Volume Law for the Entanglement Entropy in Non-local QFTs

In this paper, we present a simple class of non-local field theories whose ground state entanglement entropy follows a volume law as long as the size of subsystem is smaller than a certain scale. We will confirm this volume law both from numerical calculations and from analytical estimation. This behavior fits nicely with holographic results for spacetimes whose curvatures are much smaller than AdS spaces such as those in the flat spacetime.


Introduction
The entanglement entropy offers us a universal measure of the degrees of freedom in any quantum many-body systems. Since it is defined by focusing on an arbitrary subsystem of a given quantum system, we can probe effective degrees of freedom for any fixed length scale and position. Thus this includes quite a lot of information about any ground state.
It has been well-established that in any local quantum field theory with a ultraviolet (UV) fixed point, the entanglement entropy follows the universal rule called area law [1]. This claims that the entanglement entropy S Ω for a subsystem Ω has a UV divergence in the continuum limit of quantum field theories and that the coefficient of this divergence is proportional to the area of the boundary ∂Ω of the subsystem.
The area law was first found in free field theories [2,3]. One way to confirm the area law for interacting field theories is to employ the AdS/CFT correspondence [4]. The AdS/CFT correspondence argues that a gravitational theory on d + 1 dimensional anti de Sitter space (AdS space) is equivalent to a d dimensional conformal field theory (CFT), where the latter is typically described by a strongly coupled and large N gauge theory. The holographic formula of entanglement entropy [5,6] shows that the area law holds for such a strongly coupled CFT with a UV fixed point and this heavily relies on the geometry of the AdS space.
The AdS/CFT can be regarded as an example of a more general and earlier idea called holography [7]. This principle conjectures that a given gravity theory in a d + 1 dimensional spacetime M is equivalent to a certain quantum many-body system which lives on the d dimensional boundary of ∂M. Therefore it is natural to ask what we can say about general holography from the quantum entanglement viewpoint. If we consider, for example, a flat spacetime in any dimension, we can immediately find that its holographic entanglement entropy satisfies a volume law instead of the area law [8]. Refer also to related earlier works [9] and recent discussions [10,11] in gravity duals of non-commutative field theories, where the holographic entanglement entropy was shown to follow a volume law. Therefore it seems important to find a class of field theories whose entanglement entropy satisfies the volume law.
It is well-known that for the generic excited states in any quantum many-body systems, the entanglement entropy satisfies the volume law [12]. This means that the state which follows the area law is very special. Indeed, the local quantum field theories, which have the area law property, are clearly special in that the interactions are very short range and this property crucially helps to reduce the amount of entanglement.
In lattice models, it is not so difficult to construct models with a volume law. For example, we can consider a spin system with random interactions between any two pairs of spins. The non-local random interactions obviously lead to a highly entangled ground state which satisfies the volume law. Moreover it has been pointed out that even without local interactions, we can construct lattice models with a volume law if we give up the translational invariance [13]. The purpose of this paper is to present a field theory model with the translational invariance whose ground state satisfies the volume law. The previous arguments suggest that we may get the volume law if we consider suitable non-local field theories. There have already been suggestions of such field theories in [8,14] via heuristic discussions. Also we would like to mention that the paper [15] studied milder non-local field theories, which do not lead to the volume law but have modified coefficients of logarithmic divergent term in two dimensional field theories.
In this paper, we will present simple and concrete examples of non-local and nonrelativistic free scalar field theories. We will show manifestly that they indeed have the property of volume law both by explicit numerical calculations and by analytical estimations. We will also see that holographic calculations confirm the same behavior.
The paper is organized as follows: In section 2, we will briefly review how to calculate the entanglement entropy in free bosonic quantum many-body systems. In section 3, we will explain our models of non-local scalar field theories in two dimensions and their lattice regularizations. In section 4, we will present numerical results of entanglement entropy for our non-local scalar models in two dimensions and give their analytical explanations. In section 5, we will generalize our results into higher dimensions. In section 6 we present a holographic interpretation and find that it is consistent with our field theoretic results. In section 7 we summarize our conclusion.
When we were writing the draft of this paper, we noted the paper [16], where the authors showed that the volume law can be obtained for a non-commutative field theory on a fuzzy sphere, which is also an example of non-local field theory (refer also to [17] for earlier works on entanglement entropy in non-commutative field theories).
2 How to compute entanglement entropy: Real time approach In this section we review the method of computing the entanglement entropy in free field theories developed by Bombelli et al [2], which we will employ in this paper. Refer to [3,18,19,20,21,22,23,24,25] for examples of other useful computational methods for the entanglement entropy in free field theories. As a model amenable to unambiguous calculation we deal with the scalar field as a collection of coupled oscillators on a lattice of space points, labeled by capital Latin indices, the displacement at each point giving the value of the scalar field there. In this case the Hamiltonian can be given by where q M gives the displacement of the M-th oscillator and P M is the conjugate momentum to q M . The matrix V M N is symmetric and positive definite. The matrix V M N is independent of q M andq M . We can obtain the ground state wave function as The matrix W M N is symmetric and positive definite. Now consider a subsystem (or subregion) Ω in the space. The oscillators in this region will be specified by lowercase Latin letters, and those in its complement Ω c will be specified by Greek letters. We will use the following notation We can obtain a reduced density matrix ρ red for Ω by integrating out over q α ∈ R for each of the oscillators in Ω c , and then we have We can write the density matrix as one for non coupled degrees of freedom by making an appropriate linear transformation on q a . Finally the entanglement entropy S Ω = −trρ red ln ρ red is given by [2] S where λ n are the eigenvalues of the matrix In the last equality we have used (2.5). The last expression in (2.9) is useful for numerical calculations when Ω is smaller than Ω c , because the indices of A and D take over only the space points on Ω and the matrix sizes of A and D are smaller than those of B and E as emphasized in [26]. It can be shown that all of λ n are non-negative as follows. From It is easy to show that A, C, D and F are positive definite matrices when W and W −1 are positive definite matrices. Then AΛ is a positive semidefinite matrix as can be seen from (2.10). So all eigenvalues of Λ are non-negative. After all, we can obtain the entanglement entropy by solving the eigenvalue problem of Λ.

Two dimensional non-local scalar fields on lattices
We apply the above formalism to free scalar fields in (1 + 1)-dimensional Minkowski spacetime. As an ultraviolet regulator, we replace the continuous space coordinate x by a lattice of discrete points with spacing a. As an infrared cutoff, we allow n ≡ x/a to take only a finite integer values −N/2 < n ≤ N/2. Outside this range we assume the lattice is periodic. Later we will take N to infinity. The dimensionless Hamiltonian H 0 ≡ aH is given by where φ n and π n are dimensionless and Hermitian, and obey the canonical commutation relations [φ n , π m ] = iδ nm . (3.12) As an example, let us consider the Klein Gordon field whose mass is m. We can diagonalize the matrix V by a Fourier transform [27] and obtain where the index k is also an integer in the range of −N/2 < k ≤ N/2. We take N to infinity and change the momentum sum into an integral with the replacements q = 2πk/N and , and then we have Then the laplacian on lattices is 2(1 − cos q).
Next we turn to non-local scalar fields theories which we are interested in this paper. The Hamiltonian is defined by where A 0 , B 0 are positive constants. We define dimensionless constants We can change B into 1 by rescaling t. Thus the entanglement entropy is independent of B and we can set B = 1. We obtain W and W −1 as follows For later convenience we define When w = 1, 2, we can calculate W, W −1 analytically as we will show below.

Case1: w = 1
In the case w = 1, we obtain where J 2n is the Bessel function and E 2n is the Weber function. We can rewrite (3.23) as where I 2n is the modified Bessel function and we have used Here 1F2 is the regularized hypergeometric function. The regularized hypergeometric function is defined as where p F q [a 1 , . . . , a p ; b 1 , . . . , b q ; z] is the hypergeometric function. This expression is manifestly real and suitable for numerical calculations. We can obtain W −1 by replacing A → −A in W : In the case w = 2 we find   Figure 1: The entanglement entropy S Ω (L) of one interval whose length is L for w = 1 as a function of L. In the left picture, the blue, red, yellow and green points correspond to A = 400, 600, 800, 1000. In the right picture, the blue, red and yellow points correspond to A = 40, 60, 80.

Computations of Entanglement Entropy
Now we would like to turn to the main part of this paper: computations of entanglement entropy for our non-local scalar field theories. We will first present numerical results which support the volume law and later give an analytical explanation.

Numerical calculations
We perform matrix operations and calculate the eigenvalues λ n of the matrix Λ in (2.9). Then finally we can obtain the entanglement entropy in (2.8) with Mathematica 8. We define the subsystem Ω to be an interval on the one dimensional lattice with the length L.
Since the columns and rows of the matrix Λ describe points in Ω, the size of the matrix Λ is given by L × L. For w = 1 (Case 1), we show the computed values of S Ω (L) as a function of L in Fig.1. As can be seen, S Ω (L) is proportional to L when L << A and approaches its maximum value when L >> A. By using the data between 1 ≤ L ≤ 20 for 2000 ≤ A ≤ 3000, we obtain S Ω (L) ≃ 0.48AL for L << A. By using the data for 100 ≤ A ≤ 200, we obtain S Ω (L) ≃ 0.055A 2.0 for L >> A.
For w = 2 (Case 2), we show the computed values of S Ω (L) as a function of L in Fig.2. As can be seen, S Ω (L) is proportional to L when L << A and approaches its maximum value when L >> A. This behavior is similar to the entanglement entropy for w = 1. By using the data between 1 ≤ L ≤ 20 for 2000 ≤ A ≤ 3000, we obtain S Ω (L) ≃ 0.98AL for where c 1 and c 2 are order one constants which depends only on the value of w. Our numerical results implies the identification c 1 = w/2. In this way, we can conclude that ground states of these models satisfy the volume law as long as the subsystem size is small as L << A.

Analytical Explanation
We consider the behavior of the entanglement entropy by examining the matrix Λ in (2.9). We can write explicitly Λ as  where sinh β = n/A and n ≫ A. From the asymptotic form of I n (A), we can see that |W n | and |W −1 n | decrease rapidly when n(≫ A) increases. From (3.28), (3.29) and (4.31) we obtain Λ m,n ≃ 0 when |m − n| ≫ A. Furthermore we can see that Λ m,n ≃ 0 when A ≪ m, n ≪ L−A by the following argument. By using the identity ∞ l=−∞ W −1 m−l W l−n = δ m,n , we can rewrite Λ in (4.31) as  In the case w = 1, we can estimate the entanglement entropy in the similar way. In this case we can see numericaly that W n decrease faster than W −1 n . Via the same argument under (4.38) we can see that Λ m,n ≃ 0 when A ≪ n ≪ L − A. We show Λ in Fig3. We can rewrite the entanglement entropy in (2.7) as S Ω = Trf (Λ). From the form of the matrix Λ, we can see that TrΛ l is independent of L. Thus the entanglement entropy is constant when L ≫ A.

Higher Dimensional Generalization
Next we would like to consider a straightforward generalization of our two dimensional scalar field model (3.18) to the d dimensional one (d > 2) defined by the Hamiltonian where we defined ∂ i = ∂ ∂x i and x i (i = 1, 2, · · ·, d − 1) are the coordinates of R d−1 . We divide the coordinates of R d−1 into two parts: x 1 ∈ R and (x 2 , · · ·, x d−1 ) ∈ R d−2 . We take the Fourier transformation with respect to the latter and obtain In this way we can decompose the Hamiltonian as a sum of two dimensional scalar Hamiltonian H(k) over the transverse momenta k. Note that in the Hamiltonian H(k), |k| plays the role of mass parameter in the two dimensional theory as is familiar in the Kaluza-Klein theories. As a IR regularization, we compactify the space R d−2 into a torus with the radius Ra (a was the lattice constant and R is the size of torus in the lattice space.).
Then the momentum is quantized as We choose the subsystem Ω in the definition of entanglement entropy S Ω to be a strip with the width La defined by First consider the case w = 2. The entanglement entropy S Ω for the ground state of the Hamiltonian H(k) is identical to our original model in two dimension i.e. H(0). This is because the k dependence only appears as a factor e A 0 k 2 in front of φ 2 term, which does not change our calculation of S Ω as is clear from our previous calculations. Therefore we can estimate the total contribution when L << A as follows (5.42) Therefore we confirmed the volume law in any dimension. Now let us move on to more general w. We define p to be the momentum in the x 1 direction. Since 1 << L << R and p ∼ 1 La << 1 a , the dominant contribution comes from the region k >> p. When k >> p, we find (5.43) Thus we can approximate the calculation of S Ω by that of w = 2 with A 0 replaced with A 0 w 2 k w−2 for each k. In this way we can estimate as follows where C 1 is a certain order one constant; we employed the same definition A = A 0 a −w of A as in the two dimensional case. In this way, again we found the volume law in any dimension. We can similarly analyze S Ω in our d dimensional theory even when L >> A. In the end we find S Ω ≃ C 2 A 2 R d−2 , for a certain constant C 2 . In summary, for our d dimensional non-local scalar field model, we obtained the following behavior: In this way we confirmed that our higher dimensional model also has the property of volume law for a small size subsystem. On the other hand, when the size of Ω gets larger than the parameter A, it follows an area law.

Holographic Interpretation
Finally we would like to discuss a holographic counterpart of our field theory analysis. Originally, the non-local scalar field model defined by the Hamiltonian (5.39) was considered in [14] in the context of an interpretation of AdS/CFT correspondence as an entanglement renormalization. See also [8] for a similar but different model which was proposed as a toy model of holographic dual of gravity in a flat spacetime.
In [28], it has been conjectured that a framework of real space renormalization, called MERA (multi-scale entanglement renormalization ansatz) [29], is equivalent to the AdS/CFT correspondence. This idea allows us to relate the entanglement structure of a quantum state in MERA to the metric of its gravity dual.
Especially, by using the continuum limit of MERA (called cMERA [30]), a formula for the metric in the extra dimension has been proposed in [14]. Consider the metric in the d + 1 dimensional gravity dual: where a is the UV cut off (or lattice spacing) as in our previous sections and u is the coordinate of extra direction. We regard u = 0 as the boundary of d + 1 dimensional spacetime where its holographic dual lives. Note that we ignored any constant factor of the metric which depends only on the Hamiltonian of the theory.
If we consider the free scalar field theory defined by (5.39), then the proposed formula [14] for the dual metric from the viewpoint of cMERA computes g uu as follows Now we introduce the standard extra dimension coordinate z ≡ ae −u and then we can rewrite the metric (6.46) as Moreover, it is useful to define the coordinate y = z −w to rewrite the spacial part of the above metric into where rescaled x i by a finite amount. Now we would like to study the holographic entanglement entropy [5] for the gravity dual (6.49). We choose the subsystem Ω to be the strip defined by (5.41), with the understanding that R is infinitely large by taking the decompactifying limit. The holographic entanglement entropy is given by where the d − 1 dimensional surface γ Ω is the minimal surface which ends on ∂Ω (i.e. ∂γ Ω = ∂Ω) [5]; G N is the Newton constant of the d + 1 dimensional gravity. The area of minimal surface ending on ∂Ω can be obtained by minimizing where y ′ = ∂y ∂x 1 . By deriving a conserved quantity ('Hamiltonian'), we find where y * is the integration constant. We assumed that the minimal surface extends between y * ≤ y < y ∞ , where y ∞ = a −w is the counterpart of the UV cut off in the gravity dual, which corresponds to the boundary of d + 1 dimensional spacetime defined by u = 0. Also y = y * is the turning point of the surface where y ′ vanishes. By integrating (6.52) we find a w−1 · y∞ y * dy y 1/w y 2(d−1)/w /y In the end, the minimal area is expressed as the integral First we are focusing on the region L << A as we assumed to show the volume law in our previous sections. From (6.53), this case corresponds to y * ≃ y ∞ (= a −w ) and thus the minimal surface is always very close to the boundary y = y ∞ . Thus the minimal area (6.54) is proportional to the volume LR d−2 of the d − 1 dimensional space where the nonlocal scalar field lives. Indeed, in this case, we can estimate the holographic entanglement entropy as follows where we omitted the coefficient which only depends on the theory (or equally Hamiltonian). It is also intriguing to ask the behavior of holographic entanglement entropy when L >> A. First note that there exist two disconnected minimal surfaces which are simply given by x 1 = ±La/2 with 0 ≤ y ≤ y ∞ . The sum of these two disconnected surfaces is another candidate of minimal surfaces γ Ω for the holographic entanglement entropy S Ω . Indeed since the metric in the x i direction vanishes at y = 0, it satisfies the required condition ∂γ Ω = ∂Ω. The minimal area principle of holographic entanglement entropy tells us that we should choose γ Ω to be the sum of the disconnected surfaces when L >> A. This leads to the estimate Area(γ Ω ) ∝ AR d−2 . Notice also that in the opposite case L << A, we have to choose γ Ω to be the connected surface because the area of the connected one is clearly smaller than that of the disconnected ones. This is a typical example of 'phase transition' for the entanglement entropy as observed in a variety of holographic examples (see e.g. [31,32,33]), which is considered to be an artifact of large N limit.
In this way, our holographic results confirmed the behavior (4.30) and (5.45), assuming that the proportionality coefficient, which we are not able to fix, is given by A times a numerical constant. Especially this supports the claim that the entanglement entropy satisfies the volume law when L << A.

Conclusion
In this paper we presented a simple class of non-relativistic field theories whose entanglement entropy satisfies a volume law as long as the size of subsystem is smaller than a certain parameter (called A) of the theory, which parameterizes the magnitude of nonlocality. These field theories are highly non-local in real space and this is obviously the reason why it follows the volume law rather than the area law. This model has another parameter w which is related to the types of non-locality we are considering. We confirmed our model follows the volume law when w = 1 and w = 2 in the two dimensional scalar field theory both from numerical calculations and analytical estimates. We also extended this result into higher dimensions. The final result of entanglement entropy S Ω , when the subsystem Ω is a strip with the width L, is summarized in (5.45). Also our holographic calculation agrees with these field theory results and furthermore predicts that we will obtain the volume law for any values of w(> 0).
It will be intriguing a future problem to go beyond free field theories by taking into interactions as well as to extend our constructions to fermions. Another interesting direction is to pursuit holography for general spacetimes by using entanglement entropy. It is an important future problem to better understand our holographic relation between almost flat spacetimes and non-local field theories.