Some Results on Mutual Information of Disjoint Regions in Higher Dimensions
Rudolf Peierls Centre for Theoretical Physics
1 Keble Road, Oxford OX1 3NP, UK
All Souls College, Oxford
We consider the mutual Rényi information of disjoint compact spatial regions and in the ground state of a +1-dimensional conformal field theory (CFT), in the limit when the separation between and is much greater than their sizes . We show that in general , where is the smallest sum of the scaling dimensions of operators whose product has the quantum numbers of the vacuum, and the constants depend only on the shape of the regions and universal data of the CFT.
For a free massless scalar field, where , we show that is proportional to the capacitance of a thin conducting slab in the shape of in +1-dimensional electrostatics, and give explicit formulae for this when is the interior of a sphere or an ellipsoid. For spherical regions in and we obtain explicit results for for all and hence for the leading term in the mutual information by taking . We also compute a universal logarithmic correction to the area law for the Rényi entropies of a single spherical region for a scalar field theory with a small mass.
Since the pioneering paper of Srednicki  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 of the reduced density matrix describing the degrees of freedom inside a spherical region , which measures the entanglement of the degrees of freedom in 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 , and this prompted comparisons with black hole physics.
However in 1993 Holzhey et al.  showed that in a conformal field theory (CFT) in the entanglement entropy of an interval of length goes like with a universal coefficient proportional to the central charge . (Similar logarithms are now understood to occur whenever is even, but for higher these are non-leading with respect to the area term .) 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 . A more complete analysis of entanglement in 1+1-dimensional CFTs was given in Refs. .
More recently  these methods, which involve using the so-called replica trick of computing as limit as of the Rényi entropies
have been extended to the computation of the entanglement entropy between two disjoint intervals 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 of the difference of Rényi entropies
has an expansion of the form
Here measure the lengths of the two intervals, and is the separation between their centres. The sum is over a set of scaling operators of the CFT, labelled by , with scaling dimension , for each replica . The coefficients are universal and encode information about the correlation functions of these operators in the plane. For the leading term in (1) comes from the case when only two of the are non-vanishing and correspond to the lowest dimension operator in the CFT. It was possible  to continue this result analytically in to compute the leading term in the mutual information .
The one-dimensional case has also been studied numerically in a number of papers .
The arguments of Ref.  were based on a kind of operator product expansion first used by Headrick . 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 and are spheres of radii and . In principle our methods work in any (integer) number of dimensions, but the results are most simply expressed in and 3.
In general we find
where, in 3+1 dimensions,
For the mutual information () this gives , to be compared with a numerical result due to Shiba .
In 2+1 dimensions
which leads to , to be compared with the numerical result .
We are also able to compute the form of the corrections to the leading term, which should be and also . In principle the coefficients are calculable. Note that these are more important in than in 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 the actual expansion parameter is
In this form, our results also apply to the mutual entanglement between the interior of a sphere of radius and the exterior of a concentric sphere of radius , in the limit when .
For and general shapes we show that the coefficients are given by the capacitance of -dimensional bodies in -dimensional electrostatics.
The mutual information for a free scalar field in higher dimensions has been studied in only a few papers. Casini and Huerta  and Shiba  showed that it should decay as at large separations (and for free fermions ). This was based on the expression for the density matrix in terms of the correlation functions  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  for the case of equal spheres in and space dimensions, and also rings and shells  as well as some other shapes . 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 and .
The layout of this paper is as follows. In Sec. 2 we first recall the expressions for the various Rényi entropies as path integrals over copies of sewn together in a particular way along the boundary of the selected region to form the conifolds . We then consider the general case of the small expansion, and show how the coefficients in this expansion are related to one- and two-point functions on .
The rest of the paper is devoted to the special case of a massless free scalar field theory. In Sec. 3.1 we show that for the coefficients are related to the electrostatic capacitance of each region. In an Appendix we derive explicit formulas for the case where and are spherical, or more generally ellipsoidal, by generalising a famous method due to W. Thomson to general dimension . In Sec. 3.2 we obtain results for the spherical case for general and , using conformal invariance. We show that for odd the coefficients are polynomials in which can then be continued to to find the von Neumann entropy. For even the result is not a polynomial but explicit results may nevertheless be obtained for integer as well as the limit . In Sec. 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 .
2 General form of the expansion for .
Consider a +1-dimensional quantum field theory in a domain , in its ground state . In this paper we consider to be , although some of the results may be adapted to semi-infinite and finite domains and also to finite temperature.
If is some subdomain, we suppose that, in the presence of a suitable UV cut-off, the Hilbert space can be decomposed as . The Rényi entropies are , where
As explained in Ref. , these can be expressed in terms of a path integral on a particular conifold as follows. The ground state wave-functional is given by a path integral in imaginary time on the half-space from to , with the fields constrained to take the values on . Similarly is given by the path integral on from to . Each of these should be normalised by , where is the partition function on . [Strictly speaking we should restrict the integration to before taking the ratio, then let , and similarly with the thermodynamic limit in space.]
To get we take copies of labelled by , and sew to (mod ) along for , while we sew to for . This gives a conifold with a 1-dimensional submanifold of conical singularities along the boundary . Then
In general, for , has a leading term going like where is an ultraviolet cut-off, and the dimensionless coefficients are non-universal. This gives a term in the Rényi entropies proportional to – the famous ‘area law’ term, which is non-universal and therefore theoretically less interesting. However, in the case where consists of two disjoint compact regions the quantities
are free of these non-universal contributions, and should depend only on the geometry of and and the universal data of the renormalised QFT.
In general, however computing is difficult, and, even for , we have explicit results only for free field theories [7, 10]. However in the limit when the linear sizes of and are much smaller than their separation , it was shown in Ref.  that an expansion in increasing (in general, fractional) powers of is possible with coefficients which are calculable in principle. We now generalise this argument to dimensions .
The basic idea is that, from the point of view of an observer far from or , the sewing together of the copies along the boundaries of and should be expressible as a weighted sum of products of local operators at some conventionally chosen points inside and , where is in the algebra of local operators of the QFT defined on th copy of . That is, the sewing operation in each region can be thought of as a semi-local operator which couples together the QFTs, but can itself be expressed as a sum of a product of local operators in a direct product of the QFTs.111The idea of writing correlators of non-local objects in terms of those of local fields has been applied in the past to Wilson loops, see . We write
and similarly for . Here label a complete set of operators on the th copy. The prefactor on the rhs is inserted since we expect the leading behaviour as to come from the term when all the are the identity operator, and in this limit .
The main point now is that the coefficients 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 but we consider the correlation functions of an arbitrary set of local operators at points on outside :
Note that the rhs decomposes into a sum of products of 2-point functions on each copy of , that is we may take .
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 2-point functions are orthonormal for all separations:
where is the scaling dimension of and we have assumed scalar operators for simplicity.
Thus we find, taking the limit (that is, infinity on the the th copy of )
Note that by dimensional analysis.
In 1+1 dimensions, when is an interval of finite length, may be uniformized to by a conformal mapping and therefore the are easily computable, at least for the first few leading terms in the expansion. In higher dimensions this is more difficult. Sec. 3 of this paper will be devoted to the simplest case of a free scalar field theory.
where . Note that in the ratio all terms which contain the UV divergent area law pieces cancel.
If we now arrange the operators in order of increasing dimension , this gives an expansion in increasing powers of . The leading term is unity, and comes from taking all the to be the identity operator with . In , the leading terms with a single in general vanish because the one-point functions of primary operators in are proportional to those in where they vanish. However this is not necessarily the case for , unless the one-point function vanishes for symmetry reasons. The next contribution comes from taking two of the . Note that, unlike the case of , these do not have to be equal because orthogonality may not hold on . However, the leading correction terms will come from the smallest two non-zero , 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 magnetisation, with , and the energy operator with . The one-point function of the magnetisation on vanishes by the symmetry of the model, but there is no reason for the one-point function of the energy operator to vanish. Since, however, , the leading term in the mutual information will be proportional to , with a correction of order . Although the above inequality holds for many interacting CFTs, counterexamples exist in supersymmetric theories , in which case the leading term in the will correspond to the one-point function on of an operator with the quantum numbers of the vacuum.
A second example is free scalar field theory in all , to be considered in detail in the following section. (For the field itself is not a local operator and the leading corrections then come from exponentials and derivatives of the field .) The field has dimension but has vanishing one-point function, once again because of symmetry under . However has dimension 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 . For a massless Dirac field, which has dimension , the power in this expression is replaced by . These results agree with those of Refs. [10, 12].
The corrections to this leading behaviour come from larger values of the exponent 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 are taken to be , or when these are taken in pairs as . These all give a contribution . Note that these are more important for smaller values of . There is also a correction coming from taking one of the to be the stress tensor, which always has dimension , and taking 2 of them to be the current . These both lead to universal corrections . For they dominate the other corrections.
3 Free scalar field theory
In this section we consider the case of a free massless scalar field. The action is proportional to , and we normalise the field so that its 2-point function in is222We use to denote points in and points in .
As discussed in the previous section, we also need , which may be defined by point-splitting as
Its 2-point function in , by Wick’s theorem, is
so, to conform to our normalisation convention, we should consider .
We would like to compute the analogous correlation functions in . As in Ref. , rather than thinking of a single free field on this conifold, we think of copies on , coupled by the boundary conditions across :
Thus the coefficients we need to lowest order are
where the non-zero entries occur at , and
In the language of +1-dimensional electrostatics on , is the potential at on copy due to a unit charge at on copy , while is the excess self-energy of a unit charge at on copy . Note that
which follows from conservation of electric flux.
The leading correction in (5) is then
where we have used the cyclic symmetry to extract an overall factor of .
3.1 The case
For it is useful to define the linear combinations , which satisfy
while is continuous across . Note that
so that the correlation functions on the left hand side can be interpreted as the potential at due to a unit charge at . For the upper + sign, the potential is continuous everywhere else, and so is equal to .
For the lower sign, however, it is constrained to change sign across . We notice that in (6,7) we may take in any direction. For convenience choose it to lie in the hyperplane . Then the potential due to a unit charge at must be symmetric under reflection . Therefore the potential on vanishes. Thus, as far as is concerned, acts like a conductor, at zero electrostatic potential.
Thus we have the electrostatics problem of finding the potential at due to a unit charge at , in the presence of a conductor held at zero potential at . In general this is complicated, but since we are only interested in the far field in the limit when , we can make a simple approximation, valid in this limit. Define . Then is regular at and takes an approximately constant value on the conductor. This will induce a total charge on the conductor, where is its electrostatic capacitance. Therefore, as , . Thus
This is valid for any compact regions and . On dimensional grounds , but the coefficient depends on the shape of the regions. Very few cases are known exactly.
In the Appendix we show, generalising a result of W. Thomson (Lord Kelvin), that when is a hypersphere of radius , so that is a disc,
For this gives Thomson’s result , while for it gives .
3.2 Free field theory when and are spherical, general .
The above symmetry argument does not seem to generalise to larger values of , but further analytic progress can be made in the case when and are the interior of spheres . In that case we can exploit the conformal invariance of the free field theory to compute the coefficients .
Before doing this, we note that conformal invariance implies in this case that is a universal function of the quantity
Given any two spheres : and , we may expand them into spheres of the same radii in dimensions about their common equatorial plane . Conformal transformations in 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 where the spheres intersect this line is invariant. This is the quantity above, apart from a factor of 4 . Since by (2) 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 when is a sphere. We may regard as the intersection of a -dimensional ball of radius with the equatorial plane . Consider the effect of making an inversion in which sends a point on the boundary of to the point at infinity. This maps the boundary of the ball into a hyperplane . The plane is preserved by the mapping, and so the boundary of is mapped into a hyperplane . itself is mapped into a -dimensional half-space. This is easier to visualise in , when is mapped into a half-plane with an infinite line as its boundary. In the replicated theory this turns into a line of conical singularities. The conifold is mapped into .
The correlation functions transform covariantly under this conformal mapping:
and similarly for .
Thus we need to compute the potential on the th copy at unit distance from the hyperplane of conical singularities due to a unit charge in the same position on the th copy. As before, because of the cyclic symmetry we can take . Since this problem now has axial symmetry the calculation is simplified. One approach is to introduce cylindrical polar coordinates , where and is a -dimensional coordinate in the subspace. We need the Green’s function satisfying
with . We then have . An expression for may be found by Fourier transforming with respect to 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 .
We adopt a different approach, for which the complete answer for all may be obtained for all . Instead of considering to be initially a positive integer, suppose where is a positive integer. The solution for is then immediate by the method of images:
Specialising to , ,
For even this sum is straightforward, but more difficult for the odd case, as we illustrate below.
3.2.1 The case
In this case the sum can be evaluated explicitly in a number of ways. For example, we can regularise it and write it as
where . The sum over vanishes unless (mod ), when it gives . For we can write , and similarly for . Subtracting off the term to avoid double counting gives
Taking the limit then gives the simple result for
This valid for a positive integer. However, since it vanishes exponentially fast as , Carlson’s theorem ensures that it has a well-defined analytic continuation, to other values of , in particular to . Note however that this does not make sense before performing the sum in (13)!
First, we note that
As we show in Sec. 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 and therefore have vanishing derivative at . A similar remark applies to the 1-point function of the stress tensor .
The first term in (8) involves the sum
This may be evaluated for a positive integer by first regulating it as above, and expanding in powers of :
The sum over now gives unless (mod ), when it gives . In this case, writing , etc., as before, we get
Taking the limit gives, after some algebra,
Once again Carlson’s theorem assures us of a unique continuation to non-integer values of .
Putting all the pieces together we find, for