A Spacetime Calculation of the Calabrese-Cardy Entanglement Entropy

We calculate Sorkin's spacetime entanglement entropy of a Gaussian scalar field for complementary regions in the 2d cylinder spacetime and show that it has the Calabrese-Cardy form. We find that the cut-off dependent term is universal when we use a covariant UV cut-off. In addition, we show that the relative size-dependent term exhibits complementarity. Its coefficient is however not universal and depends on the choice of pure state. It asymptotes to the universal form within a natural class of pure states.

The Calabrese-Cardy formula for the entanglement entropy (EE) of a CFT for an interval I s of length s in a circle C of circumference is given by S = c 3 ln π + c 3 ln(sin(απ)) + c 1 where α = s/ , c is the CFT central charge, is a UV cut-off and c 1 is a non-universal constant [2]. This formula has been shown to apply to a diverse range of two dimensional systems which fall within the same universality class, including a geometric realisation by Ryu and Takayanagi [3] and others [4]. Entanglement entropy (EE) was first proposed in [5] as a possible contributor to black hole entropy. Hence understanding Eqn. (1) from a spacetime perspective is of broad interest.
As a follow up to their earlier work, Calabrese and Cardy studied the unitary time evolution of the EE for an interval I s inside a larger interval I ⊃ I s . Starting with a pure state, which is an eigenstate of a "pre-quench" Hamiltonian, and then quenching the system at t = 0, they used path integral techniques to show that the EE increases with time. It then saturates after the "light-crossing" time, in keeping with causality [6]. This corresponds to the "time" required for the domain of dependence of I s to be fully defined. Seeking out a covariant formulation of EE is therefore of interest both to understanding the results of [6] in a spacetime language as well as more generally in QFT and quantum gravity. Such a formulation is moreover in keeping with the broader framework of AQFT, where observables are associated with spacetime regions rather than spatial hypersurfaces [7].
In [8] Sorkin proposed a spacetime formula for the EE of a Gaussian scalar field Φ in a globally hyperbolic subregion O of a globally hyperbolic spacetime (M, g), with respect to its causal complement O c . It uses the restriction of the Wightmann function W (x, x ) in M to O, and the Pauli-Jordan function i∆(x, x ) which appears in the Peierl's spacetime where χ ∈ Ker( ∆) and where It is motivated by the finite system Wightmann function for a Gaussian state which is a direct sum of identical systems with two degrees of freedom [8]. The SSEE formula generalises the calculation of EE for a state at a given time to that associated with a spacetime region. In [1] the SSEE for nested causal diamonds D s ⊂ D S was shown to yield the first, cut-off dependent term of Eqn. (1) with c = 1 when s << S. Since D s is the domain of dependence of I s , this is the natural spacetime analogue of I s ⊂ I S . In this work we calculate the SSEE for the spacetime analogue of I s ⊂ C for finite and additionally, find the same α-dependence as Eqn. (1), thus explicitly demonstrating complementarity. A natural spacetime analogue of C is its (zero momentum) Cauchy completion, which is the d = 2 cylindrical spacetime (M, g) with ds 2 = −dt 2 +dx 2 , x+ ∼ x. The domains of dependence of I s and its complement I −s in (M, g) are the causal diamonds D s and D −s respectively, as shown in Fig 1. In what follows we use a mixture of analytical and numerical methods to solve the SSEE eigenvalue problem.
We will find it convenient to work with the Sorkin-Johnston (SJ) formulation [9,10,11,12,13], where the SJ spectrum provides the required (covariant) UV cut-off with which to calculate S, as was done in [1]. For a compact globally hyperbolic region (M, g) of a spacetime it follows from Ker( ) = Im(i ∆) [14] that the eigenmodes of the integral Hermitian operator i ∆ provide a covariant orthonormal basis (the SJ modes) with respect to the L 2 norm on (M, g) [7]. The SJ vacuum or Wightmann function is given by the positive part of i ∆. Since the SJ spectrum is covariant so is a UV cut-off in this basis.
For our calculation of S we will use the SJ vacuum W τ for a free massless scalar field in a slab (M τ , g) of height 2τ in the cylinder spacetime [10], and its restriction to D s ⊂ M τ , where {ψ m , m } are the L 2 normalised positive frequency SJ eigenmodes and eigenvalues in M τ [10]: The m = 0 "zero mode" in particular takes the form Unlike the standard vacuum on the cylinder, W τ is τ -dependent. Each W τ can however be viewed as a pure (non-vacuum) state in Mτ for anyτ > τ , as we will later show. To accommodate both D s and D −s in our calculations, we require 2τ ≥ s, − s.
The SJ modes in D s are naturally expressed in terms of the light cone coordinates u = 1 √ and come in the two mutually orthogonal series [15] with eigenvalues λ k = s 2 √ 2k and λ κ = s 2 √ 2κ , respectively, and with L 2 norm in D s Since i ∆ is diagonal in this basis we will use it to transform Eqn. (2) to the matrix form where Λ is the diagonal matrix {λ k , λ κ }. For X ∈ Ker(i ∆), we can invert this to suggestively write so that S can be viewed as the von-Neumann entropy of ρ. The spectrum of ρ is unbounded and hence needs a UV cut-off. As in [1] we use the covariant UV-cut off with respect to the SJ spectrum {λ k , λ κ }. For large κ the condition tan(κs/2 √ 2) = κs/ √ 2 can be approximated by κ ∼ √ 2(2n + 1)π/s, so that a consistent choice of cut-off for both sets of eigenvalues is We also need to ensure that this same cut-off is used in the causal complement, i.e., k max = 2 √ 2n max π/( − s), where n denotes the quantum number We expand the SJ modes in M τ in terms of those in D s to obtain the non-zero matrix elements for W τ | Ds for general α, γ. Suppressing the τ, D s labels, these are where x ± = (n ± αm)π, x ± = (n ± αm)π, z ± = κs/2 √ 2 ± αmπ, z ± = κ s/2 √ 2 ± αmπ, and the contribution from the zero mode is Our strategy is to construct ρ from these matrix elements and to solve for its eigenvalues using a numerical matrix solver. However, each matrix elements in Eqn. (11) is an infinite sum over the quantum number m and hence not amenable to explicit calculation. We therefore need to find a closed form expression for the above matrix elements.
We notice that when γ takes half-integer values (for which the SJ vacuum Hadamard [10]), ζ m = 1 for m = 0, which leads to a considerable simplification. Further, let α be rational, so that we can write α = p q , with p, q ∈ Z, and p, q > 0 being relatively prime. For these choices of α and γ, the infinite sums of Eqn. (11) reduce to the following finite sums over Polygamma functions Ψ(x) and Ψ (1) where γ e represents the Euler-Mascheroni constant and Ω(κ, κ , α, r) = κκ s 2 2 cos 2 (αrπ) + sin 2 (αrπ) We are now in a position to solve for the eigenvalues of ρ using Mathematica's numerical eigenvalue solver. We consider a range of values of α, γ and the cut-off n max /α given in the table below. In the list of γ values, we have also included the specific non-half-integer value of γ = 40.3 for which ζ m ∼ 1 even for m = 1. In general, we note that ζ m ∼ 1 for m >> γ −1 . The error coming from small m terms has been explicitly calculated in this case as a function of m and seen to be small. For the special case α = 0, S is trivially zero, while for α = 1, the domain of dependence of C is no longer a causal diamond, but all of M τ . Since W τ is the SJ vacuum and therefore pure, S = 0. Fig. 2 shows the results of simulations for these various α and γ values, for a fixed choice of cut-off n max /α = 2600. It is already clear that S satisfies complementarity. This is much more explicit in Fig. 3, where we vary over the cut-off. Our numerical results suggest that S takes the general form Using the best-fit curves in Figs. 3-5, and the associated data in the appendix, we find that c(γ) ∼ 1 and Thus, the first term of Eqn. (1) is reproduced for any choice of α, γ. This generalises the results of [1] where this was shown in the limit of α 1. The dependence on α, i.e., for different α fitted to S = a log (n max /α) + b for γ = 1000.  the second term of Eqn. (1) is also reproduced and hence exhibits complementarity for any α (see Fig. 2, 3). Its coefficient however is not universal and depends on γ as shown in Fig. 4. However, as γ >> 1, f (γ) does asymptote to the universal value 1/3. Finally, the non-universal constant c 1 (γ) diverges logarithmically with γ as shown in Fig. 5. This can be traced to the IR divergence in the zero modes of the massless theory.
The behaviour of f (γ) can be viewed as a dependence on the choice of the pure state W τ in Mτ , for Mτ ⊃ M τ . From Eqn. (4) we see that W τ is a state in Mτ , i.e., W τ = R τ + i ∆/2, where R τ is real and symmetric.
Expanding i ∆ in the SJ modes {ψ The solutions for this are either a mĀm + b mBm = 0 ⇒ µ = 0, or a mBm + b mĀm = 0 ⇒ µ = We end with some remarks. While we have demonstrated complementarity for certain rational values of α, an analytic demonstration using Eqn. (13) seems non-trivial, in part because the UV regulated matrices ρ α and ρ 1−α are of different dimensions. Conversely, complementarity implies that if n max > n max , ρ α = ρ 1−α ⊕ 1 N ⊕ 0 N , where 0 is the zero matrix and N = (n max − n max )/2.
In our computations we find that the eigenvalues of ρ (which always come in pairs (µ, 1 − µ)) exhibit the surprising feature that all but one pair hovers around the values 0 and 1, thus contributing most significantly to S. Indeed, the S calculated using the largest few pairs of eigenvalues accounts for most of the entropy (see appendix).
Finally, it would be interesting to calculate the non-zero mass case which is IR divergence free. While the small mass approximation of the SJ modes in D s is known [16], the challenge will be to obtain closed form expressions for the matrix elements of W as we have done.

Appendix: Supporting Data
In this appendix we present some plots with additional data which were used to compute the coefficients c(γ), f (γ) and c 1 (γ) in the Entanglement Entropy. Fig. 6 shows the dependence of S on α for different values of γ and with three different values of n max /α (1200, 2000 and 2600). The SSEE can be fitted to the form S = a 1 log(sin(απ)) + b 1 where the coefficient a 1 corresponds to f (γ) in Eqn. (15). The values of a 1 and b 1 along with their errors are given in the tables in Fig. 6. a 1 and therefore f (γ) can be seen to be independent of n max /α. It is however dependent on γ and asymptotes to the universal value of 1/3 in the Calaberse-Cardy formula for γ >> 1. We fit f (γ) values to the form f (γ) = 0.33 + a 2 /γ + b 2 /γ 2 (18) and find the a 2 ≈ −0.48 and b 2 ≈ 0.23 with the error given in the tables of Fig. 4. Fig. 7 shows the dependence of S on n max /α for different α and with three different values of γ (16, 200 and 1000). Here SSEE can be fitted to the form, S = a 3 log(n max /α) + b 3 . As is clear from the tables in this figure a 3 ≈ 0.33 ≈ 1/3 for all α and γ with the order of error given in the table. b 3 however depends on α and γ. This suggests that c(γ) ≈ 1 in Eqn. (15).
In order to extract c 1 (γ) we subtract the first term in Eqn. (15) (which depends on n max /α) using c(γ)/3 given by the values of a 3 in the table of Fig. 7 from the values of b 1 in the table of Fig. 6 for n max /α = 1200, 2000 and 2600. We find that the difference (or c 1 (γ)) is independent of the choice of n max /α which is as expected. We fit the dependence on γ by and the values for the coefficients are given in the table in the Fig. 5.
We also find that the eigenvalues (which always come in pairs (µ, 1 − µ)) exhibit the surprising feature that all but one pair hovers around the values 0 and 1 and hence contributes significantly to S. In Fig. 8 we also show the comparison of the eigenvalues obtained in the two complementary regions, we find that they differ only in the numbers of (0, 1) pairs. Further, if we calculate S for the largest pairs of eigenvalues, we find that the error is small, as shown in       g κ -eigenvalues (γ=100) g κ -eigenvalues (γ=200) g κ -eigenvalues (γ=1000) g κ -eigenvalues (γ=2000) (c) n max /α = 2600, γ = 1000 Figure 9: In order to estimate the contribution of the pairs (µ, 1 − µ), we plot the percentage error in the SSEE when only the largest pairs (one, two and three represented in blue, orange and green respectively) of eigenvalues are considered, as a function of the different parameters γ, n max /α and α. In each case, we see that the error goes down to < 1% even when only the 3 largest eigenvalues are retained.