MCTP-07-30

Holographic Entanglement Entropy at Finite Temperature

Ibrahima Bah, Alberto Faraggi,

Leopoldo A. Pando Zayas and César A. Terrero-Escalante

Michigan Center for Theoretical Physics

Randall Laboratory of Physics, The University of Michigan

Ann Arbor, MI 48109-1040

Departamento de Física, Centro de Investigación y Estudios Avanzados del IPN,

Apdo. Postal 14-740, 07000 México D.F., México

Using a holographic proposal for the entanglement entropy we study its behavior in various supergravity backgrounds. We are particularly interested in the possibility of using the entanglement entropy as way to detect transitions induced by the presence horizons. We consider several geometries with horizons: the black hole in , nonextremal Dp-branes, dyonic black holes asymptotically to and also Schwarzschild black holes in global coordinates. Generically, we find that the entanglement entropy does not exhibit a transition, that is, one of the two possible configurations always dominates.

## 1 Introduction

Given a system in a pure quantum state and density matrix , if we split the system into two subsystems and , the reduced density matrix is obtained by tracing over the degrees of freedom in the complementary subsystem, say, . The entanglement entropy is defined as the von Neumann entropy

(1.1) |

This provides a measure of how entangled or “quantum” a system is. Entanglement plays a central role in quantum information theory as it determines the ability to send quantum information [1]. The entanglement entropy also plays an important role in the study of strongly correlated quantum systems [2].

The above definition is completely field theoretical. Interestingly in the context where such field theories have supergravity duals, a prescription for the holographic computation of the entanglement entropy has been provided [3],that is, a prescription for computing completely within the holographic gravity dual. Inspired by the Bekenstein-Hawking entropy, it suggests to calculate the entanglement entropy as the area associated to a minimal surface whose boundary is the region in the field theory living at the boundary:

(1.2) |

This recipe has been successfully applied to various systems and extended in different directions [4] including its covariant formulation [5]. A slightly modified version of the entanglement entropy is

(1.3) |

The entropy is obtained by minimizing the above action over all surfaces that approach the boundary of the subsystem . In a very interesting paper [6], it was suggested that in the presence of regions with collapsing cycles, which are typical for supergravity duals of confining field theories, alternative surfaces arise (see also [7]). By comparing the entropy due to two different configurations, it was shown that the entanglement entropy could be an order parameter for the confinement/deconfinement transition. The motivation comes from the fact that the entanglement entropy jumps from a configuration with dominant term of the form to a configuration with leading term for supergravity backgrounds describing theories with gauge groups in the large and fixed ’t Hooft coupling limit.

One intriguing fact about the results of [6] is that, by analyzing a surface in the supergravity dual to the confined phase of the field theory, one is able to anticipate the existence of a deconfined phase. This prompts the natural question of whether the exploration of the deconfined phase can equally well give information about the existence of a confined phase.

According to the AdS/CFT correspondence [8], the dual of field theories at finite temperature involves black hole horizons on the supergravity side [9]. The horizon provides a natural end of the space similar to the situation discussed in [6]. In fact, in the context of the AdS/CFT is has already been established in various situations that there are phase transitions associated to different behavior of surfaces describing branes in the presence of a horizon [10, 11, 12, 13, 14, 15].

In this paper we explore the entanglement entropy for various field theories at finite temperature using its holographic definition. Namely, we study the entanglement entropy for various black hole geometries. We define our subsystem to be roughly determined by an interval of length on a curve generated by a spacelike killing vector in the conformal boundary of the geometry. Generically, there are two surfaces that satisfy the boundary conditions: a smooth one and a piece-wise smooth (see figure 1). We study the behavior of these two surfaces as a function of the distance .

In section 2 we discuss the gravity dual of 2D CFT at finite temperature – the BTZ black hole – where the smooth surface is just the geodesic anchored on . The piece-wise smooth surface is the curve composed of the lines stretching from the conformal boundary to the horizon connected by the segment of length on the horizon. We show that the smooth configuration is always dominant from the point of view of the thermodynamical comparison. Section 3 presents a general setup for the computations at hand and discusses explicitly the case of the the supergravity backgrounds describing the plasma and other field theories at finite temperature. Also in section 3 we extend the analysis to black -branes corresponding to various gauge theories on the world volume of -branes at finite temperature. Using the supergravity backgrounds for nonextremal Dp-branes for all values of , we found that the entanglement entropy is given by the smooth surface for all , except for . In the case we observe that the entanglement entropy is given by the smooth surface at large and by the piece-wise smooth surface at small . Furthermore in these geometries, we observe that the radius of the black hole factors as an overall scaling factor. This scaling implies that the entanglement entropy has the same (number of color) dependence for all the temperatures. In section 5, we study the entanglement entropy in black hole geometries in global and . The main motivation for our analysis comes from the fact that the Hawking-Page phase transition takes place only in global coordinates [16], that is, for the nonextremal Dp-brane geometries there is no Hawking-Page phase transition. In the context of the AdS/CFT this is interpreted as a transition for the dual field theory on a sphere [9]. Since this feature is absent in the Poincare patch we are motivated to study the entanglement entropy in global AdS backgrounds. However, we find that in global coordinates, the entanglement entropy is given also by the smooth surface.

Our work suggests that starting from the deconfined phase, changing the lenght of the subsystem does not allow for a transition into confinement, as opposed to what was observed in [6, 7]. According with their results, at zero temperature the length of the subsystem seems to play the role of temperature in the confined phase, while in our case this lenght corresponds possibly to some other thermodynamical quantity. It is probably worth mentioning a recent study of the topological entanglement entropy [24] where no change in the dependence on was observed. However there is mounting evidence that the entanglement entropy is a bona fide thermodynamical quantity and its precise meaning should be illuminated through further work. In section 6 we discuss some of the directions suggested by our exploration of the entanglement entropy in the context of finite temperature.

## 2 2D CFT at finite temperature from the BTZ black hole

We begin this section with a discussion of the holographic entanglement entropy using the gravitational background dual to a 2D CFT at finite temperature. Most of the calculation was explicitly done in [4], however, we reproduce it here making emphasis on the thermodynamical competition between the two configurations, something that was not considered in [4]. The relevant geometry holographically describing the field theory at finite temperature is the BTZ black hole [17]:

(2.1) |

The smooth surface is parametrized by constant and . Subsystem A is defined as the region given by where is the circumference of the boundary. Thus, is the characteristic size of A. For the BTZ black hole, the smooth surface is just the geodesic in the bulk that connects the two boundary points of A. The action of this curve is given by

(2.2) |

The equation of motion can be integrated to give

(2.3) |

This allows us to relate the length in the direction with the minimum of the curve, :

(2.4) |

The temperature is . The gravitational theory on with radius is dual to a 2D CFT living on its boundary with central charge [18].

The length of the geodesic is then

(2.5) |

where is a UV cutoff.

The entanglement entropy given by the continuous configuration is:

(2.6) |

As noted in [3], this result is in agreement with the field theory result for a 2D CFT at finite temperature [19]:

(2.7) |

where is a ultraviolet cutoff that can be thought of as a lattice spacing.

The piece-wise smooth surface is given by the parametrization gluing the 3 surfaces described as by , and . The area for this curve is then

(2.8) |

The first term of the piece-wise smooth configuration consists of two lines that go from the boundary to the horizon, their contribution to the entropy is:

(2.9) |

By construction, this entropy is independent of the length of the interval that defines the subsystem . In intrinsic field theoretic terms we could rewrite it as:

(2.10) |

The study of entanglement entropies in 1+1 system is well developed. In [20] a formula for the entanglement entropy of a system of length was obtained for conformal field theories: . Some extensions of this result have been discussed recently, including systems away from criticality [19, 21, 22]. Systems close to a phase transition (large but finite correlation length) are described by massive quantum field theories with mass inversely proportional to the correlation length [19]:

(2.11) |

The correlation length that appears above is considered to be the inverse of the mass . The correlation length and the inverse temperature can be naturally identified in our setup completing the matching of the gravity (2.10) with the field theoretic one. Interpreting the second term in the piece-wise smooth configuration is more challenging, we simply note that in field theoretic terms the contribution coming directly from the horizon takes the form: .

Now consider the difference in area:

(2.12) |

As expected, the difference is UV finite. Setting it to zero we obtain:

(2.13) |

This equation has no solutions and it follows that

(2.14) |

This implies that the entropy is always given by the smooth surface. As noted in [4] the answer matches precisely the field theory calculation [22]. More impressive agreement has also been found in the context of disconnected segments in the field theory [23].

## 3 Entanglement entropy for nonextremal Dp branes

The next geometries that we study are static geometries with a spacelike killing vector that commutes with all other killing vectors. Furthermore we assume that these geometries have conformal radius . Thus there exist a frame where the metric is given as,

(3.1) |

where is the affine parameter along and are coordinates of the internal submanifold. In general, this submanifold can have compact and non-compact directions and can have non-trivial dependence on the coordinates. However, the geometries considered here satisfy:

(3.2) |

where the ’s are non-compact coordinates and ’s are compact. The conformal boundary is obtained by taking the large limit. The region that we will consider is parametrized by:

where is the size of the subsystem . As discussed above, there are two surfaces that satisfy the minimal surface condition. The smooth surface is given by,

(3.3) |

where is the location of the holographic boundary. The second surface is piece-wise smooth and parametrized as:

(3.4) |

where is the lower bound of . Now we can proceed to study the difference in area of these surfaces.

### 3.1 Two branches of the holographic entropy

We start by computing the area of the smooth surface. The induced metric is:

(3.5) |

where prime denotes derivative with respect to . The volume element is then

(3.6) |

where and are the determinant of the metric for the non-compact and compact sub-manifolds, respectively. Since the non-compact sub manifold will contribute an infinite volume, we must work with the area density. It is given by

(3.7) |

where is the dilaton. Before we proceed, we define the following quantities:

(3.8) |

The area is:

(3.9) |

Now we want to find the that minimizes the area. Since the Lagrangian does not explicitly depend on , we can use the conserve Hamiltonian to integrate the equation of motion. It is then:

(3.10) |

The constant can be obtained by considering the turning point where . We thus obtain

(3.11) |

Physically, is the minimum of the surface. It corresponds to since the surface must be symmetric under . The quantity, , can be used to label different surfaces for different values of . This relationship is obtained by integrating the equation of motion one more time,

(3.12) |

Similarly, we can obtain the area as an integral over ,

(3.13) | |||||

We also compute the area of the piece-wise smooth surface. The induced line elements for different segments are

(3.14) | |||||

(3.15) |

The area of the piece-wise smooth is then

(3.16) |

In what follows it suffices to identify and from the geometries; the difference in area and are given by;

(3.17) | |||||

(3.18) |

It should be noted here that when , the difference in area is negative since

(3.19) |

Similarly if fast enough as , then

(3.20) |

This statement is almost always true. We will see an exception where . Thus we obtain two important results before we even start to look at specific geometries

(3.21) |

Thus, if the difference in area is monotonous, the smooth surface gives the entanglement entropy. Now we proceed to study geometries corresponding to holographic duals of the plasma at finite temperature and other field theories leaving in Dp-branes world volumes and dyonic black hole.

### 3.2 Entanglement entropy in the plasma

The holographic background dual to the plasma is a stack of nonextremal D3 branes. This background has proved to be an interesting playground for finite temperature field theories, in particular, it seems to catch some features displayed by experiments at the Relativistic Heavy Ion Collider. The corresponding background metric is:

(3.22) |

with

(3.23) |

where the parameter completely characterizes the temperature of the background: . The coordinate is related to the holographic coordinate by where is the AdS radius. The quantities and are:

(3.24) |

The difference in area and are:

(3.25) | |||||

(3.26) |

where and . The coordinate makes it clear that is just scales with respect to the temperature. We numerically plot as function of in units of .

### 3.3 Entanglement entropy for gauge theories on the world volumes of Dp branes

The backgrounds dual to finite temperature field theories are given in the string frame as [25]:

(3.27) |

where and . The dilaton is also given as:

(3.28) |

It is important to note that for , the radius of the -sphere diverges in the UV limit and shrinks in the IR limit. However, for , it shrinks to zero in the UV limit and expands in the IR limit. At , it decouples from the AdS sector. A similar feature takes place for the dilaton. The quantities and are

(3.29) |

where

(3.30) |

The difference in areas and as integrals over are

(3.31) |

We observe something interesting here. The difference in areas scales as for all while scales as . For , it is proportional to . In figure 3 we plot the difference in areas in units of with respect to in units of for . In figure 4 we show the plot for .

Here we observe a transition for and no transition for . This result is also in agreement with result 3.21.

## 4 Dyonic Black Hole

This is a very important system from the holographic point of view. It has the potential to describe a few systems that are certainly of interest from the condensed matter point of view. A partial list of interesting applications can be found in [28].

### 4.1 The solution

The dyonic black hole is a solution to Einstein Maxwell on . The solution is a consistent truncation of 11 dimensional supergravity on . Some interesting properties of this solution and its potential applications to condensed matter have been recently discussed in various papers including, [26, 27, 28]

The relevant metric is:

(4.1) |

with

(4.2) |

and electromagnetic field tensor

(4.3) |

The three parameters , and are related to the physical quantities, mass, electric charge and magnetic field of the black hole in the following way,

(4.4) |

respectively. The quantity is a constant independent of and . It is useful to also define the parameters and to characterize the effect of the electromagnetic field. In terms of the physical parameters, and are given by

(4.5) |

where . A computation of the mass using holographic renormalization can be found in [27]. Equation 4.5 has a positive real root for when

(4.6) |

which implies

(4.7) |

From these relationship, we can expect extremality at . Furthermore, we observe that even though is bounded by the mass, is allowed to take on any values. Thus if there is extremality at , we can expect different thermodynamical description of space when from when . This is discussed in the next section.

### 4.2 Comments on thermodynamics

With the coordinate change , can be rewritten

(4.8) | |||||

where satisfies

(4.9) |

We observe that has at least one positive zero at corresponding to ; and at most two positive zeros with corresponding to . We can understand this second zero by studying equation (4.9). We can rewrite it as

(4.10) |

First we observe that vanishes when and when . In addition, the function is monotonic since its derivative has no real zeros. Thus has a one to one relationship with . This justifies the above statement that has at most two real zeros. Since is not bounded, is also unbounded. The inverse of this function is then:

(4.11) |

The following picture emerges for the position of the horizon .

(4.12) |

The temperature of the horizon is given by the surface gravity,

(4.13) |

where is a timelike killing vector. With , the temperature is,

(4.14) |

We observe extremality at as expected.

In figure 5, the temperature is plotted against .

### 4.3 Entanglement Entropy

The quantities and are:

(4.15) |

The difference in area and are given as

(4.16) | |||||

(4.17) |

In the following graphs, is plotted in units of against and .

In figure 6, is plotted against for different values of ; here we observe that the difference in entropy is strictly bounded by zero from above. In figure 7(b), is plotted against for values less than 3. From this graph it is apparent that the difference in entropy approaches a constant negative value for large values . We also note that an increase in , translates to a decrease in . This feature is apparent in figure 6 and in figure 7. Furthermore, we also see that increasing , in figure 6, shifts downward. This allows us to also see that the difference in area is negative for large values of ; since small values of coincide with large values of as shown in figure 7(a). From these observations, we conclude that the difference in entropy is always negative for . Thus there is no transition.

For , we also do not observe any transition. This can be seen in figure 6 where the difference in entropy is strictly negative. In figure 8(a) the difference in entropy is plotted against . Here we observe an increase in entropy for small and then a monotonous decrease that is consistent with figure 6.

## 5 Global

### 5.1 Global

Now we explore the entanglement entropy of black holes in in global coordinates. This geometry does not satisfy the conditions of section 3 since there is no spacelike killing vector that commutes will with all killing vectors. We reformulate the problem explicitly in this case. The Schwarzschild black hole in global is given by:

(5.1) |

where the 3-sphere is written in Hopf coordinates and given by:

(5.2) |

The conformal boundary is obtained, in these coordinates, by taking the limit. The region is given by the hypersurface on the boundary parametrized as:

(5.3) |

So, measures the size of . This quantity is bounded by 1 since the period of is .

#### 5.1.1 Entanglement

The surfaces in that have the same boundary as can be parametrized as:

(5.4) |

with boundary conditions as:

(5.5) |

where is a cutoff parameter which will be taken to . The induced metric for this surface is given by:

(5.6) |

Its area is then

(5.7) | |||||

(5.8) |

The minimal surface is obtained by solving for the function that minimizes the action .

is an action for a point particle with Lagrangian:

(5.9) |

From the Euler-Lagrange equation (H=Hamiltonian):

(5.10) |

We obtain the equation of motion:

(5.11) |

where is some constant. Now we have

(5.13) |

We can determine by considering the point where . This corresponds to . It is also important to note that must be bounded by . We thus have:

(5.14) |

After integrating, we obtain the equation of motion of ,

(5.15) |

We note that this is a transcendental equation for . Thus we cannot evaluate the area explicitly. However we will be able to obtain some information when we substitute the equation of motion into the area formula.

Before we proceed, we first evaluate the piece-wise smooth surface. This surface is the sum of 3 surfaces given by , and at constant . The line element for the constant is,

(5.16) |

with area

(5.17) |

The piece has area

(5.18) |

The area of the piece-wise smooth surface is then

(5.19) |

#### 5.1.2 Comparison

Before we can compare the areas, we rewrite the area of the smooth surface as an integral over . This is done by integrating out ; and then substituting for and integrated from to . One then obtains,

(5.20) | |||||

(5.21) |

We can also write as a function of . This is done by integrating .

(5.22) |

Now we can evaluate the difference in area . It is given as: