Some results on the mutual information of disjoint regions in higher dimensions

Since the pioneering paper of Srednicki [1] there has been increasing interest in understanding and quantifying entanglement in quantum field theories. In that paper it was shown that, in a free scalar field theory, the von Neumann entropy S_A = −Tr ρ_Alog ρ_A of the reduced density matrix ρ_A describing the degrees of freedom inside a spherical region A, which measures the entanglement of the degrees of freedom in A with those in its complement, is proportional to the area of its boundary. Subsequently this 'area law' was show to be generic in space dimensions d ⩾ 2 [2], and this prompted comparisons with black hole physics.

However in 1993 Holzhey et al [3] showed that in a conformal field theory (CFT) in d = 1 the entanglement entropy of an interval of length R_A goes like log R_A with a universal coefficient proportional to the central charge c. (Similar logarithms are now understood to occur whenever d + 1 is even, but for higher d these are non-leading with respect to the area term [4].) Subsequently this logarithmic behaviour was observed in numerical studies of critical quantum spin chains whose long-distance behaviour is believed to be described by a CFT [5]. A more complete analysis of entanglement in 1 + 1-dimensional CFTs was given in [6].

More recently [7] these methods, which involve using the so-called replica trick of computing S_A as limit as n → 1 of the Rényi entropies

$$\begin{equation*} S_A=\lim _{n\rightarrow 1}S_A^{(n)}\quad \mbox{where}\quad S_A^{(n)}=(1-n)^{-1}\log {\rm Tr} \rho _A^n \end{equation*}$$

have been extended to the computation of the entanglement entropy S_A∪B between two disjoint intervals A, B and the rest of the system in a 1 + 1-dimensional CFT. It was shown that this encodes all the data of the CFT, not only the central charge. Moreover the mutual information, given by the limit as n → 1 of the difference of Rényi entropies

$$\begin{equation*} I^{(n)}(A,B)\equiv S^{(n)}_{A}+S^{(n)}_{B}-S^{(n)}_{A\cup B} \end{equation*}$$

has an expansion of the form

$$\begin{equation} I^{(n)}(A,B)=\sum _{\lbrace k_j\rbrace }C^{(n)}_A(\lbrace k_j\rbrace )C^{(n)}_B(\lbrace k_j\rbrace )\left(\frac{R_AR_B}{r^2}\right)^{\sum _{j=1}^nx_{k_j}}\,. \end{equation} \tag{ 1 }$$

Here R_A, R_B measure the lengths of the two intervals, and r is the separation between their centres. The sum is over a set of scaling operators of the CFT, labelled by k_j, with scaling dimension $x_{k_j}$, for each replica j. The coefficients are universal and encode information about the correlation functions of these operators in the plane. For R_AR_B/r² ≪ 1 the leading term in (1) comes from the case when only two of the x_k are non-vanishing and correspond to the lowest dimension operator in the CFT. It was possible [7] to continue this result analytically in n to compute the leading term in the mutual information I(A, B).

The one-dimensional case has also been studied numerically in a number of papers [8].

The arguments of [7] were based on a kind of operator product expansion first used by Headrick [9]. In this paper we argue that (1) also holds for the mutual Rényi entropies in higher dimensional CFTs. However in this case the coefficients are much more difficult to compute, and we succeeded in obtaining explicit results for the leading term only in the simple case of a free massless scalar field theory, and when A and B are spheres of radii R_A and R_B. In principle our methods work in any (integer) number of dimensions, but the results are most simply expressed in d = 2 and 3.

In general we find

$$\begin{equation*} I^{(n)}(A,B)\sim g^{(n)}_d\left(\frac{R_AR_B}{r^2}\right)^{d-1} , \end{equation*}$$

where, in 3 + 1 dimensions,

$$\begin{equation*} g^{(n)}_3=\frac{n^4-1}{15n^3(n-1)} . \end{equation*}$$

For the mutual information (n = 1) this gives $g^{(1)}_3=\frac{4}{15}=0.26\dot{6}$, to be compared with a numerical result 0.26 due to Shiba [12].

In 2 + 1 dimensions

$$\begin{eqnarray*} &&\fl g^{(n)}_2 = \frac{n}{2\pi ^2(n-1)}\int _0^\infty\!\! \int _0^\infty \left(\frac{(1-x)(1-y)(1-(xy)^{n-1})}{1-xy} + (1-x^{n-1})(1-y^{n-1})\right)\\ &&\times \frac{{\rm d}x \, {\rm d}y}{(1+x)(1+y)(1-x^n)(1-y^n)} , \end{eqnarray*}$$

which leads to $g^{(1)}_2=\frac{1}{3}$, to be compared with the numerical result 0.37 [12].

We are also able to compute the form of the corrections to the leading term, which should be O(R_AR_B/r²)^{2(d − 1)} and also O(R_AR_B/r²)^{d + 1}. In principle the coefficients are calculable. Note that these are more important in d = 2 than in d = 3 which may account for the above discrepancy.

Our methods use the conformal invariance of the massless free field theory, which in fact implies that for $R_A\not=R_B$ the actual expansion parameter is

$$\begin{equation*} \frac{R_AR_B}{r^2-(R_A-R_B)^2} . \end{equation*}$$

In this form, our results also apply to the mutual entanglement between the interior of a sphere of radius R_A and the exterior of a concentric sphere of radius R_B, in the limit when R_A ≪ R_B.

For n = 2 and general shapes we show that the coefficients $C^{(2)}_{A,B}$ are given by the capacitance of d-dimensional bodies A, B in d + 1-dimensional electrostatics.

The mutual information for a free scalar field in higher dimensions has been studied in only a few papers. Casini and Huerta [10] and Shiba [11] showed that it should decay as (R_AR_B/r²)^{d − 1} at large separations (and (R_AR_B/r²)^d for free fermions [10]). This was based on the expression for the density matrix in terms of the correlation functions [13] which holds for any system with a Gaussian wave functional. However the coefficients must still be determined numerically for finite separations and then extrapolation. Recently this was carried out by Shiba [12] for the case of equal spheres in d = 2 and 3 space dimensions, and also rings and shells [12] as well as some other shapes [14]. This shows that the mutual information is not extensive, that is, is not given by a double integral of a kernel over the boundaries or the bulk of A and B.

Free and interacting fermions at finite chemical potential (Fermi liquids) have been studied in [15], and the entanglement of the radiation field with a dielectric medium in [16].

The layout of this paper is as follows. In section 2 we first recall the expressions for the various Rényi entropies $S^{(n)}_R$ as path integrals over n copies of ${\mathbb {R}}^{d+1}$ sewn together in a particular way along the boundary of the selected region R to form the conifolds $\mathcal{C}^{(n)}_R$. We then consider the general case of the small R_{A, B} expansion, and show how the coefficients $C^{(n)}_{A,B}$ in this expansion are related to one- and two-point functions on $\mathcal{C}^{(n)}_{A,B}$.

The rest of the paper is devoted to the special case of a massless free scalar field theory. In section 3.1 we show that for n = 2 the coefficients $C^{(2)}_{A,B}$ are related to the electrostatic capacitance of each region. In an appendix we derive explicit formulas for the case where A and B are spherical, or more generally ellipsoidal, by generalizing a famous method due to W Thomson to general dimension d. In section 3.2 we obtain results for the spherical case for general n and d, using conformal invariance. We show that for d odd the coefficients are polynomials in n which can then be continued to n → 1 to find the von Neumann entropy. For d even the result is not a polynomial but explicit results may nevertheless be obtained for n integer as well as the limit n → 1. In section 3.3 we consider the case of a single spherical region in a free massive scalar theory, and confirm the existence of universal logarithmic terms in the Rényi entropies first predicted in [6].

Consider a d + 1-dimensional quantum field theory in a domain D, in its ground state |0〉. In this paper we consider D to be ${\mathbb {R}}^d$, although some of the results may be adapted to semi-infinite and finite domains and also to finite temperature.

If X is some subdomain, we suppose that, in the presence of a suitable UV cut-off, the Hilbert space can be decomposed as $\mathcal{H}_X\otimes \mathcal{H}_{\overline{X}}$. The Rényi entropies are $S^{(n)}_X=(1-n)^{-1}\log P^{(n)}_X$, where

$$\begin{equation*} P^{(n)}_X={\rm Tr}_{\mathcal{H}_X} \rho _X^n\qquad \mbox{with}\qquad \rho _X={\rm Tr}_{\mathcal{H}_{\overline{X}}} |0\rangle \langle 0| . \end{equation*}$$

As explained in [6], these can be expressed in terms of a path integral on a particular conifold as follows. The ground state wave-functional 〈{ψ(x)}|0〉 is given by a path integral in imaginary time on the half-space ${\mathbb {H}}_-={\mathbb {R}}^d\times (-\infty ,0)$ from τ = −∞ to τ = 0, with the fields constrained to take the values {ψ(x)} on τ = 0. Similarly 〈0|{ψ(x)}〉 is given by the path integral on ${\mathbb {H}}_+$ from τ = 0 to τ = ∞. Each of these should be normalized by Z^−1/2, where Z is the partition function on ${\mathbb {R}}^{d+1}$. (Strictly speaking we should restrict the integration to |τ| < T before taking the ratio, then let T → ∞, and similarly with the thermodynamic limit in space.)

To get $P^{(n)}_X$ we take n copies of ${\mathbb {H}}^{(j)}_\pm$ labelled by j = 0, ..., n − 1, and sew ${\mathbb {H}}_-^{(j)}$ to ${\mathbb {H}}_+^{(j+1)}$ (mod n) along τ = 0 for x ∈ X, while we sew ${\mathbb {H}}_-^{(j)}$ to ${\mathbb {H}}_+^{(j)}$ for x∉X. This gives a conifold $\mathcal{C}^{(n)}_X$ with a d −1-dimensional submanifold of conical singularities along the boundary ∂X∩{τ = 0}. Then

$$\begin{equation*} P^{(n)}_X=\frac{Z\big(\mathcal{C}^{(n)}_X\big)}{Z^n} . \end{equation*}$$

In general, for d > 1, $Z\big(\mathcal{C}^{(n)}_X\big)$ has a leading term going like exp (a_nVol(∂X)Λ^{d − 1}) where Λ is an ultraviolet cut-off, and the dimensionless coefficients a_n are non-universal. This gives a term in the Rényi entropies proportional to a_nVol(∂X)Λ^{d − 1}—the famous 'area law' term, which is non-universal and therefore theoretically less interesting. However, in the case where X consists of two disjoint compact regions (A, B) the quantities

$$\begin{equation} I^{(n)}(A,B)\equiv S^{(n)}_{A}+S^{(n)}_{B}-S^{(n)}_{A\cup B}=(n-1)^{-1} \log \left(\frac{Z\big(\mathcal{C}^{(n)}_{A\cup B}\big)Z^n}{Z\big(\mathcal{C}^{(n)}_A\big)\big(\mathcal{C}^{(n)}_B\big)}\right) \end{equation} \tag{ 2 }$$

are free of these non-universal contributions, and should depend only on the geometry of A and B and the universal data of the renormalized QFT.

In general, however computing $Z\big(\mathcal{C}^{(n)}_{A\cup B}\big)$ is difficult, and, even for d = 1, we have explicit results only for free field theories [7, 10]. However in the limit when the linear sizes R_{A, B} of A and B are much smaller than their separation r, it was shown in [7] that an expansion in increasing (in general, fractional) powers of R_AR_B/r² is possible with coefficients which are calculable in principle. We now generalize this argument to dimensions d > 1.

The basic idea is that, from the point of view of an observer far from A or B, the sewing together of the copies along the boundaries of A and B should be expressible as a weighted sum of products of local operators $\Phi ^{(j)}_{k_j}$ at some conventionally chosen points (r_A, r_B) inside A and B, where $\Phi ^{(j)}_{k_j}$ is in the algebra of local operators of the QFT defined on jth copy of ${\mathbb {R}}^{d+1}$. That is, the sewing operation in each region can be thought of as a semi-local operator which couples together the n QFTs, but can itself be expressed as a sum of a product of local operators in a direct product of the QFTs¹. We write

$$\begin{equation} \frac{Z\big(\mathcal{C}^{(n)}_{A\cup B}\big)}{Z^n}= \big\langle \Sigma ^{(n)}_A\Sigma ^{(n)}_B \big\rangle _{({\mathbb {R}}^{d+1})^n}\,, \end{equation} \tag{ 3 }$$

where

$$\begin{equation} \Sigma ^{(n)}_{A}=\frac{Z\big(\mathcal{C}^{(n)}_{A}\big)}{Z^n}\,\sum _{\lbrace k_j\rbrace }C^{A}_{\lbrace k_j\rbrace }\prod _{j=0}^{n-1}\Phi _{k_j}\big(r_{A}^{(j)}\big), \end{equation} \tag{ 4 }$$

and similarly for B. Here k_j label a complete set of operators on the jth copy. The prefactor on the rhs is inserted since we expect the leading behaviour as r → ∞ to come from the term when all the $\Phi _{k_j}$ are the identity operator, and in this limit $\Sigma ^{(n)}_{A\cup B}\sim \Sigma ^{(n)}_{A}\Sigma ^{(n)}_{B}$.

The main point now is that the coefficients $C^{A,B}_{\lbrace k_j\rbrace }$ should be universal and therefore independent of the other regions and other local operator insertions as long as they are far away. Thus we can compute them in the simpler situation when X = A but we consider the correlation functions of an arbitrary set of local operators at points r^(j) on $\mathcal{C}^{(n)}_A$ outside A:

$$\begin{equation*} \Bigg\langle \prod _{j^{\prime }}\Phi _{k_{j^{\prime }}^{\prime }}(r^{(j^{\prime })})\Bigg\rangle _{\mathcal{C}^{(n)}_A}=\Bigg\langle\Bigg(\prod _{j^{\prime }}\Phi _{k_{j^{\prime }}^{\prime }}(r^{(j^{\prime })})\Bigg) \Bigg(\sum _{\lbrace k_j\rbrace }C^{A}_{\lbrace k_j\rbrace }\prod _{j=0}^{n-1}\Phi _{k_j}\big(r_{A}^{(j)}\big)\Bigg)\Bigg\rangle_{({\mathbb {R}}^{d+1})^n} . \end{equation*}$$

Note that the rhs decomposes into a sum of products of two-point functions on each copy of ${\mathbb {R}}^{d+1}$, that is we may take j' = j.

This is valid for any QFT, but in the special case of a CFT we can choose the complete set of local operators so that their two-point functions are orthonormal for all separations:

$$\begin{equation*} \langle \Phi _{k^{\prime }}(r)\Phi _k(r_A)\rangle =\frac{\delta _{k^{\prime }k}}{|r-r_A|^{2x_k}} , \end{equation*}$$

where x_k is the scaling dimension of Φ_k and we have assumed scalar operators for simplicity.

Thus we find, taking the limit r^(j) → ∞_j (that is, infinity on the jth copy of ${\mathbb {R}}^{d+1}$)

$$\begin{equation*} C^A_{\lbrace k_j\rbrace }=\lim _{\lbrace r^{(j)}\rbrace \rightarrow \infty _j}|r^{(j)}|^{\sum _jx_{k_j}}\Bigg\langle \prod _j\Phi _{k_j}(r^{(j)})\Bigg\rangle _{\mathcal{C}^{(n)}_A} . \end{equation*}$$

Note that $C^A_{\lbrace k_j\rbrace }\propto R_A^{\sum _jx_{k_j}}$ by dimensional analysis.

In 1 + 1 dimensions, when A is an interval of finite length, $\mathcal{C}^{(n)}_A$ may be uniformized to $\mathbb {C}$ by a conformal mapping and therefore the $C^A_{\lbrace k_j\rbrace }$ are easily computable, at least for the first few leading terms in the expansion. In higher dimensions this is more difficult. Section 3 of this paper will be devoted to the simplest case of a free scalar field theory.

However, supposing that we have computed the $C^A_{\lbrace k_j\rbrace }$, then substituting into (3), (4) and again using orthonormality of the operators

$$\begin{equation*} \frac{Z\big(\mathcal{C}^{(n)}_{A\cup B}\big)}{Z^n}=\frac{Z\big(\mathcal{C}^{(n)}_{A}\big)}{Z^n}\frac{Z\big(\mathcal{C}^{(n)}_{B}\big)}{Z^n}\sum _{\lbrace k_j\rbrace } C^A_{\lbrace k_j\rbrace }C^B_{\lbrace k_j\rbrace } r^{-2\sum _jx_{k_j}} , \end{equation*}$$

so that

$$\begin{equation} \frac{P^{(n)}_{A\cup B}}{P^{(n)}_{A}P^{(n)}_{B}}=\sum _{\lbrace k_j\rbrace } C^A_{\lbrace k_j\rbrace }C^B_{\lbrace k_j\rbrace }\,r^{-2\sum _jx_{k_j}}\,, \end{equation} \tag{ 5 }$$

where r = |r_A − r_B|. Note that in the ratio all terms which contain the UV divergent area law pieces cancel.

If we now arrange the operators $\Phi _{k_j}$ in order of increasing dimension $x_{k_j}$, this gives an expansion in increasing powers of (R_AR_B/r²). The leading term is unity, and comes from taking all the $\Phi _{k_j}$ to be the identity operator $\bf 1$ with x = 0. In d = 1, the leading terms with a single $x_{k_j}>0$ in general vanish because the one-point functions of primary operators in $\mathcal{C}^{(n)}_{A}$ are proportional to those in $\mathbb {C}$ where they vanish. However this is not necessarily the case for d > 1, unless the one-point function vanishes for symmetry reasons. The next contribution comes from taking two of the $x_{k_j}>0$. Note that, unlike the case of d = 1, these do not have to be equal because orthogonality may not hold on $\mathcal{C}^{(n)}_{A}$. However, the leading correction terms will come from the smallest two non-zero x_k, and, barring degeneracies, these will correspond to the same operator.

As an example consider the 2 + 1-dimensional Ising field theory. The leading operators are the magnetization, with x_σ ≈ 0.52, and the energy operator with x ≈ 1.41. The one-point function of the magnetization on $\mathcal{C}^{(n)}_{A}$ vanishes by the ${\mathbb {Z}}_2$ symmetry of the model, but there is no reason for the one-point function of the energy operator to vanish. Since, however, 2x_σ < x, the leading term in the mutual information will be proportional to $(R_AR_B/r^2)^{2x_\sigma }$, with a correction of order $(R_AR_B/r^2)^{x_\epsilon }$. Although the above inequality holds for many interacting CFTs, counterexamples exist in supersymmetric theories [18], in which case the leading term in the will correspond to the one-point function on $\mathcal{C}^{(n)}_{A,B}$ of an operator with the quantum numbers of the vacuum.

A second example is free scalar field theory in all d > 1, to be considered in detail in the following section. (For d = 1 the field itself is not a local operator and the leading corrections then come from exponentials and derivatives of the field [7].) The field ϕ has dimension x = (d − 1)/2 but has vanishing one-point function, once again because of ${\mathbb {Z}}_2$ symmetry under ϕ → −ϕ. However : ϕ²: has dimension d − 1 and a non-zero one-point function (see below). The leading term in the mutual information is then a combination of these two contributions, and goes like (R_AR_B/r²)^{d − 1}. For a massless Dirac field, which has dimension d/2, the power d − 1 in this expression is replaced by d. These results agree with those of [10, 12].

The corrections to this leading behaviour come from larger values of the exponent $\sum _jx_{k_j}$ in (5), and from higher terms on the expansion of the logarithm in (2). For a free scalar field, the first type of correction comes from when either four of the $\Phi _{k_j}$ are taken to be ϕ, or when these are taken in pairs as : ϕ²:. These all give a contribution O((R_AR_B/r²)^{2(d − 1)}). Note that these are more important for smaller values of d. There is also a correction coming from taking one of the $\Phi _{k_j}$ to be the stress tensor, which always has dimension (d + 1), and taking two of them to be the current ∂_μϕ. These both lead to universal corrections O((R_AR_B/r²)^{d + 1}). For d > 3 they dominate the other corrections.

In this section we consider the case of a free massless scalar field. The action is proportional to ∫(∂ϕ)²d^{d + 1}x, and we normalize the field so that its two-point function in ${\mathbb {R}}^{d+1}$ is²

$$\begin{equation*} \langle \phi (x_1)\phi (x_2)\rangle \equiv G_0(x_1-x_2)=|x_1-x_2|^{-(d-1)} . \end{equation*}$$

As discussed in the previous section, we also need : ϕ²:, which may be defined by point-splitting as

$$\begin{equation*} :\!\phi ^2(x)\!:=\lim _{\delta \rightarrow 0}(\phi (x + \case{1}{2} \delta )\phi (x - \case{1}{2} \delta )-G_0(\delta )) . \end{equation*}$$

Its two-point function in ${\mathbb {R}}^{d+1}$, by Wick's theorem, is

$$\begin{equation*} \langle :\!\phi ^2(x_1)\!::\!\phi ^2(x_2)\!:\rangle =2G_0(x_1-x_2)^2 , \end{equation*}$$

so, to conform to our normalization convention, we should consider Φ(x) ≡ 2^−1/2: ϕ²(x):.

We would like to compute the analogous correlation functions in $\mathcal{C}^{(n)}_A$. As in [7], rather than thinking of a single free field on this conifold, we think of n copies ϕ_j on ${\mathbb {R}}^{d+1}$, coupled by the boundary conditions across τ = 0:

$$\begin{eqnarray*} \phi _j(r,0-)&=&\phi _{j+1}(r,0+)\qquad (r\in A) ;\\ &=&\phi _{j}(r,0+)\qquad (r\notin A) . \end{eqnarray*}$$

Thus the coefficients $C^A_{\lbrace k_j\rbrace }$ we need to lowest order are

$$\begin{equation} C^{A}_{jj^{\prime }}\equiv C^A_{0,\ldots ,1,\ldots ,1,\ldots ,0}=\lim _{x_1,x_2\rightarrow \infty }(x_1x_2)^{d-1}\langle \phi _{j}(x_1)\phi _{j^{\prime }}(x_2)\rangle _{\mathcal{C}^{(n)}_A}, \end{equation} \tag{ 6 }$$

where the non-zero entries occur at $j\not=j^{\prime }$, and

$$\begin{equation} C^{A}_{jj}\equiv C^A_{0,\ldots ,2,\dots ,0}=2^{-1/2}\lim _{x\rightarrow \infty } x^{2(d-1)} \big\langle\! :\!\phi _j^2(x)\!:\! \big\rangle _{\mathcal{C}^{(n)}_A}. \end{equation} \tag{ 7 }$$

In the language of d + 1-dimensional electrostatics on $\mathcal{C}^{(n)}_A$, $\langle \phi _{j}(x_1)\phi _{j^{\prime }}(x_2)\rangle$ is the potential at x₁ on copy j due to a unit charge at x₂ on copy j', while $\big\langle\! :\!\!\phi _{j^{\prime }}^2(x)\!\!:\!\big\rangle$ is the excess self-energy of a unit charge at x on copy j'. Note that

$$\begin{equation*} \sum _{j\not=j^{\prime }}\langle \phi _{j}(x_1)\phi _{j^{\prime }}(x_2)\rangle _{\mathcal{C}^{(n)}_A} + \big\langle\!\! :\!\!\phi _{j^{\prime }}^2(x)\!\!:\!\!\big\rangle _{\mathcal{C}^{(n)}_A}=0 , \end{equation*}$$

which follows from conservation of electric flux.

The leading correction in (5) is then

$$\begin{eqnarray} &&\fl r^{-2(d-1)}\Bigg ( \frac{1}{2} \sum _{j\not=j^{\prime }}C^{A}_{jj^{\prime }}C^{B}_{jj^{\prime }}+\sum _jC^{A}_{jj}C^{B}_{jj}\Bigg ) = \frac{1}{2} r^{-2(d-1)}\,n\nonumber \\ &&\times\,\Bigg[\sum _{j=1}^{n-1}\Big(\lim _{x_1\rightarrow \infty }\big(x_1^2\big)^{d-1}\langle \phi _{j}(x_1)\phi _{0}(x_1) \rangle _{\mathcal{C}^{(n)}_A}\Big)\Big(\lim _{x_1\rightarrow \infty }\big(x_1^2\big)^{d-1}\langle \phi _{j}(x_1)\phi _{0}(x_1)\rangle _{\mathcal{C}^{(n)}_B}\Big) \nonumber \\ &&+\, \Big(\lim _{x_1\rightarrow \infty }\big(x_1^2\big)^{d-1}\big\langle\!\! :\!\phi _{0}^2(x_1)\!:\!\! \big\rangle _{\mathcal{C}^{(n)}_A}\Big)\Big(\lim _{x_1\rightarrow \infty }\big(x_1^2\big)^{d-1}\big\langle\!\! :\!\phi _{0}^2(x_1)\!:\!\!\big\rangle _{\mathcal{C}^{(n)}_B}\Big) \Bigg], \end{eqnarray} \tag{ 8 }$$

where we have used the cyclic symmetry to extract an overall factor of n.

3.1. The case n = 2

For n = 2 it is useful to define the linear combinations ϕ_± = 2^−1/2(ϕ₀ ± ϕ₁), which satisfy

$$\begin{eqnarray*} \phi _-(r,0-)&=&-\phi _-(r,0+)\qquad (r\in A) ;\\ &=&+\phi _-(r,0+)\qquad (r\notin A) , \end{eqnarray*}$$

while $\tilde{\phi }_+$ is continuous across τ = 0. Note that

$$\begin{equation*} \langle \phi _\pm (x)\phi _\pm (x_1)\rangle =\langle \phi _0(x)\phi _0(x)\rangle \pm \langle \phi _1(x)\phi _0(x)\rangle , \end{equation*}$$

so that the correlation functions on the left hand side can be interpreted as the potential at x due to a unit charge at x₁. For the upper + sign, the potential is continuous everywhere else, and so is equal to G₀(x − x₁).

For the lower − sign, however, it is constrained to change sign across A∩(τ = 0). We notice that in (6), (7) we may take x₁ → ∞ in any direction. For convenience choose it to lie in the hyperplane τ = 0. Then the potential due to a unit charge at x₁ must be symmetric under reflection τ → −τ. Therefore the potential on A∩(τ = 0) vanishes. Thus, as far as ϕ₋ is concerned, A∩{τ = 0} acts like a conductor, at zero electrostatic potential.

Thus we have the electrostatics problem of finding the potential at x due to a unit charge at x₁, in the presence of a conductor held at zero potential at A∩{τ = 0}. In general this is complicated, but since we are only interested in the far field in the limit when |x₁ − r_A| ≫ R_A, we can make a simple approximation, valid in this limit. Define $\bar{\phi }(x)\equiv \langle \phi _-(x)\phi _-(x_1)\rangle -G_0(x-x_1)$. Then $\bar{\phi }(x)$ is regular at x₁ and takes an approximately constant value −|x₁|^{−(d − 1)} on the conductor. This will induce a total charge −C_A|x₁|^{d − 1} on the conductor, where C_A is its electrostatic capacitance. Therefore, as x, x₁ → ∞, 〈ϕ₋(x)ϕ₋(x₁)〉 − G₀(x − x₁) ∼ −C_A|x|^{−(d − 1)}|x₁|^{−(d − 1)}. Thus

$$\begin{equation} \langle \phi _1(x_1)\phi _0(x_1)\rangle \sim \case{1}{2} {\bf C}_A|x_1|^{-2(d-1)}\,; \end{equation} \tag{ 9 }$$

$$\begin{equation} \big\langle\! :\!\phi _0^2(x_1)\!:\!\big\rangle = \lim _{x\rightarrow x_1}(\langle \phi _0(x)\phi _0(x_1)\rangle -G_0(x-x_1))= - \case{1}{2} {\bf C}_A|x_1|^{-2(d-1)}\,, \end{equation} \tag{ 10 }$$

giving

$$\begin{equation} I^{(2)}(A,B)\sim \frac{{\bf C}_A{\bf C}_B}{2r^{2(d-1)}}\,. \end{equation} \tag{ 11 }$$

This is valid for any compact regions A and B. On dimensional grounds ${\bf C}_{A,B}\propto R_{A,B}^{d-1}$, but the coefficient depends on the shape of the regions. Very few cases are known exactly.

In the appendix we show, generalizing a result of Thomson (Lord Kelvin), that when ∂A is a hypersphere S^{d − 1}of radius R_A , so that A∩{τ = 0} is a disc,

$$\begin{equation} {\bf C}_A=\frac{\Gamma (d/2)\Gamma (1/2)}{\pi \Gamma ((d+1)/2)}\,R_A^{d-1}\,. \end{equation} \tag{ 12 }$$

For d + 1 = 3 this gives Thomson's result $\frac{2}{\pi }R_A$, while for d + 1 = 4 it gives $\frac{1}{2}R_A^2$.

3.2. Free field theory when A and B are spherical, general n

The above symmetry argument does not seem to generalize to larger values of n, but further analytic progress can be made in the case when A and B are the interior of spheres S^{d − 1}. In that case we can exploit the conformal invariance of the free field theory to compute the coefficients $C_{jj^{\prime }}^{A,B}$.

Before doing this, we note that conformal invariance implies in this case that I⁽ⁿ⁾(A, B) is a universal function of the quantity

$$\begin{equation*} \frac{R_AR_B}{r^2-(R_A-R_B)^2} . \end{equation*}$$

Given any two spheres S_{d − 1}: A and B, we may expand them into spheres S_d of the same radii in d + 1 dimensions about their common equatorial plane τ = 0. Conformal transformations in d + 1 dimensions will transform them into other spheres (counting hyperplanes as spheres through the point at infinity.) Since the system has axial symmetry about the line joining their centres, we may restrict to conformal transformations which preserve this line. Under such transformations, the cross-ratio of the points $\big(-\frac{1}{2}r-R_A,-\frac{1}{2}r+R_A,\frac{1}{2}r-R_B, \frac{1}{2}r+R_B\big)$ where the spheres intersect this line is invariant. This is the quantity above, apart from a factor of 4 . Since by (2) I⁽ⁿ⁾(A, B) is given in terms of a ratio of partition functions in which all metrical factors cancel, it should be both scale and conformally invariant.

We now use conformal invariance to compute correlation functions on $\mathcal{C}^{(n)}_A$ when A is a sphere. We may regard A as the intersection of a d-dimensional ball of radius R_A with the equatorial plane τ = 0. Consider the effect of making an inversion in ${\mathbb {R}}^{d+1}$ which sends a point on the boundary of A to the point at infinity. This maps the boundary of the ball S^d into a hyperplane ${\mathbb {R}}^d$ . The plane τ = 0 is preserved by the mapping, and so the boundary of $\mathcal{C}_A$ is mapped into a hyperplane ${\mathbb {R}}^{d-1}$. A∩{τ = 0} itself is mapped into a d-dimensional half-space. This is easier to visualize in d = 2, when A∩{τ = 0} is mapped into a half-plane with an infinite line ${\mathbb {R}}$ as its boundary. In the replicated theory this turns into a line of conical singularities. The conifold $\mathcal{C}_A^{(n)}$ is mapped into ${\mathcal C^{\prime }}_A^{(n)}=\lbrace \mbox{2-dimensional cone of opening angle}\ 2\pi n\rbrace \times {\mathbb {R}}^{d-1}$.

The correlation functions transform covariantly under this conformal mapping:

$$\begin{equation*} \langle \phi _j(x_1)\phi _{j^{\prime }}(x_2)\rangle _{\mathcal{C}_A^{(n)}}=\left|\frac{\partial x_1^{\prime }}{\partial x_1}\right|^{(d-1)/2} \left|\frac{\partial x_2^{\prime }}{\partial x_2}\right|^{(d-1)/2} \langle \phi _j(x^{\prime }_1)\phi _{j^{\prime }}(x^{\prime }_2)\rangle _{{\mathcal C^{\prime }}_A^{(n)}} . \end{equation*}$$

The mapping brings the points at x_{1, 2} = ∞ to a finite distance 1/(2R_A) from the conical singularity. The Jacobian cancels the factors of (x₁x₂)^{d − 1} in (6), (7). Thus

$$\begin{equation*} C^A_{jj^{\prime }}=\langle \phi _j(1/2R_A)\phi _{j^{\prime }}(1/2R_A)\rangle _{{\mathcal C^{\prime }}_A^{(n)}}=(2R_A)^{d-1}\langle \phi _j(1)\phi _{j^{\prime }}(1)\rangle _{{\mathcal C^{\prime }}_A^{(n)}} , \end{equation*}$$

and similarly for $C^A_{jj}$.

Thus we need to compute the potential on the jth copy at unit distance from the hyperplane of conical singularities due to a unit charge in the same position on the j'th copy. As before, because of the cyclic symmetry we can take j' = 0. Since this problem now has axial symmetry the calculation is simplified. One approach is to introduce cylindrical polar coordinates $(\rho ,\theta ,\vec{z})$, where θ ∈ [0, 2πn] and $\vec{z}$ is a (d − 1)-dimensional coordinate in the ${\mathbb {R}}^{d-1}$ subspace. We need the Green's function satisfying

$$\begin{equation*} -\nabla ^2G^{(n)}(\rho ,\theta ,\vec{z})\propto \delta (\rho -1)\delta (\theta )\delta ^{d-1}(\vec{z}) , \end{equation*}$$

with $G^{(n)}(\rho ,\theta +2\pi n,\vec{z}\,)=G^{(n)}(\rho ,\theta ,\vec{z}\,)$. We then have $\langle \phi _j(1)\phi _{0}(1)\rangle _{{\mathcal C^{\prime }}_A^{(n)}}=G^{(n)}(1,2\pi j/n,0)$. An expression for $G(\rho ,\theta ,\vec{z})$ may be found by Fourier transforming with respect to $\vec{z}$ and solving in terms of Bessel functions, but the resultant integrals and sums are ill-conditioned and we have not been able to make the continuation in n.

We adopt a different approach, for which the complete answer for all n may be obtained for all d. Instead of considering n to be initially a positive integer, suppose n = 1/m where m is a positive integer. The solution for 0 ⩽ θ ⩽ 2π/m is then immediate by the method of images:

$$\begin{equation*} G^{(1/m)}(\rho ,\theta ,z)=\sum _{k=0}^{m-1}G_0(\rho ,\theta +2\pi k/m,z) . \end{equation*}$$

Specializing to ρ = 1, z = 0,

$$\begin{equation} G^{(1/m)}(1,\theta ,0)=\sum _{k=0}^{m-1}\frac{1}{(2-2\cos (\theta +2\pi k/m))^{(d-1)/2}}\,. \end{equation} \tag{ 13 }$$

For d − 1 even this sum is straightforward, but more difficult for the odd case, as we illustrate below.

3.2.1. The case d = 3

In this case the sum can be evaluated explicitly in a number of ways. For example, we can regularize it and write it as

$$\begin{equation*} \fl \lim _{\rho \rightarrow 1-}\sum _{k=0}^{m-1}\frac{1}{(1-\rho\, {\rm e}^{{\rm i}(\theta +2\pi k/m)})(1-\rho\, {\rm e}^{-{\rm i}(\theta +2\pi k/m)})} =\lim _{\rho \rightarrow 1-}\sum _{k=0}^{m-1}\sum _{p=0}^\infty \sum _{p^{\prime }=0}^\infty \rho ^{p+p^{\prime }}{\rm e}^{{\rm i}(p-p^{\prime })(\theta +2\pi k/m)} , \end{equation*}$$

where |ρ| < 1. The sum over k vanishes unless p − p' = 0 (mod m), when it gives m. For p ⩾ p' we can write p = p' + lm, and similarly for p' ⩾ p. Subtracting off the l = 0 term to avoid double counting gives

$$\begin{equation*} \fl m\sum _{p^{\prime }=0}^\infty \sum _{l=0}^\infty \rho ^{2p^{\prime }+lm}{\rm e}^{{\rm i}lm\theta }+{\rm c.c.}-m\sum _{p^{\prime }=0}^\infty \rho ^{2p^{\prime }} =\frac{m}{1-\rho ^2}\left(\frac{1}{1-\rho ^m {\rm e}^{{\rm i}m\theta }}+\frac{1}{1-\rho ^m {\rm e}^{-{\rm i}m\theta }}-1\right). \end{equation*}$$

Taking the limit ρ → 1 − then gives the simple result for d = 3

$$\begin{equation*} G^{(1/m)}(1,\theta ,0)=\frac{m^2}{2-2\cos m\theta } . \end{equation*}$$

This valid for m a positive integer. However, since it vanishes exponentially fast as m → ±i∞, Carlson's theorem ensures that it has a well-defined analytic continuation, to other values of m, in particular to m = 1/n. Note however that this does not make sense before performing the sum in (13)!

First, we note that

$$\begin{equation} \big\langle\!\! :\!\phi _0^2(1)\!:\!\!\big\rangle _{{\mathcal C^{\prime }}_A^{(n)}}=\lim _{\theta \rightarrow 0}\left(\frac{1/n^2}{2-2\cos (\theta /n)}- \frac{1}{2-2\cos \theta }\right)=\frac{1-n^2}{12n^2}\,. \end{equation} \tag{ 14 }$$

As we show in section 3.3, this relates to the coefficient of a universal term in the Rényi entropy of a single sphere in the massive theory. Note that this does not contribute to the mutual information through (8), since this term would be O((n − 1)²) and therefore have vanishing derivative at n = 1. A similar remark applies to the one-point function of the stress tensor $\langle T\rangle _{\mathcal{C}_A^{(n)}}$.

The first term in (8) involves the sum

$$\begin{equation*} \sum _{j=1}^{n-1}G^{(n)}(1,2\pi j/n,0)^2=\frac{1}{n^4}\sum _{j=1}^{n-1}\frac{1}{(2-2\cos (2\pi j/n))^2} . \end{equation*}$$

This may be evaluated for n a positive integer by first regulating it as above, and expanding in powers of ρ:

$$\begin{equation*} \sum _{j=1}^{n-1}\sum _{p=0}^\infty \sum _{p^{\prime }=0}^\infty (p+1)(p^{\prime }+1)\rho ^{p+p^{\prime }}{\rm e}^{2\pi {\rm i}(p-p^{\prime })j/n} . \end{equation*}$$

The sum over j now gives −1 unless p − p' = 0 (mod n), when it gives n − 1. In this case, writing p = p' + ln, etc, as before, we get

$$\begin{eqnarray*} &&\fl 2n\sum _{p=0}^\infty \sum _{l=0}^\infty (p+1)(p+1+{\rm ln})\rho ^{2p+{\rm ln}}-n\sum _{p=0}^\infty (p+1)^2\rho ^{2p}-\sum _{p,p^{\prime }=0}^\infty (p+1)(p^{\prime }+1)\rho ^{p+p^{\prime }}\\ &&=\left(\frac{2n}{(1-\rho ^2)^3}-\frac{n}{(1-\rho ^2)^2}\right)\frac{1+\rho ^n}{1-\rho ^n}+\frac{2n^2\rho ^n}{(1-\rho ^2)^2(1-\rho ^n)^2} -\frac{1}{(1-\rho )^4} . \end{eqnarray*}$$

Taking the limit ρ → 1 gives, after some algebra,

$$\begin{equation*} \fl \sum _{j=1}^{n-1}G^{(n)}(1,2\pi j/n,0)^2=\frac{(n^2-1)(n^2+11)}{720n^4} . \end{equation*}$$

Once again Carlson's theorem assures us of a unique continuation to non-integer values of n.

Putting all the pieces together we find, for d = 3

$$\begin{eqnarray*} &&\fl I^{(n)}(A,B) \sim \frac{n}{2(n-1)}r^{-4}\left(\frac{(n^2-1)(n^2+11)}{720n^4}+\frac{(n^2-1)^2}{144n^4}\right)(2R_A)^2 (2R_B)^2\\ &&{\hphantom{\fl I^{(n)}(A,B) }}= \frac{n^4-1}{15n^3(n-1)}\left(\frac{R_AR_B}{r^2}\right)^2\!. \end{eqnarray*}$$

Note that for n = 2 this agrees with (11), using (12). This is a non-trivial check of our methods. Taking the derivative at n = 1 we find for the mutual information the leading term

$$\begin{equation*} I(A,B)\sim \frac{4}{15}\left(\frac{R_AR_B}{r^2}\right)^2 . \end{equation*}$$

Clearly a similar calculation can be done for any even d − 1. The result will always be a rational function of ρ and ρⁿ, which will give, on taking the limit ρ → 1, a polynomial in n.

3.2.2. The case d = 2

In this case the sum in (13) is

$$\begin{equation*} G^{(1/m)}(1,\theta ,0)=\sum _{k=0}^{m-1}\frac{1}{(2-2\cos (\theta +2\pi k/m))^{1/2}}= \sum _{k=0}^{m-1}\frac{1}{2\sin \left(\frac{\theta }{2}+\frac{\pi k}{m}\right)} . \end{equation*}$$

(Note that in the physical region 0 < θ < 2π/m the sine is always positive.) Now use the integral representation

$$\begin{equation*} \frac{1}{\sin \pi \mu }=\frac{1}{\pi }\int _0^\infty \frac{x^{\mu -1}}{1+x}{\rm d}x\qquad (0<\mu <1) . \end{equation*}$$

Inserting this and performing the sum gives

$$\begin{equation} G^{(1/m)}(1,\theta ,0)=\frac{1}{2\pi }\int _0^\infty \frac{x^{(\theta /2\pi )-1}(1-x)}{(1+x)(1-x^{1/m})}{\rm d}x\,. \end{equation} \tag{ 15 }$$

The analyticity properties of this expression in 1/m may be inferred by dividing the integration region into (0, 1) and (1, ∞) and expanding in powers of x (1/x) in each case. This gives

$$\begin{eqnarray*} &&\fl \sum _{p=0}^\infty \sum _{q=0}^\infty (-1)^p\Bigg[\frac{1}{\big(\frac{\theta }{2\pi }+p+\frac{q}{m}\big)\big(\frac{\theta }{2\pi }+p+\frac{q}{m}+1\big)} \nonumber\\ &&+\frac{1}{\big(-\frac{\theta }{2\pi }+p+\frac{q+1}{m}\big)\big(-\frac{\theta }{2\pi }+p+\frac{q+1}{m}+1\big)}\Bigg] . \end{eqnarray*}$$

In the physical region for θ this has an accumulation of poles for negative real m, but is otherwise analytic. Moreover it grows like |m| as m → ∞ except along the negative real axis. We infer from this that it has a unique continuation to the whole m plane apart from the negative real axis, in particular to m = 1/n where n ⩾ 1, which is found by simply setting m = 1/n in (15). We remark in passing that as |n| → ∞ the sum is O(n⁻²log n), consistent with there being a branch cut along the negative real axis.

We then see that

$$\begin{equation*} \fl \langle :\!\phi _0(1)^2\!:\rangle _{{\mathcal C^{\prime }}_A^{(n)}}=\lim _{\theta \rightarrow 0}\frac{1}{2\pi }\int _0^\infty \frac{x^{(\theta /2\pi )-1}}{1+x}\left( \frac{1-x}{1-x^n}-1\right){\rm d}x =-\frac{1}{2\pi }\int _0^\infty \frac{1-x^{n-1}}{(1+x)(1-x^n)}{\rm d}x . \end{equation*}$$

By substituting x → x⁻¹, we see that integral is twice its value with an upper limit of 1. For n = 2 we get −(1/2π), consistent with (9), (10), (12). As n → 1

$$\begin{equation*} \fl \langle :\!\phi _0(1)^2\!:\rangle _{{\mathcal C^{\prime }}_A^{(n)}}\sim \frac{(n-1)}{\pi }\int _0^1\frac{\log x}{1-x^2} =-\frac{(n-1)}{\pi }\sum _{p=0}^\infty \frac{1}{(2p+1)^2}=-\frac{\pi }{8}(n-1) . \end{equation*}$$

The sum in the first term in (8) is

$$\begin{eqnarray*} &&\fl \sum _{j=1}^{n-1}G^{(n)}(1,2\pi j,0)^2 = \frac{1}{4\pi ^2}\sum _{j=1}^{n-1}\int _0^\infty \!\!\!\int _0^\infty \frac{(xy)^{j-1}(1-x)(1-y)}{(1+x)(1+y)(1-x^n)(1-y^n)}{\rm d}x\, {\rm d}y\\ &&{\hphantom{\fl \sum _{j=1}^{n-1}G^{(n)}(1,2\pi j,0)^2 }}= \frac{1}{4\pi ^2}\int _0^\infty\!\!\! \int _0^\infty \frac{(1-x)(1-y)(1-(xy)^{n-1})}{(1+x)(1+y)(1-xy)(1-x^n)(1-y^n)}{\rm d}x\, {\rm d}y . \end{eqnarray*}$$

For n = 2 this is 1/4π², consistent with (9), (10), (12). As n → 1 we find

$$\begin{equation*} -\frac{n-1}{4\pi ^2}\int _0^\infty\!\!\! \int _0^\infty \frac{\log (xy)}{(1+x)(1+y)(1-xy)}{\rm d}x\, {\rm d}y . \end{equation*}$$

This integral may be done by letting y = u/x and first carrying out the x-integral, yielding

$$\begin{equation*} \frac{n-1}{4\pi ^2}\int _0^\infty \frac{(\log u)^2}{(1-u)^2}{\rm d}u= \frac{1}{6} (n-1) . \end{equation*}$$

Putting together the pieces, we then find for the leading term in the mutual information for d = 2

$$\begin{equation*} I(A,B)\sim \frac{1}{2} \cdot \frac{1}{6} r^{-2}(2R_A)(2R_B)=\frac{1}{3}\left(\frac{R_AR_B}{r^2}\right) . \end{equation*}$$

3.3. Universal correction for a single spherical region

Although the main purpose of this section is to study the mutual information of two spherical regions, it is worth noting that the result in (14) may be used to find a universal logarithmic correction to the area law for a single spherical region in d = 3.

Suppose A is the interior of a sphere S^{d − 1} of radius R centred at the origin. For the time being we keep d general. Let $\vec{R}=(R,0,\ldots )$ in the plane τ = 0, and invert $x=(r_1,\vec{r}_\perp ,\tau )$ in d + 1 dimensions, sending the point $-\vec{R}$ to infinity:

$$\begin{eqnarray*} r_1^{\prime }&=&\frac{r_1+R}{(r_1+R)^2+\vec{r}_\perp {\!}^2+\tau ^2}-\frac{1}{2R} ,\\ \vec{r}{\,}^{\prime }_\perp &=&\frac{\vec{r}_\perp }{(r_1+R)^2+\vec{r}_\perp {\!}^2+\tau ^2} .\\ \tau ^{\prime }&=&\frac{\tau }{(r_1+R)^2+\vec{r}_\perp {\!}^2+\tau ^2} . \end{eqnarray*}$$

As before, this maps the conifold $\mathcal{C}^{(n)}_A$ into ${\mathcal C^{\prime }}^{(n)}_A=\lbrace \mbox{cone in$(r_1^{\prime },\tau ^{\prime })$}\rbrace \times {\mathbb {R}}_\perp ^{d-1}$, and

$$\begin{equation*} \langle :\!\phi ^2(x)\!:\rangle _{\mathcal{C}^{(n)}_A}=\left|\frac{\partial x^{\prime }}{\partial x}\right|^{d-1} \langle :\!\phi ^2( x^{\prime })\!:\rangle _{{\mathcal C^{\prime }}^{(n)}_A} . \end{equation*}$$

The Jacobian pre-factor is

$$\begin{equation*} (x^2)^{-(d-1)}= [(r_1+R)^2+\vec{r}_\perp {}^2+\tau ^2]^{-(d-1)} , \end{equation*}$$

and

$$\begin{equation*} \big\langle\!\! :\!\phi ^2_j(x^{\prime })\!:\!\!\big\rangle _{{\mathcal C^{\prime }}^{(n)}_A}=a_n \big({r^{\prime }_1}^2+{\tau ^{\prime }}^2\big)^{-(d-1)/2} \end{equation*}$$

where, from (14),

$$\begin{equation*} a_n=\big\langle\!\! :\!\phi ^2_j(1)\!:\!\!\big\rangle _{{\mathcal C^{\prime }}^{(n)}_A}=(1-n^2)/(12n^2) . \end{equation*}$$

After some algebra we find

$$\begin{equation} \big\langle \!\!:\!\phi ^2_j(x)\!:\!\!\big\rangle _{\mathcal{C}^{(n)}_A} =a_n\left(\frac{4R^2}{[r^2+\tau ^2-R^2]^2+4R^2\tau ^2}\right)^{(d-1)/2}\,. \end{equation} \tag{ 16 }$$

Note that the final answer depends only on $r^2=x_1^2+x_\perp ^2$ and τ, as it should. At large |x| it decays as (x²)^{−(d − 1)} so is integrable at infinity if d > 3. On the other hand, close to the conical singularity (r ∼ R, τ ∼ 0), writing r = R + u with |u| ≪ R, it behaves as (u² + τ²)^{−(d − 1)/2}, so is integrable if d < 3.

Therefore for d = 3 the integral of $\langle :\!\phi ^2_j(x)\!:\rangle _{\mathcal{C}^{(n)}_A}$ over the conifold is logarithmically divergent in the massless theory both in the UV and the IR. The former divergence necessitates a short-distance cut-off Λ⁻¹. The IR divergence may be regulated by assuming the theory has a small mass m ≪ R⁻¹. However, since both divergences are logarithmic, to leading order we may still use the expression (16). This gives from large distances

$$\begin{equation*} a_n\int ^{1/m}\frac{4R^2}{x^4}{\rm d}^4x\sim a_n\cdot (4R^2)(2\pi ^2)\log (1/m) , \end{equation*}$$

and from short distances

$$\begin{equation*} a_n(4\pi R^2)\int _{1/\Lambda }\frac{{\rm d}u\ {\rm d}\tau }{u^2+\tau ^2}\sim a_n(4\pi R^2)(2\pi )\log \Lambda . \end{equation*}$$

Happily the two coefficients agree, so we get

$$\begin{equation*} \int \langle :\!\phi ^2(\vec{x})\!:\rangle _{\mathcal{C}^{(n)}_A}{\rm d}^3r{\rm d}\tau \sim na_n(8\pi ^2R^2)\log (\Lambda /m) . \end{equation*}$$

The extra factor of n comes from summing over j.

The action for the massive theory is

$$\begin{equation*} S=\frac{1}{8\pi ^2}\int ((\partial \phi )^2+m^2\phi ^2)\,{\rm d}^4x , \end{equation*}$$

(note the factor of (4π²)⁻¹ inserted to conform with our field normalization convention so that the propagator is 1/x² when m = 0.)

Then

$$\begin{equation*} (\partial /\partial m^2)\log \big (Z\big(\mathcal{C}^{(n)}_A\big)/Z^n\big )=-(8\pi ^2)^{-1}\int \langle :\!\phi ^2(x)\!:\rangle _{\mathcal{C}^{(n)}_A} {\rm d}^4x , \end{equation*}$$

which gives a universal logarithmic correction term to the area law for mR ≪ 1:

$$\begin{equation*} \Delta S_A^{(n)}=-\frac{n+1}{12n} R^2m^2\log (\Lambda /m) . \end{equation*}$$

Since the dependence on Λ comes from the region near the conical singularity, we may conjecture that this is a special case of

$$\begin{equation*} \Delta S_A^{(n)}=-\frac{n+1}{12n} \frac{\mbox{Area}(\partial A)}{4\pi } m^2\log (\Lambda /m) , \end{equation*}$$

valid for any region A with a smooth boundary ∂A. This agrees with a calculation in [6] for the case when A is a half-space (see also [10]) based on the 1 + 1-dimensional result:

$$\begin{equation*} S_A^{(n)}\sim \frac{n+1}{12n}\mbox{Area}(\partial A)\int \log (k^2+m^2)^{1/2} \frac{{\rm d}^2k}{(2\pi )^2} , \end{equation*}$$

after differentiation with respect to m².

We emphasize that this universal logarithmic term is present only in the massive theory, and is not the same as the term O(log (ΛR)) discussed in [4] (see also [10]). This can be obtained using our approach by computing the one-point function of the stress tensor on the conifold $\mathcal{C}_A^{(n)}$ [19]. Although we have carried out our calculation for d = 3, it may be generalized to any even d + 1. For d + 1 odd, logarithms of the type we have discussed are absent.

We also point out that in lattice regularizations we also expect 'unusual' corrections coming from (possibly relevant) operators, which do not break the symmetry, localized on the conical singularity [20]. In 1 + 1 dimensions these have scaling dimensions (x/n) and lead to corrections O(r^−2x/n) in the mutual Rényi entropies, where x is the usual bulk scaling dimension of these operators. For a general CFT in higher dimensions, the n-dependence is more complicated. However for a free field theory the 1 + 1-dimensional result should still hold, and the : ϕ²: operator on the conical singularity has dimension (d − 1)/n. This will lead to corrections O(r^{−2(d − 1)/n}) in the mutual Rényi entropies, which, if present, actually dominate the universal terms we have found for n > 1. However the amplitude is O((n − 1)²) and therefore they do not contribute to the mutual information.

We have presented analytic calculations of the mutual Rényi information I⁽ⁿ⁾(A, B) of two disjoint regions in the ground state of a conformal field theory in general space dimension d. They are given as an expansion in powers of the ratio R_AR_B/r² with powers which are integer combinations of the scaling dimensions of the local operators in the theory, with universal coefficients, which factorize between A and B.

Unlike the case of d = 1 we are able to obtain explicit results only for a free scalar field theory, although similar methods should work for other free theories, and even interacting theories using the -expansion. For n = 2 we showed that the coefficient of the leading term is proportional to the product of the capacitances of A and B in d + 1-dimensional electrostatics. For spheres we computed this by generalizing an argument of W Thomson to general d.

When A and B are spheres we can obtain explicit results for the coefficient of the leading term for all n and d, although the computations increase in complexity with d. For d + 1 even the results are polynomials in n, while for d + 1 odd they may be expressed as Dirichlet series. We remark that from a technical point of view it was simpler first to compute the Rényi entropies for n = 1/m where m is a positive integer, and then perform the analytic continuation. This approach may be more generally useful.

It would be interesting to compare our results with those predicted by the holographic approach of Ryu and Takayangi [21], which were hypothesized to extend to the case of more than one region in [22] (however, see [23]). We note that our result for n = 2 that the mutual Rényi information is given by the product of the capacitances suggests that it cannot be interpreted as an extensive quantity.

Although we have considered only the massless case for the mutual information, our results extend straightforwardly to QFTs with a mass scale m, as long as R_A, R_B ≪ m⁻¹, by replacing massless propagator r^{−(d − 1)} by its massive version. We also confirmed the existence of a universal correction O(R²m²log m) for the Rényi entropies of a single spherical region in the massive theory with mR ≪ 1.

I would like to thank Noburo Shiba for sending me an early version of his paper and for further correspondence. I thank Pasquale Calabrese for his comments on a draft of the present paper, and Mark Srednicki and Erik Tonni for discussions. I also thank Slava Rychkov for drawing my attention to the result in [18].

Consider electrostatics in D = d + 1 dimensions. What is the capacitance of a hollow ellipsoidal shell with semi-axes a₁, ..., a_D? In the limit a_D → 0 this will give the formula for a flat disc in the shape of a d-dimensional ellipsoid. We need to know what charge distribution produces zero field inside. We generalize an argument due to W Thomson [24] (see [25] for a modern account.)

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Start from Newton's observation [26] that for a sphere, when all the a_j are equal, the field vanishes when the charge distribution is uniform³. This may be understood as follows: consider a point P inside the sphere, and a bi-cone of small solid angle dΩ which intersects the sphere in two regions of areas proportional to |IH|^d and |KL|^d shown in figure A1. The contribution to the electric field at P from these two regions is proportional to

$$\begin{equation*} \frac{|IH|^d}{|{PH}|^d}-\frac{|KL|^d}{|PK|^d} \end{equation*}$$

directed along HP. But this vanishes by similarity of the two triangles.

Note that this extends to the case of any spherically symmetric charge distribution. In particular, we can consider a uniform charge distribution between two shells of radii a and a + δa.

Now consider the effect of making a uniform shear transformation $x_j\rightarrow x_j^{\prime }=\lambda _jx_j$ where ∏_jλ_j = 1. This distorts the two spheres into similar ellipsoids with axes λ_ja and λ_j(a + δa). Gauss' law remains true as long as we rescale the electric field components E_j → λ_jE_j, so the electric field inside the ellipsoid still vanishes. The bulk charge density between the ellipsoids remains uniform.

Now take the limit δa → 0. We need to work out the thickness of the shell at a point $\lbrace x_j^{\prime }\rbrace$ on the inner shell, which will be proportional to the surface charge density. This came from a point $\lbrace x_j^{\prime }/\lambda _j\rbrace$ on the sphere, and hence its image on the outer shell is $x_j^{\prime }(1+\delta a/a)$. To get the thickness we need to form the inner product of x_j(δa/a) with the normal at the point $x_j^{\prime }$ . The equation of the ellipsoid is

$$\begin{equation*} \sum _j\big(x_j^2/\lambda _j^2\big)=a^2 , \end{equation*}$$

so the normal vector satisfies ∑_jn_jdx_j = 0 where dx_j is any vector satisfying $\sum _jx_j{\rm d}x_j/\lambda _j^2=0$. The solution is to take $n_j\propto x_j/\lambda _j^2$, and so the thickness at x_j is proportional to

$$\begin{equation*} x_jn_j=\frac{\sum _j\big(x_j^2/\lambda _j^2\big)}{\big(\sum _j\big(x_j^2/\lambda _j^4\big)\big)^{1/2}}\propto \bigg(\sum _j\big(x_j^2/\lambda _j^4\big)\bigg)^{-1/2} . \end{equation*}$$

This is the surface charge density required to ensure that the field within an ellipsoidal shell vanishes.

In our case we need to take the limit λ_D → ∞. Rescale x_D = bλ_D where |b| < 1. Then the ellipsoid degenerates into

$$\begin{equation*} \sum _{j=1}^d\frac{x_j^2}{a_j^2}=1-b^2<1 , \end{equation*}$$

and the charge density is

$$\begin{equation*} \propto \frac{\lambda _D^2}{x_D}\propto \frac{1}{|b|}=\frac{1}{\sqrt{1-\sum _{j=1}^d\frac{x_j^2}{a_j^2}}} . \end{equation*}$$

For the case of a spherical disc when all the a_j = a we get $\sigma \propto 1/\sqrt{a^2-r^2}$ for any d. In this case the total charge is

$$\begin{equation*} Q\propto \int _0^a\frac{r^{d-1}}{\sqrt{a^2-r^2}}{\rm d}r=a^{d-1}\int _0^{\pi /2}(\sin \theta )^{d-1}{\rm d}\theta =a^{d-1}\frac{\Gamma (d/2)\Gamma (1/2)}{2\Gamma ((d+1)/2)} . \end{equation*}$$

On the other hand the potential is (with the same constant of proportionality)

$$\begin{equation*} V=\int _0^a\frac{r^{d-1}}{r^{d-1}\sqrt{a^2-r^2}}{\rm d}r=\int _0^{\pi /2}{\rm d}\theta =(\pi /2) , \end{equation*}$$

so

$$\begin{equation*} Q=\frac{\Gamma (d/2)\Gamma (1/2)}{\pi \Gamma ((d+1)/2)}a^{d-1} V={\bf C}V . \end{equation*}$$

For D = 3 this reduces to Thomson's result C = (2/π)a.