Advances in High Energy Physics

Volume 2015, Article ID 523408, 11 pages

http://dx.doi.org/10.1155/2015/523408

## Weak Minimal Area in Entanglement Entropy

^{1}Department of Physics, Utkal University, Bhubaneswar 751004, India^{2}Centre of Excellence in Theoretical and Mathematical Sciences, Siksha ‘O’ Anusandhan University, Khandagiri Square, Bhubaneswar 751030, India

Received 18 November 2014; Accepted 8 February 2015

Academic Editor: Sally Seidel

Copyright © 2015 Shesansu Sekhar Pal and Shubhalaxmi Rath. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We revisit 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 unable to cross the horizon, which is in agreement with that studied earlier by Engelhardt et al. and 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, .

#### 1. Introduction

The recent conjecture on the holographic formulation of the entanglement entropy by Ryu-Takayanagi (RT) [1, 2] has given a new direction to do explicit calculations in the field theory provided that it admits a dual gravitational description (in a recent development in [3], the authors have conjectured the existence of a geometric entropy in a theory of quantum gravity that includes it in the entanglement entropy) [4–7]. In order to compute the entanglement entropy of a given region, , with its complement in the field theory, it proposes with a fixed time slice to consider a codimension two-hypersurface, , in the bulk in such a way that its boundary coincides with the boundary of the region under study; that is, . Moreover, we need to consider the hypersurface that minimizes the area. In this case, the entanglement entropy is simply given by the area of the hypersurface divided by , where is Newton’s constant and it reads as

Recall that the area of a codimension two-hypersurface is given by where and 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 [8] and is further studied (some other interesting studies are reported in [9, 10]) in [11–14]: where is the inverse of the induced metric, . and 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 ’s in (3) for a given bulk geometry and substitute that into the area integral. However, it is not* a priori* clear that the solution of (3) will necessarily give us a minimum area. It can give a maximum, a minimum, or a point of inflection/saddle point. It is suggested in [15] 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 needs to opt for the solution that gives a minimum area.

In this paper, we want to study the (weak) minimal condition on the entanglement entropy functional with the Minkowski signature for generic that follows from (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 (2), which gives where the column vector 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 to the boundary, also a term proportional to the equation of motion. If we want the area to be a minimum then the determinant of the matrix should be positive. The Jacobi test talks about the positivity of the matrix 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 (3) and (5) so as to draw any useful conclusion. (However, it is certainly very interesting to find connection between (5) and the extrinsic curvature as proposed in the context of black holes in [16], if there is any.) Instead, in what follows, we will calculate the quantity (5) in different examples and check whether the area is (weak) minimum or not.

In this paper, we study the consequence of 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 minimal area condition of the RT conjecture gives us a very interesting consequence; that is, the spacelike hypersurfaces do not cross the horizon. This conclusion matches precisely as studied in [17], where the author did not find any solution to the embedding field, , of (3) inside the horizon and further studied in [12] at finite ‘t Hooft coupling and more generally in [16].

By studying different examples, we find that the second variation of the area functional can be written as where and is one of the coordinates on the hypersurface. The integral is over the world volume coordinates of the codimension two-hypersurface.

The weak minimality condition states that the second variation of the area functional with respect to becomes positive: and the Jacobi test says that the determinant of the matrix should be positive, . In this paper, we will be checking these conditions by studying several examples.

It is also very interesting to ask the minimal nature of the entanglement entropy functional even in the finite ‘t Hooft coupling limit. (A prescription is given in [18, 19] to construct the entanglement entropy functional in such cases.) In this context, it is argued in [18, 20] based on the strong subadditivity property that the first possible higher derivative correction of the entanglement entropy functional indeed obeys the minimality condition. For our purpose, we consider the following entanglement entropy functional, as also studied in [21–23]: where denotes the Ricci scalar made out of the induced metric. We found the following constraint upon demanding the weak minimality of the entanglement entropy functional:

Note that we denote as the radius of the AdS spacetime. The constraint on the Gauss-Bonnet coupling, , does not coincide with the result obtained in [24, 25]. So it means that the hypersurface under study does not have either minimal or maximal entanglement entropy. The maximal entanglement entropy is ruled out; otherwise, the Gauss-Bonnet coupling can be as large as infinity. Through this study, there follows an important result; that is, theories without higher derivative terms do admit a minimal hypersurface but not with. (The caveat is that the weak minimality analysis is performed only to leading order in the coupling.) Hence, the nature of the hypersurface with the higher derivative term remains to be seen in the future.

#### 2. Example: Strip Type

In this section, we will 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 (3). The strip on the field theory is defined as and . Moreover, the bulk geometry is assumed to take the following form:

With the following embedding fields , the induced metric is

In this case, the area takes the following form: , whose second variation gives the following column vector, , and the matrix, :

This means

In order to have a minimum area functional, should be positive and . Note that the determinant of the matrix is and , where the expressions for these quantities are

The meaning of the derivative is as follows: and . Generically, it is very difficult to draw any conclusion on the determinant of matrix . However, it is easy to show that the quantity is positive. This follows by considering the solution that follows, in fact as constructed in [12], , in which case and the expressions for are very cumbersome to write down explicitly. The quantity, , is determined by requiring that vanishes there.

Note that . In order to check the weak minimality condition on the area functional, we need to look at the condition , which is obeyed automatically. Now moving on to determine the sign of the determinant of the matrix , generically, it is very difficult to draw any conclusion. Nevertheless, we will check it on case-by-case basis.

*AdS*. To begin with, let us consider the AdS spacetime with radius and the boundary is at , in which case
where we have considered . The quantity
We know that the surface under study starts from the boundary and goes all the way to but does not go past , which means the above quantity is positive only close to , whereas close to UV, it becomes negative. This result suggests that the weak minimum is not a sufficient condition.

*HSV*. For hyperscale violating (HSV) solution in the convention of [26] with where is a constant, the positivity of is easy to observe whereas the is

It is easy to see again that, close to UV, the becomes negative and becomes positive close to for both positive and negative ’s.

*Black Hole*. Let us consider a black hole; for simplicity, we assume it asymptotes to AdS spacetime with the boundary to be at . In this coordinate system the horizon is located at . Moreover, is positive for all values of and it takes the following form:

It follows from (15) that as the hypersurface goes inside the black hole, the quantity, , becomes negative, whereas outside the horizon, it stays positive. So, we see that if the hypersurface stays outside the horizon, as suggested in [16, 17], then it follows naturally that there exists a (weak) minimality condition on the area functional.

In order to check the sign of the determinant of the matrix , let us take the following choice of the metric components: In this case, we get where is the turning point of the solution, which is the maximum reach of the hypersurface in the bulk.

Let us rescale and , so that and are dimensionless. For simplicity, we take , in which case Generically, is a function of two variables and . It is very easy to see that, close to UV, that is, for very small values of , the function becomes negative. It means that close to the boundary the determinant of matrix is not positive. So the weak minimality condition is not a sufficient condition.

Let us recall that . It means that when we are inside the horizon and when we are outside the horizon. If the turning point is inside the horizon, then . This means that . Similarly, for outside the horizon, , which means that . In summary, For simplicity, we will restrict to stay from for outside the horizon which means , whereas, for inside the horizon, we will take which means .

The quantity is plotted inside the horizon for AdS black hole in -dimensional spacetime in Figure 1. It is easy to notice that this quantity is always negative inside the horizon of the AdS black hole.