We study universal features in the shape dependence of entanglement entropy in the vacuum state of a conformal field theory (CFT) on . We consider the entanglement entropy across a deformed planar or spherical entangling surface in terms of a perturbative expansion in the infinitesimal shape deformation. In particular, we focus on the second order term in this expansion, known as the entanglement density. This quantity is known to be non-positive by the strong-subadditivity property. We show from a purely field theory calculation that the non-local part of the entanglement density in any CFT is universal, and proportional to the coefficient appearing in the two-point function of stress tensors in that CFT. As applications of our result, we prove the conjectured universality of the corner term coefficient in CFTs, and the holographic Mezei formula for entanglement entropy across deformed spheres.
Thomas Faulkner, Robert G. Leigh and Onkar Parrikar
Department of Physics, University of Illinois, 1110 W. Green St., Urbana IL 61801-3080, U.S.A.
Entanglement entropy has a central position in the study of quantum field theories. It is a powerful tool to probe the structure of quantum states, primarily because: (i) it is sufficiently non-local to capture certain global properties, and (ii) it is geometric by definition and hence universal in its applicability. As a result, entanglement entropy has provided great insights in a wide class of systems such as relativistic field theories [1, 2, 3], conformal field theories (CFTs) [4, 5, 6], topologically ordered phases of matter [7, 8, 9, 10], strongly-coupled theories with holographic duals [11, 12, 13], etc. It has also become clear that entanglement will play a crucial role in understanding the emergence of geometry in the AdS/CFT correspondence [14, 15]. Despite this, computing entanglement entropy for arbitrary shaped regions in general dimension still remains a non-trivial task, especially outside the arena of quantum field theories with classical gravitational duals. While much progress can be made in special symmetric cases such as the entanglement entropy across planar surfaces in relativistic quantum field theories, spherical surfaces in CFTs, etc., it is desirable to develop a larger theoretical toolkit.
In this paper, we study entanglement entropy for deformed half-spaces and ball-shaped regions in the vacuum state of a conformal field theory on . To be concrete, we first explain our construction for deformed half-spaces (see Fig. 1). Let us pick global coordinates on , where is the time coordinate. Pick the Cauchy surface , and consider the reduced density matrix on the half space given by
The entanglement entropy between and its complement is defined as the Von Neumann entropy of . Next, deform the region slightly to
where is a smooth function of the transverse spatial coordinates (parametrizing the entangling surface) which we denote collectively as , and is a positive infinitesimal parameter. This corresponds to deforming the entangling surface to within the original Cauchy surface. We can also generalize this and deform the entangling surface by the infinitesimal vector field , which we take to lie in the plane perpendicular to the original surface (i.e., the overlined indices run over ), and which also includes, for instance, time-like deformations. The entanglement entropy across the deformed surface can be written as a perturbative expansion in
The quantity is known as the entanglement density [16, 17, 18]111Sometimes entanglement density is defined with an extra minus sign to make it a naturally positive quantity, see (5). It is also not clear why one should think of it as a density - entanglement susceptibility would probably be a more appropriate name; however we will follow [16, 17, 18] in using the term entanglement density. A similar quantity was studied in ., and will be the primary focus of the present paper. In the case where is spacelike and , there is another nice way of thinking about the entanglement density: start with the half space and glue on to it small area elements and at the points and on the entangling surface such that and are non overlapping. Then to lowest order in , the entanglement density is proportional to the conditional mutual information between and given the state on
where is the mutual information between the regions and . The strong subadditivity property then implies
Consequently, the entanglement density provides a natural notion of a metric on the space of geometries of the entangling surface. In theories with holographic duals, the Ryu-Takayanagi proposal maps this space into the space of mimimal-area surfaces in the bulk, and so the entanglement density provides a natural metric on the latter space as well (see  for more details in the /CFT case). It has also been argued in  that in holographic theories, equation (5) applied to a special class of deformations maps to the integrated null-energy condition on the bulk minimal-area surface.
In general, entanglement density in conformal field theories can contain two types of terms: (1) contact terms which arise in the coincident limit , and (2) a non-local term which is finite and well-defined when and are separated. For the most part, we will be interested in the latter. This non-local term is isolated via the definition (4) in terms of the conditional mutual information which makes it is clear this term should be independent of the UV cutoff. The main result of the present paper is as follows: for any conformal field theory, the non-local term in the entanglement density for a planar surface is universal and given by
where is the numerical coefficient appearing in the two-point function of stress tensors in the CFT. Equation (6) was obtained in  for a class of holographic theories using the Ryu-Takayanagi formula. However, we emphasize that in this paper we are working with completely general CFTs.222Actually it is enough to invoke the conformal symmetries that leave the entangling surface fixed, including the boosts in the transverse plane, to argue that the cut-off independent part of the entanglement density should take the form as in (6). Here we will be tracking down the overall coefficient. We will employ purely field theoretic techniques (developed in [21, 22, 23, 6]) to prove equation (6), thus extending the validity of this formula to arbitrary conformal field theories with or without holographic duals, and further providing a non-trivial check on the Ryu-Takayanagi and the Hubeny-Rangamani-Takayanagi proposals for entanglement entropy in holographic theories.
An analogous formula also holds in the case where we take to be a ball-shaped region of radius (see Fig. 2). Let denote the spatial coordinates on the Cauchy slice and take to be the region . For and two well separated points on the entangling surface, the non-local term in the entanglement density is given by
A number of results follow as corollaries: (i) in [24, 25], it was conjectured based on holographic and numerical evidence that the coefficient of the corner term contribution to the entanglement entropy in CFTs has the universal behaviour
where is the opening angle of the corner. We will prove this conjecture as a special case of our results. (ii) We also prove the Mezei formula for the universal part of the entanglement entropy across deformed spheres
which was conjectured in  based on holographic calculations in a large class of theories. In (9), are the coefficients of the expansion of the shape deformation in terms of real hyperspherical harmonics on the entangling surface. The Mezei formula is meant to apply to the universal term in CFT entanglement entropy for a deformed sphere, and the positivity of the overall coefficient demonstrates that the sphere locally minimizes this universal term in the space of shapes, suggesting that the sphere is somehow the optimal measure of degrees of freedom in a CFT for use as an RG monotone. Further, the above formula was used in  to compute universal corner contributions to entanglement entropy in higher dimensions. Therefore, our proof of the Mezei formula also completes the proof of universality of corner contributions in higher dimensions. In this way, our CFT calculation fits nicely into the triangle of recent studies and conjectures [17, 23, 26, 27, 28, 24, 25, 29, 30] (see also [31, 32, 33, 34, 35]) on entanglement density, corner contributions and the Mezei formula.
The rest of the paper is organized as follows: in section 2, we review some elementary facts about entanglement across planar and spherical surfaces, which will be relevant for our subsequent calculations. In section 3, we present the CFT calculation of the universal non-local term in the entanglement density for planar and spherical surfaces. In section 4, we will then use our result for the entanglement density to prove the universality of corner contributions in CFTs and the Mezei formula. Finally, we will end with some discussion about prospects for future work.
Entanglement entropy is defined as follows – consider the density matrix corresponding to a pure state defined on the Cauchy surface . In this paper, we will take to be the ground state of a conformal field theory. Let us partition into two subregions and . For local quantum field theories, we expect the Hilbert space to factorize into the tensor product . If this is the case, we can trace over to obtain the reduced density matrix
which contains all the relevant information pertaining to the subregion . Then the entanglement entropy between and is defined as the von Neumann entropy of
In this context, the boundary of is referred to as the entangling surface. It is also useful to define the modular Hamiltonian (also known as the entanglement Hamiltonian) in terms of as
In general, the modular Hamiltonian is not a local operator, in the sense that the modular evolution does not map local operators to local operators. However, there are a few special cases where symmetry forces the modular Hamiltonian to be local. The simplest such case is when we take to be the half-space . In this case, the modular Hamiltonian takes a simple form in terms of the CFT defined on Euclidean space :
where is the generator of rotations around the entangling surface in the plane ( is Euclidean time)
and the constant in (13) is chosen such that . This is known as the Bisognano-Wichmann theorem . The fact that the modular Hamiltonian for planar entangling surfaces in the vacuum state of a conformal field theory on is local, and can be written as an integral over the stress tensor will play a crucial role in our calculation of the entanglement density. In fact, the statement of the Bisognano-Wichmann theorem is true for the vacuum state of any relativistic quantum field theory, irrespective of conformal symmetry, and so it should be possible to extend our calculation to the more general class of relativistic quantum field theories. However, in this paper we will restrict ourselves to CFTs, because the calculation simplifies greatly in this case.
In conformal field theories, the modular Hamiltonian for a ball-shaped region (of radius ) is also local . This happens because the conformal transformation
with , maps the half-space to the ball-shaped region . Since conformal transformations are symmetries in a CFT, such a map leaves the ground state invariant, and the reduced density matrix on the ball-shaped region can be related to the reduced density matrix on the half-space by a unitary transformation. Additionally, one can transplant the modular Hamiltonian from the half-space to the ball-shaped region by pushing forward the modular flow by , which gives
For this reason, the calculation of the entanglement density for ball-shaped regions is no more difficult than the calculation for half-spaces in CFTs.
3 The CFT computation
Let us now delve into the calculation of the entanglement density in conformal field theories. For simplicity, we will describe the computation for half spaces in some detail, and then derive the corresponding result for ball-shaped regions by using the conformal transformation mentioned previously. So take to be the half-space . Consider now the entanglement entropy of the deformed region given by . In order to compute this entropy, we can use the coordinate transformation
to map the deformed entangling region back to the half-space . However, we must bear in mind that such a coordinate transformation has a non-trivial action on the metric. In terms of the new coordinates, the metric is given by
Therefore, to compute the entanglement entropy for in flat space, we may equivalently compute the entanglement entropy for the half-space but with the deformed metric 333 Note that (19) is true (even for the UV divergent terms) if we use a “covariant” regulator to define EE [38, 39, 40]. However since we are ultimately interested in a UV finite quantity the regulator used at intermediate stages in the calculation should not matter.
For our purpose it suffices to keep only the term linear in in equation (18) because we are interested in computing the non-local contribution to the entanglement entropy at second order in the perturbation series, while the terms in (18) can at most generate a local contribution at this order. The shape deformation in (17) is somewhat special in that it preserves the Cauchy surface . In our calculation we will relax this and consider the more general deformation
which also includes time-like deformations of the entangling surface, and of which equation (17) is a special case.444We need not include components along the transverse directions because these simply amount to reparametrizations of the entangling surface, which do not change the entanglement entropy.
The advantage of trading the original problem of computing with that of computing , is that it is possible to use conformal perturbation theory to write an expansion for the latter in terms of the deformation [21, 22, 23]. To see how this works, consider the reduced density matrix on in the presence of the metric deformation . A straightforward calculation shows (see Appendix A for details)
where are now coordinates on Euclidean space , and
is the original reduced density matrix on in the absence of the metric perturbation.555From now on, by we mean unless otherwise specified. Further, is the angular-ordering operator in the plane, i.e., if is the angular coordinate in the plane, then
where is the Heaviside step function. The next step is to perturbatively expand the entanglement entropy using eqaution (3). However, care must be taken in expanding the logarithm, because and do not commute in general. In order to deal with this, we use the following integral representation for the entanglement entropy
Expanding this out to second order in , we obtain
coming respectively from the first and the second term in equation (25). The first term (after performing the integration) is given by
where is the modular Hamiltonian corresponding to and is the connected trace. From the above equation, we see that can be interpreted as the change in the expectation value of the (original) modular Hamiltonian; we will henceforth refer to this term as the modular Hamiltonian term. Given that all the operators inside the trace are naturally -ordered, we can rewrite the above traces in terms of connected Euclidean correlation functions
Now, the second term in (26) is given by
This term is in fact the negative of the relative entropy of with respect to at second order in
and will henceforth be referred to as the relative entropy term. The non-negativity of relative entropy then implies
Unfortunately, the operators appearing in equation (3) are not manifestly -ordered, and so the trace in this form cannot be written as a Euclidean correlation function. However, it is possible to perform the integral and manipulate this expression further to bring it to a more convenient form (see Appendix B)
where is a rotation in the plane by the angle . This manipulation essentially involves steps similar to passing from old-fashioned perturbation theory to time dependent perturbation theory in quantum mechanics. This is usually achieved using Schwinger parameters and the appearing above can be thought of as such.
The trade-off however is the additional integral with the attendant measure. Interestingly, note that the way appears in the above correlation function corresponds to a relative boost between the two stress tensor insertions, with being the boost angle/rapidity. Equivalently, from the point of view of the modular Hamiltonian, we are forced into “real time” evolution. Indeed, we can rewrite the above equation in the following way to make this point manifest
Having written this term in the above form we can now use the -ordering to rewrite the trace in terms of the Euclidean two-point correlation function to obtain
From equations (28) and (3) we see that the entanglement density can be computed in terms of the two-point and three-point Euclidean correlation functions of the stress tensor. Indeed, in any conformal field theory, these correlators are universal and fixed by conformal invariance modulo finitely many parameters . The two-point function in particular takes the form
and is determined entirely by specifying the single parameter . The three-point function is more complicated, and in general dimension depends on three independent parameters. Nevertheless, it is clear from the above discussion that the (non-local part of the) entanglement density in a CFT is uniquely determined in terms of the parameters appearing in the two- and three-point correlators.
All that remains now is to explicitly evaluate the integrals in equations (28) and (3). Doing so, one encounters the following surprising result – the modular Hamiltonian term (28) does not contribute to the non-local part of the entanglement density. Since the explicit computation is somewhat tedious, we will defer the details to Section 3.2. We also give a quicker more sketchy proof of the vanishing of the modular Hamiltonian term, using a slightly different setup, in Appendix E. The non-trivial contribution to the entanglement density then comes entirely from the relative entropy term. Indeed, this is why the result (6) for the non-local part of the entanglement density depends only on the single parameter . We now proceed to compute the relative entropy term.
3.1 Relative entropy term
In order to compute the integrals in (3), it is much more efficient to use the conformal transformation from to to pull-back and evaluate the integrals on . To see how this works, let us coordinatize by , where is periodic with period , and are Poincaré coordinates on the hyperbolic space . The metric on in these coordinates is given by
The map given by
is a conformal transformation, i.e.
with being the Weyl factor (and being the pullback). This implies that the stress tensors on the two spaces are related by
where denotes additional Schwarzian derivative-type terms, which vanish in odd dimensions, but are present in even dimensions. The integral (3) then pulls back to
where we have defined , and is the measure on . Further
where the vector field on is the push-forward of the vector field on by
Note that the Schwarzian terms have dropped out of equation (41) because of the connectedness of the correlation function. Additionally, we note that the second term in (43) can also be dropped by the tracelessness of the stress tensor (more precisely, the trace Ward identity) in conformal field theories.666In even dimensions, there are contributions coming from the trace anomaly. However, these contributions are local at the present order. Since we are interested in the non-local part of the entanglement density, we can drop these terms. So we obtain
Since the above integrals include integration over hyperbolic space, there are potential divergences coming from the conformal boundary of hyperbolic space at . These divergences in the entanglement entropy of course correspond to the short-range entanglement coming from the region close to the entangling surface. One way to regulate such potential divergences is to put a cut-off at (which corresponds to cutting out a tubular neighbourhood around the entangling surface in the original description on Euclidean space). We denote the resulting regulated space as , and rewrite the above integral as
Next, integrating by parts and using the diffeomorphism Ward identity,777 Which says that and for separated points. we arrive at
where is the boundary of the regulated space at , is the outward pointing unit-normal on the boundary, and is the measure induced on the boundary
The next thing to compute is the two-point function of stress tensors on . This can be done efficiently using the embedding space formalism developed in . In this formalism (see Section 2 of  for a review relevant for this calculation), one considers the larger embedding space (or ambient space) on which the (Euclidean) conformal group acts linearly. Let us pick global coordinates on this space, with the coordinate being time-like. One then embeds (more generally, any space which is conformally equivalent to ) as a section of the upper light-cone . Here, we pick the embedding
where are auxiliary variables, and the right hand side above is evaluated on the section (49). The index-free function is defined as,
Using this formalism, we compute the required two-point correlation functions
where the overlined indices run over , and
We have also defined
with given by the two dimensional matrix