An inverse mass expansion for entanglement entropy in free massive scalar field theory
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Abstract
We extend the entanglement entropy calculation performed in the seminal paper by Srednicki (Phys Rev Lett 71:666, 1993) for free real massive scalar field theories in \(1+1\), \(2+1\) and \(3+1\) dimensions. We show that the inverse of the scalar field mass can be used as an expansion parameter for a perturbative calculation of the entanglement entropy. We perform the calculation for the ground state of the system and for a spherical entangling surface at third order in the inverse mass expansion. The calculated entanglement entropy contains a leading area law term, as well as subleading terms that depend on the regularization scheme, as expected. Universal terms are nonperturbative effects in this approach. Interestingly, this perturbative expansion can be used to approximate the coefficient of the area law term, even in the case of a massless scalar field in \(2+1\) and \(3+1\) dimensions. The presented method provides the spectrum of the reduced density matrix as an intermediate result, which is an important advantage in comparison to the replica trick approach. Our perturbative expansion underlines the relation between the area law and the locality of the underlying field theory.
1 Introduction
Quantum entanglement is the physical phenomenon that appears when a composite quantum system lies in a state such that no description of the state of its subsystems is available. In the presence of quantum entanglement, measurements in the entangled subsystems are correlated. The most well known example of an entangled system, the so called EPR paradox [2], requires just two spinors; it was initially conceived as contradictory to causality, and, thus, as an adequate theoretical experiment to question the completeness of the quantum description of nature. However, later on, the corresponding correlations were verified experimentally.
In a seminal paper [1], Srednicki performed a numerical calculation of entanglement entropy for a real free massless scalar field theory at its ground state, considering as subsystem A the degrees of freedom inside a sphere of radius R. The surprising result shows that entanglement entropy is not proportional to the volume of the sphere, but rather to its area. This profound similarity to the black hole entropy [25, 26, 27], discussed even before Srednicki’s calculation [28], became even more intriguing after the development of the holographic dualities [29, 30, 31] and the RyuTakayanagi conjecture [16, 17], which interrelates entanglement entropy in the boundary conformal field theory to the geometry of the bulk. The latter may allow the perspective of understanding the black hole entropy as entanglement entropy, and the gravitational interactions as an entropic force associated with quantum entanglement statistics [32, 33, 34, 35].
In this context, the further investigation of the similarities between gravitational and quantum entanglement physics and the development of appropriate tools for their study presents a certain interest. In this work, we extend the original entanglement entropy calculation presented in [1] to massive free scalar field theory and develop a perturbative method for the calculation of entanglement entropy in such systems.
The majority of entanglement entropy calculations in field theory are based on the replica trick [14, 36, 37, 38, 39, 40, 41]. This technique is based on the calculation of the entanglement Rényi entropies \(S_q\) for an arbitrary positive integer index \(q>1\).^{1} Although the entanglement Rényi entropies \(S_q\) in principle contain the whole information of the reduced density matrix spectrum, the process of deriving the latter from the former is complicated. Relevant calculations are usually restricted to the specification of the largest eigenvalue and its degeneracy. The same holds for holographic calculations. The original prescription by Ryu and Takayanagi [16, 17] provides only the entanglement entropy. In the case of spherical entangling surfaces, the reduced density matrix can be considered thermal, allowing the holographic calculation of the Rényi entropies as the black hole entropy of topological black holes with hyperbolic horizons [42, 43]. A more general framework for the holographic calculation of Rényi entropies has been provided by Lewkowycz and Maldacena in [44] towards a derivation of the Ryu–Takayanagi formula. An important feature of Srednicki’s calculation is the fact that it is not limited to the calculation of entanglement entropy; on the contrary the full spectrum of \(\rho _A\) is an intermediate result. As we discussed above, quantum entanglement is encoded into the spectrum of \(\rho _A\); the entanglement entropy is just one piece of information. Therefore, although they are old, the methods of [1] present a certain advantage.
The structure of this paper is as follows: in Sect. 2, we review the derivation of entanglement entropy in systems of coupled harmonic oscillators lying at their ground state and extend the calculation in free scalar field theory including a mass term, closely following [1]. In Sect. 3, we show that the inverse of the scalar field mass can be used as an expansion parameter allowing a perturbative calculation of entanglement entropy and develop the basic formulae of this perturbation theory. In Sect. 4, we perform the perturbative calculation for massive free scalar field theory in \(1+1\), \(2+1\) and \(3+1\) dimensions and show that the leading contribution to the entanglement entropy for large entangling sphere radii obeys an area law; we specify the relevant coefficients and the first subleading corrections and we compare with numerical calculations. In Sect. 5, we discuss our results. “Appendix A” contains the details of Srednicki’s regularization scheme. “Appendix B” contains the details of the perturbative calculation of entanglement entropy at second and third order. Finally, “Appendix C” contains the code used for the numerical calculations of entanglement entropy.
2 Entanglement entropy in free scalar QFT
2.1 Entanglement entropy of coupled oscillators
The first step towards the calculation of entanglement entropy in free scalar field theory at its ground state is the calculation of the latter in a system of coupled harmonic oscillators. This problem has been solved long ago; here, we briefly sketch its solution. More details are provided in [1].
The entanglement entropy is a valuable measure of entanglement, however, it does not contain the whole information. The latter is contained in the full spectrum of the reduced density matrix \(\rho _A\). An important advantage of the approach followed in this work is the fact that it allows the direct calculation of the latter through Eq. (2.5), as an intermediate step towards the calculation of entanglement entropy.
2.2 Entanglement entropy in free scalar field theory
In the approach of [1], the degrees of freedom of the scalar field theory are discretized via the introduction of a lattice of spherical shells, and, thus, the introduction of a UV cutoff. Furthermore, an IR cutoff is imposed, putting the system in a spherical box. This inhomogeneous discretization may appear disadvantageous, as it breaks some of the symmetries of the theory; although it preserves rotations, it breaks boosts and translations. However, the consideration of the stationary entangling sphere, which separates the degrees of freedom to two subsystems, has already broken these symmetries. This approach reduces the problem of the calculation of entanglement entropy in field theory to a similar quantum mechanics problem with finite degrees of freedom. Since we are studying free scalar field theory, the latter quantum mechanical system is simply a system of coupled oscillators with a quadratic Hamiltonian at its ground state. More details on this discretazation scheme are provided in Appendix A.
2.2.1 \(3+1\) Dimensions
For large \(\ell \), the Hamiltonian \(H_\ell \) becomes almost diagonal. Therefore, for large \(\ell \), the degrees of freedom are almost decoupled, and, thus, the system (2.12) at its ground state is almost disentangled. It can be shown that \({S_{\ell }}\left( {N,n} \right) \) decreases with \(\ell \) fast enough so that the series (2.11) is converging [1, 45].
2.2.2 \(2+1\) Dimensions
2.2.3 \(1+1\) Dimensions
3 An inverse mass expansion for entanglement entropy
The discretized Hamiltonians (2.12), (2.16) and (2.18) are describing a system of N coupled harmonic oscillators that falls within the class of systems studied in Sect. 2.1. We may thus proceed to calculate the entanglement entropy following the scheme of this section.
3.1 An inverse mass expansion
In order to calculate the desired entanglement entropy, we need to calculate the square root \(\varOmega \) of the matrix K, then the matrices \(\beta \), \(\gamma \) and finally the eigenvalues of \(\gamma ^{1}\beta \), perturbatively in \(\varepsilon \). There is one important detail that has to be taken into account in these perturbative calculations. Since the lowest order elements of K are the diagonal ones, this is also going to be the case for its square root \(\varOmega \). However, the matrix B, being an offdiagonal block of the matrix \(\varOmega \), does not contain such elements. The lowest order elements of B are the first subleading elements that appear in \(\varOmega \). As a result, preserving a certain order in perturbation theory requires the calculation of the square root of K at one order higher than the desired order. In the following, we will present the calculation at first nonvanishing order, therefore we will keep two nonvanishing terms in the expansion of \(\varOmega \).
3.2 The expansion at higher orders

Only odd powers of \(\varepsilon \) appear in the expansion of \(\varOmega \).

The leading term in any element in the kdiagonal is of order \(\varepsilon ^{2k1}\). Therefore, the matrix \(\varOmega \) calculated with n nonvanishing terms in the perturbation theory contains nonvanishing elements up to the \(\left( n1\right) \)diagonal.

Any subleading term in the elements of the matrix \(\varOmega \) is four orders higher than the previous one. Thus, an element in the kdiagonal is written as a series of the form
The calculation at the next to the leading order is analytically presented in Appendix B. It turns out that the second largest eigenvalue vanishes at this order, whereas the largest eigenvalue receives corrections at order \(\varepsilon ^8\). At third order the calculation is straightforward but more messy. The result is presented in the appendix only in the appropriate limit for the specification of the “area law” contribution to the entanglement entropy that we will discuss in next section. At this order, the largest eigenvalue receives another correction at order \(\varepsilon ^{12}\), while one more nonvanishing eigenvalue emerges, with a leading contribution at the same order. As a general rule, a new nonvanishing eigenvalue emerges every second order in the perturbation theory. The corrections to the largest eigenvalue at a given order in the expansion have a more important effect to the entanglement entropy than the emergence of new eigenvalues at the same order.
4 Area and entanglement in the inverse mass expansion
4.1 The leading “Area Law” term
4.1.1 \(3+1\) Dimensions
The divergence of the numerical results from the expansive formula for entangling sphere radii close to Na is an effect induced by the IR cutoff that has been imposed since the theory has been defined in a finite size spherical box.
The numerical calculation requires the introduction of a cutoff in the values of \(\ell \). The convergence of the series (2.11) gets slower as the mass parameter increases. Thus, the perturbative expansion has an additional virtue; it provides a result for entanglement entropy in cases that the numerical calculation is more difficult.
4.1.2 \(2+1\) Dimensions
4.1.3 \(1+1\) Dimensions
Especially in the massless case, the perturbative formulae fail completely to capture the logarithmic behaviour of entanglement entropy (Fig. 3 topleft). Technically, this happens due to the structure of the couplings matrix K. In all cases this matrix is diagonally dominant, i.e. the sum of the absolute value of all nondiagonal elements does not exceed the diagonal one, in all rows and columns. As a result, the perturbative calculation of its square root and its inverse converges. Only in \(1+1\) dimensions and only in the massless case, the matrix saturates the diagonally dominant criterion. Not unexpectedly, the saturating case, lying between convergence and divergence, leads to a logarithmic dependence on the cutoff scale. However, this logarithmic dependence cannot be evident in a finite number of terms of the perturbation series. We will return to this kind of behaviour in the Sect. 4.2 on the subleading contributions to entanglement entropy.
The area law is the leading contribution to the entanglement entropy for large entangling sphere radii in all number of dimensions. The reason for this fact can be attributed to the locality of the underlying field theory [47, 48, 49]. The locality is depicted to the fact that the matrix K contains interaction elements only in the subdiagonal and superdiagonal. As a result, no matter what is the size of the entangling sphere (the value of n), there is only one element of K connecting a degree of freedom of subsystem A to a degree of freedom of subsystem \(A^C\). This property is inherited to the leading corrections in matrix B, and, thus, to the eigenvalues of the reduced density matrix. Had the theory been nonlocal, the number of leading contributions to entanglement entropy, would be a complicated function of the entangling sphere radius in general, leading to large divergences from the area law. In a more geometric phrasing, the area law emerges from locality, since the pairs of strongly correlated degrees of freedom (i.e. neighbours) that have been separated by the entangling surface, are proportional to its area.
4.2 Beyond the area law
The “area law” term of entanglement entropy is the leading contribution to the entanglement entropy for large radii of the entangling sphere. Beyond that, there are subleading terms, which can also be calculated in the inverse mass expansion that we developed in Sect. 3.

The reflections of these “edge effects” will lead to matrices \(\varOmega \), \(A^{1}\), etc, that depend on all the elements of the matrix K. Therefore, at high orders in the perturbation theory, such reflections introduce contributions to the entanglement entropy that depend on the global characteristics of the entangling surface. Such “universal” terms cannot be captured at any finite order in our perturbation series. They are rather nonperturbative effects in this expansion. The logarithmic term in even number of spacetime dimensions [16, 17, 42, 57, 58, 59, 60], as well as the constant term in odd number of dimensions [57, 58] are known to be exactly this kind of universal terms, and, thus, our inability to capture them in the inverse mass expansion should not be considered surprising. Of course such effects are visible in the numerical calculations.

The terms we capture in our perturbation series cannot sense the global properties of the region defined by the entangling surface. They have the property to depend on the local characteristics of the entangling surface. In a more technical language, this is depicted to the fact that the perturbative expressions for the elements of the matrices \(\varOmega \), \(A^{1}\) and \(C^{1}\) depend on a finite number of the elements of matrix K. This is the reason our method is appropriate to capture the “area law”, as well as subleading terms that scale with smaller powers of the entangling sphere radius. Therefore, our method is appropriate to study the dependence of such terms on geometric characteristics of the entangling surface, such as curvature [59], for more general entangling surfaces.

The introduction of the field mass exponentially dumps the propagation of the “edge effects” through the matrix elements [61]. As a result, our expansive calculations accurately converge to the numerical results in this case.
4.3 Dependence on the regularization scheme
Finally, we would like to comment on the dependence of the area law term, as well as the subleading terms of entanglement entropy on the regularization scheme. In our analysis, we have applied a peculiar, inhomogeneous regularization. Namely, we have imposed a cutoff in the radial direction, but not in the angular directions. Thus, the measurables that we have calculated, are those measured by a peculiar observer who has access to radial excitations of the theory up to an energy scale 1 / a and to arbitrary high energy azimuthal excitations.
We could have applied a different more homogeneous regularization imposing an azimuthal cutoff by constraining the summation series in \(\ell \) to a maximum value equal to \(\ell _{\max }\). Such a prescription would make our approach more similar to a traditional square lattice regularization. Notice however, that even in the square lattice case, the imposed cutoff is a cutoff to each of the momentum components and not strictly an energy cutoff that would allow direct comparison with formulae like (4.21).
As we discussed above, locality enforces the area law term to depend on the characteristics of the underlying theory in the region of the entangling surface. Therefore, a natural selection for an azimuthal cutoff \(\ell _{\max }\), when considering a ddimensional entangling surface should have the following property: the total number of harmonics with \(\ell \le \ell _{\max }\) should equal the area of the entangling surface divided by \(a^d\). In \(3+1\) dimensions this argument implies that a natural selection for the azimuthal cutoff is \(\ell _{\max } = 2 \sqrt{\pi } R / a\), whereas in \(2+1\) dimensions it implies \(\ell _{\max } = \pi R / a\). In all number of dimensions such a cutoff is of the form \(\ell _{\max } = c R / a\), where c is a constant. It is not difficult to repeat our analysis including this azimuthal cutoff. The only extra necessary steps are the introduction of a finite upper bound in the definite integral (4.4) and similarly the inclusion of the terms calculated at \(x=\ell _{\max }\) in the Euler–Maclaurin formula (4.2).

An azimuthal cutoff of the form \(\ell _{\max } = c R / a\) preserves the dominance of the area law term in entanglement entropy. This is not the case when a more general azimuthal cutoff is chosen (e.g. \(\ell _{\max } = c\)). The inverse mass expansion is still a good approximation when such a regularization scheme is chosen.

The area law term, as well as the subleading terms, are strongly affected by the selection of the dependence of the azimuthal cutoff \(\ell _{\max }\) on the radial cutoff a. This is the expected behaviour comparing with calculations in CFT or holographic calculations via the RyuTakayanagi conjecture. The only terms that do not depend on the regularization scheme are the universal terms, which cannot be captured by our perturbation theory.

The introduction of an azimuthal cutoff would also set the perturbative calculation of the entanglement entropy finite at higher number of dimensions, where the respective integral term diverges as \(\ell _{\max } \rightarrow \infty \).

Srednicki’s calculation, which is equivalent to the specific choice \(c \rightarrow \infty \), is an upper bound for the area law coefficient. The fact that the integral terms in more than \(3+1\) dimensions diverge, implies that such an upper bound exists only in \(2+1\) and \(3+1\) dimensions.
5 Discussion
The calculation of entanglement entropy in the ground state of oscillatory systems, which include free scalar field theories, at their ground state is in general a difficult, nonperturbative calculation, since the ground state is highly entangled. We managed to find a perturbative method to calculate it, using as expansive parameter the ratio of the nondiagonal to diagonal elements of the couplings matrix of the system. This parameter in the case of free scalar field theory is being played by the inverse mass of the field.
The calculation of entanglement entropy in the inverse mass expansion indicates that the major contribution to entanglement entropy is a term proportional to the area of the entangling surface, i.e. the “area law” term, a wellknown fact since [1, 28]. The perturbative calculation of the coefficient of this term agrees with the numerical calculation of entanglement entropy, based on the techniques of [1], and provides an analytic method for the specification of such coefficients. Subleading terms in the expansion of entanglement entropy for large entangling sphere radii can also be perturbatively calculated. The inverse mass expansion and the entangling sphere radius expansions can be performed simultaneously, but they are not parallel in any sense. The leading term in the entangling sphere radius expansion, i.e. the area law term, as well as the subleading terms, receive contributions at all orders in the inverse mass expansion.
The area law term, as well as the subleading ones are dependent on the regularization scheme, in line with analogous replica trick calculations. Universal terms that appear in the massless limit and depend on the global characteristics of the entangling surface (logarithmic terms in even dimensions and constant terms in odd dimensions) are nonperturbative contributions in this expansive approach. Furthermore, in this approach, the coefficient of the area law term in \(2+1\) and \(3+1\) dimensions has an upper bound, for any regularization scheme. The latter does not exist in higher dimensions.
An interesting feature of the inverse mass expansion is the following: the perturbation parameter is not exactly the inverse mass, but rather the quantity \(1 / \sqrt{\mu ^2 a^2 + 2}\), where a is the UV cutoff length scale imposed in the radial direction. This fact allows the application of the perturbation series even in the massless field case. Not surprisingly, the perturbation series converges more slowly than in the massive case; however, the values of the first terms strongly suggest that it still converges to the numerical results. In the case of free massless scalar field in \(3+1\) dimensions the inverse mass series for the coefficient of the area law term approaches the value 0.295 found in [1, 46].
An important advantage of the presented perturbative method is that it is not limited to the calculation of entanglement entropy, but it provides the full spectrum of the reduced density matrix. The latter, unlike entanglement entropy, contains the full information of the entanglement between the considered subsystems. This is clearly an advantage in comparison to holographic (the latter of course can be applied to strongly coupled systems, where it is impossible to apply our perturbative method) or replica trick calculations, which naturally allow the specification of Rényi entropies \(S_q\) for all q. Although in principle it is possible to reconstruct the spectrum of the reduced density matrix from the latter, in practise this process is very complicated and usually only the specification of the largest eigenvalue and its degeneracy may be easily achieved.
This perturbative method is an appropriate tool to expose the connection between the “area law” and the locality of the underlying field theory. Locality is encoded into the couplings matrix K as the absence of nondiagonal elements apart from the elements of the superdiagonal and subdiagonal. This in turn results in an hierarchy for the eigenvalues of the reduced density matrix system, leading to the area law. This hierarchy in the spectrum of the reduced density matrix depicts the fact that locality enforces entanglement between the interior and the exterior of the sphere to be dominated by the entanglement between pairs of neighbouring degrees of freedom that are separated by the entangling surface. The latter are clearly proportional to the area and not the volume of the entangling sphere.
It would be interesting to extend the applications of this perturbative expansion to other geometries, e.g. dS or AdS spacetimes, to cases where the overall system does not lie at its ground state (e.g. systems at a thermal state) or to other field theories containing fermionic fields or gauge fields. Furthermore, application of the above techniques for nonspherical entangling surfaces may shed light to the dependence of entanglement entropy on the geometric features of the latter, such as the curvature.
Footnotes
 1.
The entanglement Rényi entropy \(S_q\) is defined as \(S_q := \frac{1}{{1  q}}\ln \mathrm {Tr}\rho _A ^q\). Then, the entanglement entropy is recovered from the analytic continuation of \(S_q\) as the limit \(S_{\mathrm {EE}} = \mathop {\lim }\limits _{q \rightarrow 1} S_q\).
Notes
Acknowledgements
The research of G. Pastras is funded by the “Postdoctoral researchers support” action of the operational programme “human resources development, education and long life learning, 20142020”, with priority axes 6, 8 and 9, implemented by the Greek State Scholarship Foundation and cofunded by the European Social Fund  ESF and National Resources of Greece. The authors would like to thank M. Axenides, E. Floratos and G. Linardopoulos for useful discussions.
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