hypersuface should be determined in such a way for which it minimizes the entanglement entropy functional. We have checked by studying various examples like confining solution, hyperscale violating solution and the black holes in AdS spacetime for the strip type entangling region that it obeys such (weak) minimality condition. For our purpose, the outside of the black hole is described by the radial coordinate that stays from the boundary r = 0 to the horizon, r = r h , whereas the inside is described by r > r h . Let us recall from the second variation of the area functional eq (19) that it is the sign of the quantity, A, that determines whether the area functional is a minimum or a maximum. It is easy to notice using the property of g rr as mentioned in section 2 and from eq (18), that as long as we stay outside of the horizon, it gives a minimum. Once we are inside the horizon, it gives a maximum. So, we may interpret that its the horizon that acts as a surface which separates the minimum area functional from the maximum. Now upon combining this result with the RT conjecture leads to the conclusion that we better stay outside of the horizon since we want a minimum area. This finally leads us to say that the minimality of the area functional does not allow the co-dimension two hypersurface to enter into the black hole horizon. The same conclusion is obtained 10 in [12].
In a recent study in [11], it is argued that regions with negative extrinsic curvature cannot be accessed by any probes, in particular the hypersurfaces of spacelike, timelike or null type.
Let us recall that the imposition of the (weak) minimality condition gives us eq(5), which is negative inside the horizon and a priori, it is not clear whether there exists any relationship 11 between the extrinsic curvature studied in [11] and eq (5). We do expect there should exist some kind of relation between these quantities because of the similarity in their behavior, in particular, for the black hole geometry. The quantity, A, as written in eq (18) shows for r < r h (Outside the horizon) 0 for r = r h (On the horizon) −ve for r > r h (Inside the horizon).
The extrinsic curvature shows precisely the similar type of behavior [11]. The connection between these two quantities are worth studying, which we leave for future studies.
It is argued in [13] that the hypersurface should be minimal when the entanglement entropy functional is described by eq(51). Upon applying such a minimality condition imposes an important restriction on the coupling, λ 1 . This is given in eq(26) and eq (27), which essentially gives an upper bound on the coupling. Generically, it is not clear whether one can have such a minimal hypersurface for the Jacobson-Myers action functional as in [28] or the ones constructed in [14,15], which we leave for future studies.