A Appendix: Einstein-Dilaton Janus

Abstract

We discuss the computation of holographic entanglement entropy for interface conformal field theories. The fact that globally well defined Fefferman-Graham coordinates are difficult to construct makes the regularization of the holographic theory challenging. We introduce a simple new cut-off procedure, which we call “double cut-off” regularization. We test the new cut-off procedure by comparing the results for holographic entanglement entropies using other cut-off procedures and find agreement. We also study three dimensional conformal field theories with a two dimensional interface. In that case the dual bulk geometry is constructed using warped geometry with an factor. We define an effective central charge to the interface through the Brown-Henneaux formula for the factor. We investigate two concrete examples, showing that the same effective central charge appears in the computation of entanglement entropy and governs the conformal anomaly.

A note on entanglement entropy and regularization in holographic interface theories

Michael Gutperle and Andrea Trivella

Mani L. Bhaumik Institute for Theoretical Physics

Department of Physics and Astronomy

University of California, Los Angeles, CA 90095, USA


gutperle@ucla.edu; andrea.trivella@physics.ucla.edu



1 Introduction

The AdS/CFT correspondence provides the most well understood example of holography. The degrees of freedom of a theory of gravity in a geometry that includes an asymptotically space are encoded in the degrees of freedom of a dual conformal field theory, living on the boundary of the asymptotically space [1, 2, 3].

Figure 1: The top surface represents the field theory side, the two different colors identify the two sides of the interface (purple line). The vertical dimension represents the holographic direction, there are two Fefferman-Graham coordinate patches (represented with different colors) that do not cover the entire bulk geometry. In the gray wedge originating from the interface the Fefferman-Graham coordinate expansion breaks down.

The correspondence is mostly studied in the large and large t’Hooft coupling limit, when the the bulk side can be treated using semi-classical gravity. For example, the Ryu-Takayanagi formula relates entanglement entropy on the field theory side to the area of the minimal bulk co-dimension two surface anchored at the boundary of on the entangling surface [4]

(1.1)

One should note that the entanglement entropy on the field theory and gravity side are infinite and both require regularization. On the CFT side the divergence comes from the short distance degrees of freedom entangled across the entangling surface, for this reason a UV cut-off is required. On the gravity side the divergence arises from the fact that the minimal surface is anchored on the boundary of the asymptotic space, which has an infinite volume. For that reason we need to regulate it by introducing a cut-off on the holographic coordinate, this process is called holographic renormalization (for a review see [5]). The regularization is based on the fact that an asymptotically metric can be expressed in terms Fefferman-Graham coordinates [6].

(1.2)

Where has a leading independent term and terms falling off as , whose exact form depend on the dimensionality and details of the theory.

The boundary of the asymptotic metric is located at and the theory is regulated by imposing a cut-off at .

Unfortunately the construction of Fefferman-Graham coordinates which cover all of the boundary can be difficult. One example are systems with an interface (ICFT) or a defect (DCFT). In the present paper we consider holographic interface or defect solutions which are commonly known as Janus solutions, where one solved the bulk gravitational equations for a metric which is warped with an factor. For some other approaches to describe interface, defect or boundary CFTs holographically see e.g. [7, 8, 9, 10].

In these cases the small expansion used for the Fefferman-Graham construction turns out to be an expansion in small , where denotes the field theory direction perpendicular to the defect. This dependence is dictated by scale invariance. The expansion breaks down close to the defect, where . Thus there is a wedge bulk region originating from the defect that cannot be covered. In the case of a co-dimension one defect we have two different Fefferman-Graham coordinates patches that cover some portion of the bulk on the two sides of the defect and a region just behind the defect that cannot be covered. A schematic representation is given in figure 1.

This problem has been faced in literature in different ways. The authors of [11] connected the two Fefferman-Graham patches with an arbitrary curve, showing that any universal quantity would not depend on the details of this curve. To avoid dealing with Fefferman-Graham coordinates the authors of [12] simply imposed a cut off on the factor of the metric that diverges as one moves to the boundary. We refer to this regularization procedure as “single cut-off regularization”.

Recently, a third regularization procedure has been used in literature in the computation of the quantum information metric of a conformal theory which is deformed by a primary operator. Such a set up shares a lot of similarities with a DCFT [13, 14, 15] since it is natural to express the bulk metric using an slicing. In such coordinates one encounters a divergence associated to the infinite volume of the slice and a divergence associated to the coordinate that slices the bulk geometry. It is then natural to introduce two cut offs. We name this regularization procedure “double cut-off regularization”. Note that an analogous cutoff was also used to regulate holographic duals of surface operators, i.e. defects of higher co-dimensionality in [16, 17].

The purpose of this paper is to study the double cut-off regularization in more detail. We will test it against several examples to show that it provides the same results as the other regularization methods but involve much simpler computations.

The paper is organized as follows: after reviewing and discussing the main features of different cut-off procedures in section 2, we move on to discuss specific examples provided by ICFTs with a co-dimension one planar interface. In section 3 we discuss systems with an interface extended along at least two spatial dimensions. The computation of the entanglement entropy in these cases has been carried out in [11] and we find agreement between the calculations which utilize the old and new regularization methods. In section 4 we focus on three dimensional CFTs with a two dimensional conformal interface. The bulk geometry dual to this systems is given by a warped space with a factor. We associate to the interface an effective central charge through the Brown-Henneaux formula for the factor. We study two concrete examples, showing that the effective central charge obtained holographically appears also in the computation of the entanglement entropy and it is the same quantity that governs the conformal anomaly associated with a two dimensional CFT living on the interface.

2 Regularization prescriptions

In this paper we mainly focus on the computation of entanglement entropy for a ball shaped region in a CFT with a co-dimension one interface. This quantity is divergent because of the UV degrees of freedom entangled across the entangling surface. The regularization is achieved by introducing a UV cut-off. Once this is done if we want to isolate the interface contribution we need to subtract the entanglement entropy for the vacuum of the theory without interface. In this way we are able to compute a quantity that is intrinsic to the interface. To better explain this statement let us discuss in detail the divergence structure of entanglement entropy. For the vacuum state of a pure CFT and a ball shaped region of radius we have:

(2.1)

where we have introduced the UV cut-off [18]. Notice that in odd dimensions a rescaling of the cut-off does not affect constant , while in even dimension it is the coefficient of the logarithmic term, , that is not sensitive to any rescaling of . For this reason and are independent of regularization and are universal. Let us discuss how the presence of a defect affects the structure of entanglement entropy. For definiteness we start with the vacuum state of an even dimensional CFT. We then turn on a co-dimension one interface that breaks the full conformal symmetry group down to , interpreted as the conformal symmetry restricted to the interface. When this is done we expect the entanglement entropy to show terms typical of both even and odd dimensional CFTs [19]. That creates a problem in isolating the universal term characterizing the interface. In fact since the interface is odd dimensional we expect that the universal term should be a constant, however since the original CFT is even dimensional we have a logarithmic term in the divergence structure of the entanglement entropy and we are free to change the additive constant by a rescaling of the cut-off . The way to bypass this problem is to use the same cut-off for both the pure CFT and the ICFT, once that is done we can isolate the interface contribution by subtracting the vacuum component. We refer to this procedure as vacuum subtraction.

Now that we have discussed regularization and vacuum subtraction on the CFT side of the duality let’s focus on the bulk side, where all the computations will be performed. First of all we need to identify a bulk geometry dual to the interface CFT. This is realized by a metric that is invariant under transformations. The natural way to do that is to consider a bulk geometry that can be written in slices:

(2.2)

The coordinate is taken to be non compact and as we have and such that the gets enhanced to . Unless otherwise stated we will work in Poincaré coordinates for the slices

(2.3)

The boundary is approached in different ways. Taking we recover the CFT region on the right/left side of the interface, while taking we approach the CFT on the interface itself. A schematic illustration is given in figure 2.

Figure 2: Schematic representation of the slicing of the bulk geometry . Each colored line corresponds to a single slice located at a fixed value of the coordinate .

We will now describe how to regularize divergent quantities on the bulk side using three different methods.

interface

Left

FG patch

Right

FG patch

Figure 3: Schematic representation of the Fefferman-Graham regularization. Where the Fefferman-Graham coordinates are available (red and blue regions) the cut off surface is chosen to be . In the middle region a Fefferman-Graham coordinate patch is not available. The cut off surface for this region is an arbitrary curve that continuously interpolates between the left and right patches, this is represented by a black arc in the picture.
  • Fefferman-Graham regularization: The traditional approach is to make use of Fefferman-Graham coordinates. As mentioned in the introduction this is problematic in a bulk geometry that is dual to a CFT with a defect or interface. There are two Fefferman-Graham patches which do not overlap, so one cannot simply glue them together. A possibility is then to interpolate with an arbitrary curve between these two patches, this is the approach used in [11] where the authors were able to compute universal quantities that do not depend on the interpolating curve. Even though this approach is very rigorous it requires a heavy computational effort. For this reason we want to explore other regularization procedures. A schematic representation of this procedure is given in figure 3.

  • single cut-off regularization: we follow the idea of [12], regularizing all the divergent integrals by putting a cut-off at . This is motivated by the study of pure . In fact for pure with unit radius one has , we can then change coordinates to recover Poincaré by choosing:

    (2.4)

    where is the holographic coordinate and is the coordinate perpendicular to the fictitious interface. The natural cut-off procedure corresponds, in the slicing coordinates, to . For the interface solution which can be viewed as a deformation away from the vacuum we keep the same regularization procedure.

  • double cut-off regularization: this procedure is based on the observation that, after one performs the vacuum subtraction, one should be left with a quantity that is intrinsic to the interface. In that sense a cut-off should be imposed not on the full bulk geometry but on the slices, at . Of course that cut-off does not regulate all the possible divergences, since the metric factor in (2.2) diverges as as . What one should do is to introduce a second cut-off , such that , that regulates any dependent divergence. Once we subtract the vacuum contribution to the particular physical quantity in consideration we will be allowed to take , the result will be independent. To sum up, the double cut-off procedure makes use of two cut-offs and . is interpreted as a physical cut-off in the usual sense, it regulates the bulk divergence associated to the integration and it is interpreted as a UV cut-off for the degrees of freedom localized on the interface. On the other side the cut-off is a purely mathematical tool. It is used only to make any quantity that appears in the intermediate steps finite, any physical quantity should be independent.

This discussion applies to any divergent quantities that can be computed in a holographic ICFT. Let us now focus on the computation of holographic entanglement entropy. We take the entangling surface to be a ball shaped region of radius centered on the interface (see figure 4). The holographic entanglement entropy for these systems has been studied in [19], where the authors were able to show that the RT surface is simply given by , giving the following expression for the entanglement entropy

(2.5)

This equation can be adapted also for by taking .

Let us discuss how to regulate the entanglement entropy using the single and double cut-off regularizations. For the double cut-off procedure we cut-off the integral at , defined as the two roots of . In most examples is an even function, in that case , we can then focus only on and we will drop the subscript. Generally speaking the form of might be very complicated, however since eventually goes to zero we can assume large, allowing us to find . We introduce a cut-off for the integration at . We then get:

(2.6)

where the symbol denotes the vacuum subtraction. At this point we will take . The divergence will come exclusively from the integral and the result will be independent.

We will now discuss the single cut-off procedure for the entanglement entropy. In this case we put a cut-off at . We will always proceed by performing the integral first and then the integral. To do so we start by fixing and integrating in over , where are the solutions to . At this point we might be tempted to take small, however that is not possible. The reason for it is that the integration over runs over , where denotes the minimum of (in most examples that corresponds to ). Nonetheless we can expand as a Laurent series in . Once this is done we will proceed to the integration, whose details depend on the concrete examples we will examine.

Notice that one could work in different coordinates than (2.2). In particular one could change coordinates from to another coordinate, say . The function will then be replaced with another function, say . In that case the regularization procedures just described will go through without any change, one would simply put a cut-off for the integration at for the double cut-off procedure and at for the single cut-off procedure.

Figure 4: Representation of a time slice of the field theory side. Two regions (blue and red) are separated by a interface (purple). We compute the holographic entanglement entropy for a ball centered on the interface. The Ryu-Takayanagi surface is represented in green.

3 Higher Dimensional Examples

In this section we discuss the computation of the holographic entanglement entropy for ICFT that present an interface extended on at least two spatial dimension. We will leave the discussion of lower dimensional cases in section 4.

3.1 Supersymmetric Janus

In this section we discuss the entanglement entropy for a ball shaped region for a Yang-Mills interface that preserves 16 supercharges [20, 21]. That is realized in the bulk by a metric that explicitly exhibits symmetry where the first factor is associated to the conformal symmetry preserved on the interface and the other two factors are related to unbroken R-symmetry. The full supergravity solution also has the dilaton, the three-form and the five-form are turned on in the bulk, see [20] for details. In the following we will only need the metric which is given by:

(3.1)

The coordinates and parametrize a two dimensional Riemann surface with boundary. The functions , , and depend on , and they can be obtained from two functions and in the following way:

(3.2)

where

(3.3)

For the supersymmetric Janus solution we have:

(3.4)

with and and . The asymptotic regions located at correspond to the two sides of the interface, where the dilaton assumes different values corresponding to different values of the Yang Mills coupling constant . The constants and are reals and they are related to the radius and to the Yang Mills coupling constant by:

(3.5)

Equation (2.5) gives the following expression for the entanglement entropy of a ball shaped region centered on the interface:

(3.6)

We now need to specify the cut-off procedure. We dedicate the next two sections to two different regularizations.

Single cut-off

For the single cut-off procedure we have:

(3.7)

We start by fixing letting varying from to , with defined by:

(3.8)

Notice that even though we are going to let , we cannot assume to be large, since . Nonetheless we can expand in Laurent series of . We have:

(3.9)

thus:

(3.10)

Of course the coefficients in the sum are going to be different with respect to the one of the previous equations, but since we are not really interested in those coefficients we will adopt a loose notation. We can now perform the integral over :

(3.11)

We proceed with the integration over :

(3.12)

Integrating over and taking leads to:

(3.13)

for some constant . Subtracting the vacuum contribution leads to1:

(3.14)

for some constant , however note that is non universal. The universal contribution is given by the first term in (3.14):

(3.15)

Double cut-off

We introduce two different cut-offs and . We will use to regulate the integration over and to regulate the integration over . Remember that by vacuum subtraction we are going to obtain a result that is -independent.

Let’s start with the integration. We regularize it by cutting off the integral at , where is defined by:

(3.16)

Notice that since , , thus we can use the following asymptotic expression for :

(3.17)

We get:

(3.18)

We then have:

(3.19)

For the integration we put a cut-off at . We have:

(3.20)

The integration is finite and gives a factor. We obtain:

(3.21)

Remember that in ICFT the physical information can be extracted only after a background subtraction. We obtain:

(3.22)

The universal contribution is

(3.23)

Notice that we get the same result independently of the regularization procedure adopted. Moreover our result matches the expression found in literature using the Fefferman-Graham regularization [11].

3.2 Non Supersymmetric Janus

The Non Supersymmetric Janus [22, 25] is a solution of type IIB supergravity where the vacuum solution is deformed into the following metric

(3.24)

where

(3.25)

and is the -Weierstrass function obeying , with and . The deformation depends on a real number called Janus deformation parameter. corresponds to the vacuum solution. The metric is supported by a non trivial dilaton and RR five-form. This solution breaks all supersymmetries. Notice that diverges as , defined by . The dilaton takes two different values in these asymptotic regions and the metric asymptotes to . We interpret the bulk configuration as being dual to a deformation of SYM, where an interface is present and the Yang Mills coupling constant takes different values on the two sides of the interface.

Once the metric is available we can use equation (2.5) to write the entanglement entropy for a ball shaped region of radius centered on the interface. We have:

(3.26)

We now discuss in detail the two regularization procedures explained in 2.

Single cut-off

We introduce the cut-off by

(3.27)

We start with the integration over . The cut-off for the integral is given by . Notice that we cannot simply take small, since eventually is going to be when performing the integral. Nonetheless we can perform a Taylor expansion in , we find:

(3.28)

We then get:

(3.29)

for some coefficient and

(3.30)

We have now to perform the integral, in particular :

(3.31)

Let’s look at the last term of . When we integrate the generic -th term we obtain two terms, one behaving like and the other as , this means that the third term in contribute to the divergence structure of with a term of the form . Let’s now focus on the remaining terms, the integration is straightforward, one gets

(3.32)

where we have dropped the terms that vanish as we take .
The vacuum entanglement entropy is given by taking :

(3.33)

We then have:

(3.34)

the universal contribution is given by:

(3.35)

Double cut-off

We regulate the integral and the integral using two different cut-offs. Let’s start with the integral over . This integral is divergent because blows up at , defined by . In order to regularize this integral we introduce a cut-off at , defined in the following way:

(3.36)

solving for one gets:

(3.37)

Expanding in we get:

(3.38)

At this point we perform the integration over we get:

(3.39)

where has been defined in equation (3.30). and we have introduced the Weierstrass and functions. For the integral we place a cut-off at we finally obtain

(3.40)

The holographic entanglement entropy for the vacuum is found by considering :

(3.41)

After vacuum subtraction we obtain:

(3.42)

The universal contribution is given by:

(3.43)

Notice that we get the same result independently of the regularization procedure adopted. Also in this case our result matches the expression found in literature using the Fefferman-Graham regularization [11].

4 Two dimensional holographic interfaces

In this section we are going to focus on gravity solutions representing a two dimensional interface. It has been observed in various contexts that in a three dimensional CFT with a two dimensional conformal defect one can associate an effective central charge to the defect [23, 24, 17]. This central charge appears both in the entanglement entropy and in the Weyl-anomaly of the theory.

The fact that we can identify an effective central charge can be understood holographically. The argument is that when a 1+1 dimensional interface enjoys conformal symmetry we expect the dual bulk geometry to present an factor, we can thus associate an effective central charge to the interface through the Brown-Henneaux formula [26]. This was first done in [17] in the context of type IIB supergravity solutions dual to half-BPS disorder-type surface defects in Super Yang-Mills theory. It was also observed that the effective central charge arising from the Brown-Henneaux formula was the same quantity that appears in the computation of the entanglement entropy. In this section we explore other examples of a 1+1 dimensional interface which enjoys conformal symmetry.

In particular we focus on examples where the 1+1 interface is embedded in a 3 dimensional theory. In addition to the computation of entanglement entropy we calculate the conformal anomaly and show that it is governed by the same central charge appearing in the entanglement entropy computation and arising from the Brown-Henneaux formula. Before going over explicit examples we prove the following statement: in an ICFT with an even dimensional interface embedded into an odd dimensional spacetime the universal contribution of entanglement entropy for a spherical entangling surface centered on the interface is equal to minus the universal term of free energy on a sphere.

We explicitly prove this statement for a 3 dimensional theory with a 2 dimensional interface. The generalization to arbitrary dimensions is straightforward. The proof follows closely section 4 of [28]. The field theory lives on a three dimensional spacetime given by:

(4.1)

where we have chosen polar coordinate for the spatial slice. The interface is located at . We perform the following change of coordinates:

(4.2)

The spacetime is then given by

(4.3)

which, after removing , corresponds to the static patch of de Sitter space with curvature scale R. It can be shown (for details see [28]) that the new coordinates cover the causal development of the ball on the surface (which is exactly our entangling region). In addition one can show that the modular flow generated by the modular Hamiltonian in the causal diamond corresponds to time flow in this new coordinate system and that original density matrix can be written as a thermal density matrix with temperature . This implies that the entanglement entropy of the ball shaped region can be written as a thermal entropy:

(4.4)

where is the free energy and is the expectation value of the operator which generates time evolution, explicitly:

(4.5)

where is a constant slice, is the unit normal and is the Killing vector that generates translations .
To compute we need to write an expression for . A powerful tool to do that is symmetry. In fact we know that the interface is extended along the surface which corresponds to a two dimensional de Sitter spacetime. The isometry of de Sitter space forces the stress tensor to satisfy the following relations:

(4.6)

where and denote any of the coordinates and . This suffices to show that is finite. On the other side, since the interface is even dimensional we expect a logarithmic divergence in both and . This means that does not contribute to the universal terms in equation (4.4), thus:

(4.7)

In order to find we go to imaginary time with periodicity . The metric becomes

(4.8)

which we recognize as the metric of once we identify . Thus:

(4.9)

as anticipated.

We would like to relate this quantity to an effective central charge (since we are in presence of a two dimensional conformal field theory living on the interface). To do that we focus on . For definiteness let’s say we locate the interface at the equator of the sphere. By the same symmetry arguments as in the de Sitter case we have:

(4.10)

where and denotes the directions along the interface and is the metric of the sphere

(4.11)

with and corresponding to the location of the interface. If we change the radius of the sphere by we have:

(4.12)

where we have used equations (4.10) to get the final result. This shows that the coefficient of the logarithmic term of entanglement entropy is related to the coefficient of the Ricci scalar in the conformal anomaly2.

Notice that a priori this is a non trivial fact. In a two dimensional CFT the only central charge is the coefficient of the Ricci scalar in the trace anomaly, but in a ICFT the situation is more complicated. In fact the 1+1 dimensional interface is embedded in a higher dimensional spacetime where the theory lives, thus other terms, such as the trace of the extrinsic curvature, could contribute to the trace anomaly.

In the following we are going to focus on specific examples. We are going to compute both entanglement entropy and free energy holographically and we will show that equation (4.9) holds. To find the free energy holographically write the metric in the same form as in equation (2.2), replacing with its Euclidean counterpart, named

(4.13)

where and we have sliced using spheres. The free energy can then be computed holographically as the on shell action . We are going to use ony the double cut off procedure, one can obtain the same results using the single cut off regulator.

4.1 3 dimensional Einstein-Dilaton Janus

The first example we discuss is a bottom up system. We can construct an ICFT from a CFT by considering a marginal operator and assigning to it a coupling constant that jumps across a 1+1 dimensional plane. We construct the bulk theory dual to this deformation by solving the equations of motion derived from the action of a massless field , dual to , minimally coupled to the metric. In particular one has

(4.14)

from which one finds:

(4.15)

where and is defined by . The parameter quantifies the strength of the Janus deformation, and one recovers for . Notice that the bulk geometry is covered using two different patches, the patches smoothly join at while the boundary is located at . There are two boundary regions (glued together at ) that correspond to the two different sides of the interface.

Holographic Entanglement Entropy

As usual we take the entangling region to be a ball or radius centered on the interface. From equation (2.5) we get:

(4.16)

Working with the double cut-off regulator requires to compute

(4.17)

The expression of as a function of is:

(4.18)

We change variable of integration by introducing :

(4.19)

Using the fact that one can write

(4.20)

where , and . Using the change of coordinate , we write the integral in the following form:

(4.21)

with and . We note that for we have . This is an elliptic integral and can be found in [27]. It evaluates to:

(4.22)

We expand (4.1.1) for small , we get: