This can be written as S(ρout) =−Tr
log(1−Ξ) + Ξ
1−Ξlog Ξ
, Ξ = β0
1 +p
1−β02 . (14.17) All square roots and inversions are well-defined and the eigenvalues of Ξ, ξj, obey 0 ≤ ξj ≤1 for all j= 1, . . . , N−n.
It is also shown in Ref. [71] that the above result for N coupled harmonic oscillators can be generalized to the case of a free quantum field theory (cf. Sec. 15 below).
14.2 Entanglement entropy in quantum field theory
We now consider a free massless real field φ(t, ~x) defined in four-dimensional spacetime (witht denoting time). We work in the Hamiltonian formalism and assume that at t= 0 the system is in its ground state, the vacuum. We wish to eliminate the quantum degrees of freedom associated withφ(~x) and its conjugate momentumπ(~x) located in the spherical region |~x| ≤ R in space. We eliminate these degrees of freedom by tracing over all wave functionals of φ(~x) with |~x| ≤ R. Vacuum expectation values of operators O depending only on φ(~x) and π(~x) with|~x|> R, denoted asφout,πout, respectively, can be expressed with the help of the density matrix operatorρout(φout, φ0out),
hOi= Tr (Oρout) . (14.18)
The reduced density matrixρout represents a mixed state and a measure of its “distance”
from a pure state may again be taken as the von Neumann entropy
Sout=S(ρout) =−Trρoutlogρout. (14.19) One can trace out the outside degrees of freedom instead which results in an entropy Sin = S(ρin) that is equal to Sout since the reduced density matrices ρin and ρout are obtained from a pure state (the ground state) and therefore have the same eigenvalues, up to zeros that do not contribute to the entropy (cf. also Eq. (14.5) above). Hence, it is not surprising that the entanglement entropy is not an extensive quantity and in general depends only on the geometric properties of the surface separating the regions “in” and
“out”.
The entropy Sin=Sout is non-zero because the operatorsφ(~x) are coupled for points
~
x infinitesimally close to the two sides of the surface|~x|=R. Were it not for the spatial derivative terms in the Hamiltonian, the ground state would be a single tensor product over ~x of functionals of φ(~x) and the elimination of the degrees of freedom inside the sphere would leave a pure state describing the outside degrees of freedom. Thus, one can view Sout as an entanglement entropy where the reference basis is made out of single tensor products of functionals of φ(~x). Since the reason for Sin =Sout 6= 0 is due to the coupling of the fields at the surface of the sphere, this also indicates that the entropy should depend only on the surface of the sphere and its embedding in flat spacetime. As the coupling causing the entanglement occurs at infinitesimal separation, it is natural that a complete definition of the entropy will require, at the least, an ultraviolet cutoff, i.e., a small distance a. Without any cutoff, there could be no dependence on R since the entropy is a pure number. For the same reason, only a logarithmic dependence onR can have an a-independent meaning. The main result presented in Ref. [70] is a numerical estimate for this coefficient of logR inSout (cf. Sec. 15 below).
In Ref. [74], a general formula for Sout is derived for Hamiltonians that are quadratic in the fields. Furthermore, it is shown that with the addition of a mass term to the
Hamiltonian, the entropy per unit surface for a cavity of the form of a three-dimensional slab of finite thickness is finite in the a → 0 limit after the subtraction of a divergent term which does not depend on the thickness. First, ultraviolet and infrared cutoffs are introduced, and then, the appropriate limits are taken.
Ref. [74] also outlines the calculation for more general cavities. In the spherical case with massless free fields, the entropy cannot be finite and R-dependent because R is the single available scale. The spherical case was first studied numerically by Srednicki in Ref. [71]. Srednicki independently arrived at the same setting of the problem as Bombelli et al. in Ref. [74] and took the next step and evaluated Sout for the case of the sphere with a specific regularization. It turns out that one only needs to discretize the spatial radial direction and that there are no infrared divergences. The short distance structure in the spatial angular directions does not require any ultraviolet regularization, in agreement with the expectation that only the coupling in the normal direction to the sphere surface is relevant. If the lattice spacing in the radial direction is denoted by a, the leading term in Sout is proportional to (R/a)2 forR/a → ∞. The coefficient of this leading term was computed numerically in Ref. [71], but it clearly is not a universal number, i.e., it is not independent of the regularization procedure.
The spherical case is particularly interesting since tracing out the degrees of freedom inside the imaginary sphere (in the vacuum of flat space) results in an entanglement en-tropy that is somewhat reminiscent of the Bekenstein-Hawking enen-tropy SBH of a black hole, being proportional to the area A of the event horizon, SBH = 14MPlanck2 A. (The classical law that the surface area of the event horizon of a black hole can only increase with time led Bekenstein to the suggestion that it might be related to an entropy. This interpretation was supported by other analogies between classical black holes and ther-modynamics found by Bardeen, Carter, and Hawking, and finally by Hawking’s discovery that applying quantum mechanics to matter fields in the background geometry of a black hole metric leads to the emission of particles corresponding to a thermal spectrum with a certain temperature (determined by the mass of the black hole), which enables the black hole to remain in equilibrium with thermal radiation at the same temperature, see, e.g., Ref. [75].) The observation that the entanglement entropy in free field theory is also pro-portional to the area of the (imaginary) boundary surface led to the interpretation that the amount of missing information in the black hole case, quantified by the entropySBH, can be viewed in analogy to the entropy resulting from restricting the access of an observer to the outside of a sphere in flat spacetime [71, 74].
In Ref. [70], we have followed Srednicki and pushed his numerical analysis further, looking for terms in Sout that are subleading in R/a. We found subleading terms of the form
clog(R/a) +d . (14.20)
We determined the values c = −1/90 and d = −0.03537 with a precision of about two tenths of a percent (cf. Sec. 15 below). Like the coefficient of the leading (R/a)2 term, the constant d is non-universal, but the value −1/90 for c is expected to be a universal number (cf. Sec. 15.4 for a more detailed discussion).
Since c might be universal, there ought to be other, analytical, ways to derive it. An attractive method to do this is based on an analogue of the so-called replica method, using the identity
Sout=− ∂
∂nTrρnout
n=1. (14.21)
First, the vacuum wave functional is represented by a functional integral over the Euclidean half spacet <0. The reduced density matrixρout is then obtained by gluing two copies of the half space along the space region complementary to the “outside” region (the interior
188 14.2 Entanglement entropy in quantum field theory
of the sphere) at t = 0. For integer n, the trace operation can then be implemented by taking n copies of the Euclidean space, which are cut along the complement of the imaginary sphere, and cyclically gluing together successive copies along the two sides of the cut. At the end, Trρnoutis obtained from a partition function on a complicated Riemann surface, an n-sheeted manifold with conical singularities located at the boundary of the sphere (see Ref. [72] for details). One advantage of this method is that the universal term can be obtained from the conformal anomaly, perhaps in closed form and for arbitrarily shaped cavities, not just a spherical one. However, handling the singularity and the needed analytic continuation in nmake the application of this method somewhat uncertain.
In the ’t Hooft large-N limit of a conformal field theory, one may try to use the AdS/CFT correspondence in order to calculate the entanglement entropy for various cav-ities in the context of strongly interacting conformal field theories. One needs a prescrip-tion for the quantity corresponding toSout. An ansatz that seems to work is reviewed in Ref. [76]. This ansatz can be applied to N = 4 U(N) supersymmetric YM theory and leads to an entropy given by −N2logR for the sphere.
The result of applying the replica method to the spherical case for a real scalar field in flat four-dimensional spacetime is quoted in the review [72] and relevant references are given. The answer they quote isc=−1/90 (cf. Eq. (281) in Ref. [72]). In this calculation, originally presented in Ref. [77], a missing coefficient in the generic four-dimensional case is calibrated by comparing the replica method to the holographic ansatz of Ref. [76] for the entanglement entropy in superconformal gauge theories (based on the assumption that the coefficient does not depend on the field content).
The resultc=−1/90 is in agreement with the numerical result presented below. Our numerical work is presented in greater detail in the next section since the application of the replica method in conjunction with conformal anomaly calculations encounters some subtleties in the case that the surface enclosing the cavity has extrinsic curvature, as is the case for the sphere, cf. Ref. [78]. Our numerical work is a check of the logarithmic coefficient for a real scalar field in the free case using a specific regularization (introduced in Ref. [71]). We expect that a similar approach could be used to determine the coeffi-cients for electromagnetic fields and massless Weyl fermions (for which the predictions of the replica method are c = −62/90 and c = −11/180, respectively, cf. Ref. [72]). Any general conclusions about the validity of the replica method, the associated conformal anomaly calculation, and the related AdS/CFT correspondence prescription for entan-glement entropy in four dimensions, in the presence of cavities with surfaces possessing extrinsic curvature, are left for future work. More examples might have to be numerically worked out before matters can be clarified. In this context, we conclude that reasonable numerical results can be obtained in sufficiently simple cases with the accuracy attainable in reasonable amounts of time on today’s consumer-level desktop computers.
15 Numerical computation for a sphere
15.1 Setup of the problem
In the following, we summarize the setup of the problem in Ref. [71], which is the starting point for our numerical approach. The Hamiltonian of the free and massless scalar field is
H= 1 2
Z
d3x[π2(~x) +|∇φ(~x)|2]. (15.1) It is convenient to expand π and φ in spherical harmonics, labeled by integersl ≥0 and m=−l, . . . , l, which amounts to a canonical transformation to
[φlm(x), πl0m0(x0)] =iδll0δmm0δ(x−x0), (15.2) where x ≡ |~x| ≥0. The new variables can still be separated into “inside” and “outside”
sets. Now H=P
The variable x is discretized to j ·a, where a is our short distance cutoff and j = 1,2, . . . , N. The number N provides an infrared cutoff which will be taken to infinity at the end. The range of l is kept infinite and it will be shown that the sum over l and m converges for fixed N. This means that one does not need to discretize also the angular degrees of freedom; no ultraviolet divergences are generated in the directions tangential to the surface of the sphere. The finite, regularized Hlm is given by
Hlm= 1 where we set φlm,N+1 ≡0. Focusing on a specific (lm)-sector, we drop the l, mindices of the field variables and write
Hlm= 1
The real, symmetric, semipositive, tridiagonalN×N matrixK(l) is independent ofmand has non-vanishing entries given by
K11(l)= 9 This means that we can proceed exactly as in the simple example of N coupled harmonic oscillators for every l and m independently (see Sec. 14.1). We trace out the degrees of freedom at radial coordinates 1 ≤j ≤n, which corresponds to a separation into “inside”
and “outside” regions by a sphere of radius R= n+12 a.