NSF-KITP-06-31

KUNS-2021

hep-th/0605073

May, 2006

Aspects of Holographic Entanglement Entropy

Shinsei Ryu111e-mail: and Tadashi Takayanagi222e-mail:  333Moved from Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA

Kavli Institute for Theoretical Physics, University of California,

Santa Barbara, CA 93106, USA

Department of Physics, Kyoto University, Kyoto 606-8502, Japan

This is an extended version of our short report hep-th/0603001, where a holographic interpretation of entanglement entropy in conformal field theories is proposed from AdS/CFT correspondence. In addition to a concise review of relevant recent progresses of entanglement entropy and details omitted in the earlier letter, this paper includes the following several new results : We give a more direct derivation of our claim which relates the entanglement entropy with the minimal area surfaces in the AdSCFT case as well as some further discussions on higher dimensional cases. Also the relation between the entanglement entropy and central charges in 4D conformal field theories is examined. We check that the logarithmic part of the 4D entanglement entropy computed in the CFT side agrees with the AdS result at least under a specific condition. Finally we estimate the entanglement entropy of massive theories in generic dimensions by making use of our proposal.

## 1 Introduction

When we study properties of a given quantum field theory (QFT), it is common to first investigate behaviors of correlation functions of local operators in the theory. However, properties of non-local quantities are equally important, especially for understanding of its quantum mechanical phase structure. One basic such example of non-local physical quantities is the Wilson loop operators in gauge theories, which is a very useful order parameter of confinement.

In a more generic class of QFTs, we can instead consider a quantity called entanglement entropy (or geometric entropy). This is defined as the von Neumann entropy when we ‘trace out’ (or smear out) degrees of freedom inside a -dimensional space-like submanifold in a given dimensional QFT. Its complement is denoted by the submanifold . It measures how a given quantum system is entangled or strongly correlated. Intuitively we can also say that this is the entropy for an observer in who is not accessible to as the information is lost by the smearing out in region .

As its name suggests, we expect that the entanglement entropy is directly related to the degrees of freedom. Indeed, the entanglement entropy is proportional to the central charge in two dimensional conformal field theories (2D CFTs) as first pointed out in . Recently, this property was also confirmed in  in which a general prescription of computing the entropy in 2D CFTs is given. Also in the mass perturbed CFTs (massive QFTs) the same conclusion holds [3, 4, 2]. Furthermore, as we will discuss later, the similar statement is also true in 4D CFTs. The entropy is related to the 4D central charges.

In higher dimensional (more than two dimensional) QFTs, it is not easy to compute the entanglement entropy for arbitrary submanifolds even in free field theories. Motivated by this, we would like to consider a holographic estimation of the quantity by applying AdS/CFT correspondence (or duality) [5, 6]. We can find pioneering works [7, 8] that discuss related issues from slightly different viewpoints. Recently, the authors of the present paper proposed a holographic computation of entanglement entropy in CFTs from the AdS/CFT . This reduces the complicated quantum calculation in QFTs to a classical differential geometrical computation.

The AdS/CFT correspondence relates a dimensional AdS space (AdS) to a dimensional CFT (CFT), which is sitting at the boundary of the AdS. The claim is that the entropy in a dimensional CFT can be determined from the dimensional minimal surface whose boundary is given by the dimensional manifold . The entropy is given by applying the Bekenstein-Hawking entropy formula to the area of the minimal surface as if were an event horizon. This is motivated by the idea of the entropy bound [10, 11, 12] and by the similarity between the black hole horizons and the minimal surface . They become equivalent in the special cases such as those in AdS black holes  and in black holes of brane-world , as the minimal surfaces wrap the horizons (see also [7, 8, 14] for earlier related discussions444Also refer to  for recent arguments on the relation between the entanglement entropy in three qubit systems and the entropy of BPS black holes, based on similarities of their symmetries, though this does not seems to be related to our issues directly.). In  we have shown that our proposal, when applied to the lowest dimensional AdSCFT example, correctly reproduces the known results of entanglement entropy in 2D CFT. Also it is easy to see that the Bekenstein-Hawking like formula is consistent with the known ‘area law’ in entanglement entropy [16, 17] for the CFTs (also QFTs) in any dimensions.

In the present paper we would like to study the entanglement entropy in higher dimensional CFTs, especially CFT from both the CFT and gravity sides. In particular, we find the computations of the logarithmic term from both sides agree at least when the second fundamental form of embedded in the dimensional space vanishes. In addition, we present a review of the required knowledge of the entanglement entropy in conformal field theories and the details of our short report.

We would also like to mention recent interests in entanglement entropy in condensed matter physics. One of main foci in modern condensed matter physics is to understand quantum phases of matter which are beyond the Ginzburg-Landau paradigm. Many-body wavefunctions of quantum ground states in these phases look featureless when one looks at correlation functions of local operators; They cannot be characterized by classical order parameters of some kind. Indeed, they should be distinguished by their pattern of entanglement rather than their pattern of symmetry breaking . Thus, the entanglement entropy is potentially useful to characterize these exotic phases.

Indeed, this idea has been pushed extensively in recent couple of years for several 1D quantum systems. It has been revealed that several quantum phases in 1D spin chains, including Haldane phases, can be distinguished by different scaling of the entanglement entropy. See, for example, [3, 4, 19, 20, 21] and references in .

For higher dimensional condensed matter systems, there has been not many works in this direction yet. Recently, the entanglement entropy was applied for so-called topological phases in 2+1 D [22, 23]. Typically, these phases have a finite gap and are accompanied by many exotic features such as fractionalization of quantum numbers, non-Abelian statistics of quasi-particles, topological degeneracy, etc. They can be also useful fault tolerant quantum computations.

On the other hand, unconventional quantum liquid phases with gapless excitations, such as gapless spin liquid phases, seem to be, at least at present, more difficult to characterize in higher dimensions. Our results from AdS/CFT correspondence can be useful to study these gapless spin liquid states (some of these phases have been suspected to be described by a relativistic gauge field theory of some sort ) .

The organization of the present paper is as follows. In section 2 we present a review of definition and basic properties of entanglement entropy. Section 3 is devoted to computations of entanglement entropy in 2D CFTs. In section 4 we first summarize the known facts on entanglement entropy in higher dimensional CFTs and perform explicit computations especially for 4D CFTs. Next we relate the central charges in a given 4D CFT to its entanglement entropy. In section 5 we present our proposal of holographic computations of entanglement entropy from AdS/CFT. We also give an explicit proof of this claim in AdSCFT based on the well-known relation [24, 25] and discuss its extension to higher dimensional cases. Based on our proposal, in section 6, we compute the entanglement entropy in 2D CFTs from the AdS side and find agreements. Higher dimensional cases are considered in section 7 where we compute the entropy from the analysis of AdS spaces. We compare it with the CFT results especially for AdSCFT case and find an agreement under a specific condition for simplification. We also estimate entanglement entropy in massive or non-conformal theories. In section 8 we summarize our results and discuss future problems.

## 2 Basics of Entanglement Entropy

We start with a review of basic ideas and properties of entanglement entropy.

### 2.1 Definition of Entanglement Entropy

Consider a quantum mechanical system with many degrees of freedom such as spin chains. More generally, we can consider arbitrary lattice models or QFTs including CFTs. We put the system at zero temperature and then the total quantum system is described by the pure ground state . We assume no degeneracy of the ground state. Then, the density matrix is that of the pure state

 ρtot=|Ψ⟩⟨Ψ|. (2.1)

The von Neumann entropy of the total system is clearly zero .

Next we divide the total system into two subsystems and . In the spin chain example, we just artificially cut off the chain at some point and divide the lattice points into two groups. Notice that physically we do not do anything to the system and the cutting procedure is an imaginary process. Accordingly the total Hilbert space can be written as a direct product of two spaces corresponding to those of subsystems and . The observer who is only accessible to the subsystem will feel as if the total system is described by the reduced density matrix

 ρA=trB ρtot, (2.2)

where the trace is taken only over the Hilbert space .

Now we define the entanglement entropy of the subsystem as the von Neumann entropy of the reduced density matrix

 SA=−trAρAlogρA. (2.3)

This quantity provides us with a convenient way to measure how closely entangled (or how “quantum”) a given wave function is. Notice also that in time-dependent backgrounds the density matrix and are time dependent as dictated by the von Neumann equation. Thus we need to specify the time when we measure the entropy. In this paper, we always study static systems and we can neglect this issue.

It is also possible to define the entanglement entropy at finite temperature . This can be done just by replacing (2.1) with the thermal one , where is the total Hamiltonian. When is the total system, is clearly the same as the thermal entropy.

### 2.2 Properties

There are several useful properties which the entanglement entropy satisfies generally. We consider the zero temperature case. We summarize some of them as follows:

• (i) When is the complement of as before, we obtain

 SA=SB. (2.4)

This manifestly shows that the entanglement entropy is not an extensive quantity. This equality (2.4) is violated at finite temperature.

• (ii) When is divided into two submanifolds and , we find

 SA1+SA2≥SA. (2.5)

• (iii) For any three subsystems , and that do not intersect each other, the following strong subadditivity inequality holds :

 SA+B+C+SB ≤ SA+B+SB+C. (2.6)

Equivalently, we can have a more strong version of (2.5) as follows

 SA+SB ≥ SA∪B+SA∩B, (2.7)

for any subsystems and . When and do not intersect with each other, this relation is reduced to the subadditivity (2.5.)

More details of properties of the entanglement entropy can be found in e.g. .

### 2.3 Entanglement Entropy in QFTs and Area Law

Consider a QFT on a dimensional manifold , where and denote the time direction and the dimensional space-like manifold, respectively. We define the subsystem by a dimensional submanifold at fixed time . We call its complement the submanifold . The boundary of , which is denoted by , divides the manifold into two submanifolds and . Then we can define the entanglement entropy by the previous formula (2.3). Sometimes this kind of entropy is called geometric entropy as it depends on the geometry of the submanifold . Since the entanglement entropy is always divergent in a continuum theory we introduce an ultraviolet cut off (or a lattice spacing). Then the coefficient in front of the divergence turns out to be proportional to the area of the boundary of the subsystem as first pointed out in [16, 17],

where is a constant which depends on the system. This behavior can be intuitively understood since the entanglement between and occurs at the boundary most strongly. This result (2.8) was originally found from numerical computations [17, 16] and checked in many later arguments (see e.g. recent works [27, 28, 29] ).

The simple area law (2.8), however, does not always describe the scaling of the entanglement entropy in generic situations. As we will discuss in details in the later sections, the entanglement entropy of 1D quantum systems at criticality scales logarithmically with respect to the linear size of , where is the central charge of the CFT that describes the critical point. It has been also recently pointed out that the area law is corrected by a logarithmic factor as (subleading terms) for fermionic systems in the presence of a finite Fermi surface, where is the characteristic length scale of the dimensional manifold [30, 31, 32, 33]. Since we mainly consider relativistic QFTs (without a finite Fermi surface) in this paper, the area law (2.8) applies to our examples for as we will see.

Before we proceed to further analysis of entanglement entropy, it might be interesting to notice that this area law (2.8) looks very similar to the Bekenstein-Hawking entropy (BH entropy) of black holes which is proportional to the area of the event horizon

 SBH=Area of horizon4GN, (2.9)

where is the Newton constant. Intuitively, we can regard as the entropy for an observer who is only accessible to the subsystem and cannot receive any signals from . In this sense, the subsystem is analogous to the inside of a black hole horizon for an observer sitting in , i.e., outside of the horizon. Indeed, this similarity was an original motivation of the entanglement entropy [16, 17] (earlier related idea can also be found in ). Even though this analogy is not completely correct as it is, the one-loop quantum correction to the BH entropy in the presence of matter fields is known to be equal to the entanglement entropy . This interesting relation is an important hint to find the holographic dual of the entanglement entropy discussed later. Indeed, the connection between this relation and our proposal has been found recently in  by employing the brane-world setup instead of backgrounds.

## 3 Entanglement Entropy in 2D Cft

Here we review and slightly extend existing computations of entanglement entropy in D CFTs. The central charge of a given CFT is denoted by . Such a computation was initiated in [1, 36] and a general prescription how to calculate the quantity was given in a recent work  (see also ), which we will explain in an orbifold theoretic manner. We separately discuss this lowest dimensional CFT since only in this case we can exactly compute the entropy for general systems at present.

### 3.1 How to Compute Entanglement Entropy

In order to find the entanglement entropy, we first evaluate , differentiate it with respect to and finally take the limit (remember that is normalized such that )

 SA = limn→1trAρnA−11−n (3.10) = −∂∂ntrAρnA|n=1=−∂∂nlogtrA ρnA|n=1. (3.11)

This is called the replica trick. Therefore, what we have to do is to evaluate in our 2D system. The first line of the above definition (3.10) without taking the limit defines the so-called Tsallis entropy, . 555 The Tsallis entropy is related to the alpha entropy (Rényi entropy) through . The and limits of the alpha entropy give the von Neumann entropy and the single-copy entanglement entropy, respectively.

This can be done in the path-integral formalism as follows. We first assume that is the single interval at in the flat Euclidean coordinates . The ground state wave function can be found by path-integrating from to in the Euclidean formalism

 Ψ(ϕ0(x))=∫ϕ(tE=0,x)=ϕ0(x)tE=−∞Dϕ e−S(ϕ), (3.12)

where denotes the field which defines the 2D CFT. The values of the field at the boundary depends on the spacial coordinate . The total density matrix is given by two copies of the wave function . The complex conjugate one can be obtained by path-integrating from to . To obtain the reduced density matrix , we need to integrate on assuming when .

 [ρA]ϕ+ϕ−=(Z1)−1∫tE=∞tE=−∞Dϕ e−S(ϕ)∏x∈Aδ(ϕ(+0,x)−ϕ+(x))⋅δ(ϕ(−0,x)−ϕ−(x)), (3.13)

where is the vacuum partition function on and we multiply its inverse in order to normalize such that . This computation is sketched in Fig. 1 (a).

To find , we can prepare copies of (3.13)

 [ρA]ϕ1+ϕ1−[ρA]ϕ2+ϕ2−⋅⋅⋅[ρA]ϕn+ϕn−, (3.14)

and take the trace successively. In the path-integral formalism this is realized by gluing as () and integrating . In this way, is given in terms of the path-integral on an -sheeted Riemann surface (see Fig. 1 (b))

 trAρnA=(Z1)−n∫(tE,x)∈RnDϕ e−S(ϕ)≡Zn(Z1)n. (3.15)

To evaluate the path-integral on , it is useful to introduce replica fields. Let us first take disconnected sheets. The field on each sheet is denoted by (). In order to obtain a CFT on the flat complex plane which is equivalent to the present one on , we impose the twisted boundary conditions

 ϕk(e2πi(w−u))=ϕk+1(w−u),   ϕk(e2πi(w−v))=ϕk−1(w−v), (3.16)

where we employed the complex coordinate . Equivalently we can regard the boundary condition (3.16) as the insertion of two twist operators and at and for each (th) sheet. Thus we find

 trAρnA=n−1∏k=0⟨Φ+(k)n(u)Φ−(k)n(v)⟩. (3.17) Figure 1: (a) The path integral representation of the reduced density matrix [ρA]ϕ+ϕ−. (b) The n-sheeted Riemann surface Rn. (Here we take n=3 for simplicity.)

### 3.2 Derivation of Entanglement Entropy in an Infinitely Long System

When is a real scalar field, this is a non-abelian orbifold. To make the situation simple, assume that is a complex scalar field. Then we can diagonalize the boundary condition by defining new fields . They obey the boundary condition

 ~ϕk(e2πi(w−u))=e2πik/n~ϕk(w−u),   ~ϕk(e2πi(w−v))=e−2πik/n~ϕk(w−v). (3.18)

Thus in this case we can conclude that the system is equivalent to disconnected sheets with two twist operators and inserted in the th sheet for each values of . In the end we find

 trAρnA=n−1∏k=0⟨σk/n(u)σ−k/n(v)⟩∼(u−v)−4∑n−1k=0Δk/n=(u−v)−13(n−1/n), (3.19)

where

 trAρnA=n−1∏k=0⟨σk/n(u)σ−k/n(v)⟩∼(u−v)−c6(n−1/n), (3.20)

setting the central charge .

To deal with a general CFT with central charge , we need to go back to the basis (3.16). The paper  showed that the result (3.20) is generally correct (see also ). The argument is roughly as follows. Define the coordinate as follows

 z=(w−uw−v)1n. (3.21)

This maps to the -plane . In this simple coordinate system we easily find . Via Schwartz derivative term in the conformal map we obtain a non-vanishing value of . From that result, we can learn that twist operators in (3.17) have conformal dimension . Thus we find the same result (3.20) for general CFTs as follows from (3.17).

Applying the formula (3.11) to (3.20), we find666Here we neglect a constant term which does not depend on , and . the famous result 

 SA=c3 logla, (3.22)

where is the UV cut off (or lattice spacing) and we set .

It is possible to extend the above result to the general case where consists of multi intervals

 A={w|Imw=0,Rew∈[u1,v1]∪[u2,v2]∪⋅⋅⋅∪[uN,vN]}. (3.23)

We obtain the value of the trace 

 trAρnAw∼⎛⎝∏1≤j

Thus the entanglement entropy is given as follows 

 SA=c3∑1≤i,j≤Nlogui−vja−c3∑1≤i

### 3.3 Derivation of Entanglement Entropy on a Circle

We assume the space direction is compactified as a circle of circumference . The system is defined by the subsystem by the union

 A={x|x∈[r1,s1]∪[r2,s2]∪⋅⋅⋅∪[rN,sN]}, (3.26)

where we assume . This subsystem is related to the previous one (3.23) via the conformal map

 w=tan(πw′L). (3.27)

This maps the previous -sheeted Riemann surface to the -sheeted cylinder . We find and .

To compute in this cylinder coordinates, we can apply the conformal transformations (3.27). This leads to the extra factor

 N∏i=1[Lπcos(πriL)cos(πsiL)]−c6(1−n−2), (3.28)

which should be multiplied with (3.24). In this way, the entanglement entropy is given by

 SA = c3∑1≤i,j≤Nlog(Lπasin(π(ri−sj)L)) − c3∑1≤i

When we only have one interval with the length , (LABEL:entamiult) is reduced to the known result [1, 2]

 SA=c3⋅log(Lπasin(πlL)). (3.30)

Notice that in the small limit, (3.30) approaches to (3.22) as expected. Also the expression (3.30) is invariant under the exchange and thus satisfies the property (2.4).

### 3.4 Derivation of Entanglement Entropy at Finite Temperature

It is also possible to calculate at finite temperature when its spacial length is infinite . In this case we need to compactify the Euclidean time as . We can map this system to the previous one (3.23) via the conformal map

 w=e2πβw′. (3.31)

We find and . This conformal map leads to the extra factor

 N∏i=1[β2π e−πβ(ri+si)]−c6(1−n−2), (3.32)

in addition to (3.24). Thus we obtain as follows

 SA = c3∑1≤i,j≤Nlog(βπasinh(π(ri−sj)β)) − c3∑1≤i

If the subsystem is a single length segment, it becomes the known result 

 SA=c3⋅log(βπasinh(πlβ)). (3.34)

In the zero temperature limit , this reduces to the previous result (3.22). On the other hand, in the high temperature limit , it approaches

 SA≃πc3lT. (3.35)

This is the same as the thermal entropy for the subsystem as expected.

### 3.5 Massive Theories

When we are away from a critical point, the logarithmic scaling law Eq. (3.22) does not persist for , where is the correlation length (inverse of the mass gap). For large (), the entanglement entropy saturates to a finite value [3, 2]

 SA=A⋅c6logξa, (3.36)

where is the number of boundary points that separate from its complement. Thus, unlike critical (1+1)D systems, the area law holds for the massive case. This behavior was studied in details in several 1D quantum spin chains [3, 2, 21, 19], and QFTs [2, 37, 39]. In , the result (3.36) is derived from an argument similar to Zamolodchikov’s -theorem. We will mention this proof briefly in section 4.2.2.

## 4 Entanglement Entropy in Higher Dimensional CFTs

Now we would like to move on to the computations of entanglement entropy in higher dimensional conformal field theories CFT. This was initiated in [16, 17] and a partial list of later results can be found in [40, 27, 2, 30, 31, 39, 28, 32, 41]. In spite of many progresses, the calculation of the entropy is too complicated to find exact results. This is one motivation to consider the holographic way of computing the quantity as we will discuss later.

As in the 2D CFT case explained in section 3, we assume the CFT is defined on the dimensional manifold . We define the subsystem as the submanifold of at a fixed time . The strategy of calculating the entanglement entropy is the same as in the 2D case. First find the reduced trace and then plug this in (3.11) to obtain . We can compute from the partition function on the -sheeted dimensional manifold as in the 2D case (3.15)

 trAρnA=Zn(Z1)n. (4.1)

The -sheeted manifold can be constructed as follows. First we remove the infinitely thin dimensional slice from . Then the boundary of such a space consists of two s, which we call and . Next we prepare copies of such a manifold. Their boundaries are denoted by and . Now we glue with for every . As we take the trace of , is glued with . In the end this procedure leads to a manifold with conical singularities where all cuts meet.

It is not straightforward to calculate for an arbitrary choice of even in free field theories. This is because the conformal structure is not as strong as in the 2D CFT case. Thus below we mainly restrict our arguments to specific forms of given by the following two examples. We also simply assume .

The first one is the straight belt of width

 AS={xi|x1∈[−l/2,l/2],x2,3,⋅⋅⋅,d∈[−∞,∞]}, (4.2)

as depicted in Fig. 2. Since the lengths in the directions of are infinite, we often put the regularized length . Taking the limit and looking at the region near , we obtain the subsystem which covers a half infinite space of . The boundary in this case is given by the straight surface .

The second example is the circular disk of radius defined by

where (see Fig. 2.). Figure 2: Two different shapes of the submanifold A considered in this paper. (a) The straight belt AS and (b) the circular disk AD. (Here, d=3 for simplicity.)

### 4.1 Entanglement Entropy of d+1 D Massless Free Fields

As an explicit example, we consider the entanglement entropy of dimensional CFT on defined by massless free fields such as a massless scalar field or Dirac (or Majorana) fermion. This can be regarded as the infinite volume limit of the CFT on , where the volume of torus is .

Because this theory is free, we can perform the dimensional reduction on and obtain infinitely many two dimensional free massive theories whose masses are given by

 m2=d∑i=2k2i=(2πL)2⋅d∑i=2n2i, (4.4)

where are the quantized momenta such that in the torus directions.

To take this advantage we concentrate on the case where the subsystem is defined by the straight belt with radius (4.2). The point is that the computation of the entanglement entropy in this case is now reduced to the calculations of in massive 2D QFTs.

#### 4.1.1 Rough Estimation

As we reviewed in section 3, we know the formulas of entanglement entropy both in the massless limit (i.e. ) (3.22) and the massive limit (3.36) (i.e. ). The correlation length is estimated as , where is defined in (4.4). This leads to the following rough estimation of by replacing the summations of infinitely many modes with the integral of in the limit

 SroughA = ξ≤l∑k2,⋯,kdc3logξa+ξ≥l∑k2,⋯,kdc3logla (4.5) = (L2π)d−1c3[∫a−1l−1dd−1klogξa+∫l−10dd−1klogla] = c3(d−1)⋅2d−1πd−12Γ(d+12)[Ld−1ad−1−Ld−1ld−1].

If we set (i.e. massless fields in 4 dimension), we obtain

 SroughA=c24π(L2a2−L2l2). (4.6)

Notice that is the two dimensional central charge and thus for a 4D real scalar field and (or ) for a 4D Majorana (or Dirac) fermion. As can be seen from the exact computation discussed in the next subsection, this rough estimation already captures the correct functional form of the entanglement entropy.

The first term in (4.5) represents the leading divergence which indeed obeys the area law (2.8). This part can be found by taking the limit i.e. when is the straight surface . It is also possible to compute this term analytically as done in [40, 2]. On the other hand, the second term does not depend on the cutoff and thus is an interesting quantity to examine in more detail.

The violation of the area law for systems with a finite Fermi surface can be also understood from this rough estimation of the entanglement entropy. For simplicity, we assume a spherical Fermi surface with being the Fermi momentum. For the momentum outside and close to the Fermi surface, the gap is given by . Thus, as before, the entanglement entropy is estimated as

 SA = (L2π)d−1c3[∫a−1|k|=kF+l−1dd−1klogξa+∫|k|=kF+l−10dd−1klogla]. (4.7)

We thus find, for ,

where note that . A more precise calculation based on the Widom conjecture in  gives the prefactor in front of as the double integral over the fermi surface in the momentum space and the region in real space.

#### 4.1.2 Exact Estimation from Entropic c-function

The previous approximation (4.5) uses the formulas which are exact only in the two opposite limits and . To perform an exact estimation, we need to be precise about the intermediate region . In other words, we need to use a sort of -function under the massive deformation corresponding to the interpolating region instead of the UV central charge in (4.5). To make this more explicit we can employ the entropic -function introduced in [42, 37, 39]. It is defined for 2D CFTs as follows

 ldSA(l)dl = C(lm), (4.9)

where is the length of the subsystem and is the mass of the field. For massive free fermions and scalar fields, the function is characterized as a solution to a differential equation of Painleve V type and its numerical form can be found in [37, 39]. Unfortunately, its analytical expression is not known.

This function is positive and is also a monotonically decreasing function  with respect to as in the Zamolodchikov’s -function . These properties are indeed true in explicit examples , which we reproduced in Fig. 3 for a free massive real scalar boson and free Dirac fermion in 1+1 D. The function is normalized such that in the UV limit it is related to the ordinary central charge via . Note that if we set , we recover from this equation the well-known result (3.22). We will also show this later independently in (4.24). It was argued that the positivity of is connected to a majorization relation for local density matrices [3, 4, 44, 45, 46]

In our example of the dimensional free field, we can reduce it to infinitely many massive fields in two dimensions. Thus in this case we again just have to sum over the discrete quantum numbers . In the limit we can replace the sum with an integral

 ldSA(l)dl = [Ld−1(2π)d−1]∫dk2⋯dkd C(l|k|) (4.10) = ⎡⎢ ⎢⎣Ld−12d−2πd−12Γ(d−12)⎤⎥ ⎥⎦∫∞0dkkd−2C(lk).

The merit of the quantity instead of itself is that it does not include UV divergences and thus we can set in . After the integration of we find

where we determine the integral constant by requiring that should be vanishing777It is possible that this requirement is not absolute, i.e. this choice of the cutoff may depend on the theory we consider. Thus only the constant in front of the second term (i.e. finite term) in (4.11) has a qualitative meaning. at since we are cutting off degrees of freedom below the energy scale . It is straightforward to find analogous formula for the free massive fields. This is given just by replacing in (4.10) or (4.11) with .

The second term in (4.11) does not depend on the cutoff . Thus we are interested in its coefficient which is proportional to the integral of the function . In principle, we can compute it numerically based on the numerical results of . Indeed by this method the coefficient was computed for three dimensional free fields in . We extend it to four dimensions which we are interested in later discussions and present the result as follows

 K = ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩1π∫∞0dtC(t)≃0.039,for d+1=3 dimensional real scalar % boson1π∫∞0dtC(t)≃0.072,% for d+1=3 dimensional Dirac fermion14π∫∞0dttC(t)≃0.0049,for d+1=4 dimensional real scalar boson14π∫∞0dttC(t)≃0.0097,for d+1=4 dimensional Majorana fermion. (4.12)

To find the coefficient in higher dimensions it is useful to notice that when is large the entropic -function behaves as ( is the deformed Bessel function)

 Cscalar(x)≃14xK1(2x),   and   CDirac(x)≃12xK1(2x), (4.13)

for a 2D free scalar field and a 2D Dirac fermion. When the dimension is large, the contribution of the integral mainly comes from the large region. Thus can be well approximated by plugging (4.13) into (4.11). This leads to888For example, from this approximation we find and for and these are rather close to the previous results in (4.12). It may also be interesting to compare this with the our rough estimation done in the previous subsection. There we found when .

 Kscalar≃2−d−3π(1−d)/2Γ(d−12),   Kfermion≃2Kscalar, (4.14)

where corresponds to a dimensional real scalar field while to the dimensional fermions which is reduced to a 2D Dirac fermion999In higher dimension we need to multiply an appropriate degeneracy with to obtain the result for a ordinary fermion such as Dirac fermion.. Figure 3: The entropic c-functions C(x) for free massive real scalar boson and free Dirac fermion in 1+1 D reproduced from .

Finally we would like to stress again that our result (4.11) was obtained by assuming a free field theory. In the presence of interactions we no longer have the simple sum over infinitely many massive fields in two dimensions (4.10) due to interactions between two different massive fields.

#### 4.1.3 Entanglement Entropy of 4D Gauge Field

As we will consider 4D gauge theories later, we would also like to examine the entanglement entropy of 4D gauge field. We neglect the interactions as before and thus we can concentrate on the abelian gauge theory. Its gauge fixed action with the ghost is given by

 S = ∫d4x[14FμνFμν+12α(∂μAμ)2+¯c(−□)c]. (4.15)

In order to compute the entanglement entropy, we consider the gauge theory on an -sheeted manifold as before. We can rewrite the gauge field action as follows (we fix the gauge by setting )

 SAμ = ∫d4x[14FμνFμν+12(∂μAμ)2] (4.16) = ∫d4x12∂μAν∂μAν+[∫d3x[−(∂νAν)A0+(∂μA0)Aμ]]surf,

where denotes the surface term when we performed an partial integration. If we neglect the surface term, the theory is equivalent to four real scalar fields and a ghost field which is a complex scalar. Since the complex ghost scalar field cancels two real scalars, the theory is equivalent to two real scalar fields. However, there is a subtle issue on the surface term that appears when we do the partial integration. Since fields are discontinuous along the time direction in some region of the spacetime, we got surface contributions. In this paper we assume such a term is not relevant for the computation of the entanglement entropy as101010Indeed if we include such a contribution we find the total entropy of the gauge field becomes negative in the particular case discussed in , which looks strange if we remember the original definition (2.3). in .

### 4.2 Entanglement Entropy and Central Charges in 4D CFT

As we have seen, the entanglement entropy in 2D CFTs is proportional to the central charge . Since the central charge roughly measures the number of degrees of freedom , we find the entanglement entropy is also proportional to . This fact is very natural as its name of ‘entropy’ shows. Therefore we may expect that a similar story is true also in the higher dimensional theories. As such an example, below we consider 4D CFTs. Indeed we will find that an important part of the entanglement entropy is proportional to the central charges. See also [47, 41] for an earlier discussion.

In principle, it is possible to extend the relation between central charges and entanglement entropy to higher dimensions as far as the spacetime dimension is even. When we consider odd dimensional spacetime, we do not have any clear definition of central charges due to the absence of the Weyl anomaly. Under this situation, the entanglement entropy may play an important alternative role 111111We are grateful to Anton Kapustin for pointing out this possibility to us..

#### 4.2.1 Entanglement Entropy from Weyl Anomaly

Central charges in CFTs can be defined from the Weyl anomaly (or conformal anomaly) . Define the energy-momentum tensor in terms of the functional derivative of the (quantum corrected) action with respect to the metric

 Tμν=4π√gδSδgμν. (4.17)

In 2D CFTs, the Weyl anomaly is given by the well-known formula

 ⟨Tμμ⟩=−c12R, (4.18)

where is the scalar curvature. We can regard this as a definition of the central charge in 2D CFTs.

Now we move on to 4D CFTs. In our normalization of (4.17), the Weyl anomaly can be written as

 ⟨Tαα⟩=−c8πWμνρσWμνρσ+a8π~Rμνρσ~Rμνρσ. (4.19)

in a curved metric background , where and are the Weyl tensor and the dual of the curvature tensor. Notice that the second term is the Euler density. In terms of the ordinary curvature tensor, we can express the curvature square terms in (4.19) as follows

 WμνρσWμνρσ = RμνρσRμνρσ−2RμνRμν+13R2, ~Rμνρσ~Rμνρσ = RμνρσRμνρσ−4RμνRμν+R2. (4.20)

The coefficients and in (4.19) are called121212 The central charge should not be confused with a UV cutoff. To avoid confusion, is used to denote the UV cutoff in this subsection. the central charges of 4D CFTs [48, 49, 50]. This is the original definition of the central charges in 4D CFTs. The central charge is believed to decrease monotonically under the renormalization group (RG) flow, while for this is not true and indeed counter examples are known; these properties of the central charges and are confirmed in many supersymmetric examples e.g. .

To compute the entanglement entropy, we first consider the partition function on the dimensional -sheeted manifold . Then we find the trace of reduced to the subsystem is given by the formula (4.1). The entanglement entropy can be found by taking the derivative of with the
{"ft_lang_label":"__label__en","ft_lang_prob":0.9140572,"math_prob":0.9730708,"size":34057,"snap":"2023-14-2023-23","text_gpt3_token_len":7824,"char_repetition_ratio":0.16803218,"word_repetition_ratio":0.027772956,"special_character_ratio":0.22444725,"punctuation_ratio":0.10144927,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9950932,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-03-22T12:44:47Z\",\"WARC-Record-ID\":\"<urn:uuid:62cc3921-20bc-4f4a-ad6e-ba6b11963b97>\",\"Content-Length\":\"1049412\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0ba2ee82-21f3-4d81-bc63-9284db6ee494>\",\"WARC-Concurrent-To\":\"<urn:uuid:6fbc0c92-9ce7-4c45-8363-c6ae041f82e0>\",\"WARC-IP-Address\":\"172.67.158.169\",\"WARC-Target-URI\":\"https://www.arxiv-vanity.com/papers/hep-th/0605073/\",\"WARC-Payload-Digest\":\"sha1:IWXKYCQFW2K747IMO2ASHT2ZKGBWHH35\",\"WARC-Block-Digest\":\"sha1:QTZ7FUZVYWEN7KBJUV6YMDGAVQVCYWYQ\",\"WARC-Truncated\":\"length\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-14/CC-MAIN-2023-14_segments_1679296943809.76_warc_CC-MAIN-20230322114226-20230322144226-00056.warc.gz\"}"}