Entanglement Rényi entropies in holographic theories
Ryu and Takayanagi conjectured a formula for the entanglement (von Neumann) entropy of an arbitrary spatial region in an arbitrary holographic field theory. The von Neumann entropy is a special case of a more general class of entropies called Rényi entropies. Using Euclidean gravity, Fursaev computed the entanglement Rényi entropies (EREs) of an arbitrary spatial region in an arbitrary holographic field theory, and thereby derived the RT formula. We point out, however, that his EREs are incorrect, since his putative saddle points do not in fact solve the Einstein equation. We remedy this situation in the case of two-dimensional CFTs, considering regions consisting of one or two intervals. For a single interval, the EREs are known for a general CFT; we reproduce them using gravity. For two intervals, the RT formula predicts a phase transition in the entanglement entropy as a function of their separation, and that the mutual information between the intervals vanishes for separations larger than the phase transition point. By computing EREs using gravity and CFT techniques, we find evidence supporting both predictions. We also find evidence that large- symmetric-product theories have the same EREs as holographic ones.
The concept of holography originated as an idea about quantum information, that the number of qubits that can be stored in a region of space is fundamentally limited by its surface area in Planck units. Modern holographic theories go beyond a mere counting of states, and posit that the physics governing certain spacetimes can be fully described by a quantum field theory residing on its boundary. However, the way that those qubits are organized remains unclear on both sides of the correspondence. On one side, we don’t yet understand how the states are organized in quantum gravity; on the other, despite an in-principle understanding of the state space of quantum field theories, in practice we have to deal with a strongly coupled theory with a large number of degrees of freedom. And, of course, the map between the two descriptions remains deeply mysterious.
A useful probe of physical information in quantum systems is the entanglement entropy (EE). Here we imagine decomposing a system into two subsystems, , with a corresponding decomposition of the Hilbert space . Given a density matrix for the full system, the reduced density matrix , which acts on , is defined by tracing over and represents the effective density matrix for an observer who has access only to the subsystem . The EE for is then the von Neumann entropy of : . A non-zero EE may be due to the full system being in a mixed state, to information about the state being lost by the inability to observe the rest of the system, or to a combination of the two effects. The degree of correlation (both classical and quantum) between disjoint subsystems may be quantified by their mutual information , which puts an upper bound on correlators between operators in and in .
In a quantum field theory, it is natural to consider subsystems that are spatial regions. Their EEs and mutual informations then tell us about the spatial distribution and correlations of quantum information in a given state. Unfortunately, EEs in quantum field theories are notoriously difficult to calculate, mainly because one does not have a good way to represent the operator . On the other hand, if the density matrix for the full system can be represented by a path integral (as in the vacuum or a thermal ensemble, for example), then both the reduced density matrix and its positive integer powers can also be represented in a fairly simple way by path integrals. If those path integrals can be computed explicitly for all , then one can obtain the EE indirectly as follows. Defining the entanglement Rényi entropy (ERE) for , one analytically continues in and takes the limit to obtain the EE. This procedure is called the replica trick. Aside from being easier to calculate than the EE, the EREs are of interest in their own right, as a more refined characterization of the reduced density matrix . In fact, knowing for all is equivalent to knowing the full eigenvalue distribution of . In Section 2, we review the basic properties of entanglement and Rényi entropies.
Even given the replica trick, exact results for the EE in field theories are known only in very simple cases, such as a single interval in the vacuum of an arbitrary two-dimensional conformal field theory . For two disjoint intervals, the EE, and hence mutual information, remain unknown even for a theory as simple as that of a compact free scalar . It might therefore seem hopeless to dream of knowing the EE in a strongly coupled, large- field theory. Remarkably, however, Ryu and Takayanagi (RT) proposed a simple, elegant, and universal formula for the EE of an arbitrary spatial region in an arbitrary holographic field theory [4, 5]. Their formula, which applies to any state described by a static classical geometry, says simply that the EE equals one quarter the area in Planck units of the minimal surface in the bulk ending on the boundary of the region . If correct, the RT formula is not only very useful as a calculational tool, but also a significant hint regarding quantum information in holographic theories, and probably in quantum gravity more generally (see for example ).
The RT formula passes several non-trivial checks. For example, it correctly reproduces the EE for a single interval in a two-dimensional CFT. A general derivation, using the replica trick, was offered by Fursaev . He found that the ERE equaled one-quarter the minimal-surface area, independent of . The analytic continuation in was thus trivial, giving agreement between the resulting value of and the RT formula. In computing the ERE, Fursaev performed the necessary path integrals using Euclidean quantum gravity. Unfortunately, as we show, the bulk geometries that he used to evaluate the partition function are not actually saddle points of the gravitational action. As a result, the ERE he derived is incorrect, as we can see by comparing it to the known exact result in the case of a single interval in a two-dimensional CFT. We show how the latter result can be reproduced using the correct saddle-point action. The RT formula and Fursaev’s proof are reviewed and discussed in Section 3.
The question thus arises of whether, in cases where the correct value is not already known, we can compute the ERE in a holographic theory, both for its own sake and in order to confirm or refute the RT conjecture. Unfortunately, to do so in complete generality, as Fursaev attempted, appears to be quite difficult. Therefore, in Section 4, we focus on a simple but non-trivial case: two disjoint intervals in a two-dimensional CFT. The RT formula predicts a rather interesting phase transition for the mutual information between the two intervals as a function of their separation. In particular, for separations larger than a certain critical value, the mutual information vanishes, implying a decoupling between the degrees of freedom in the two regions. (This behavior of the mutual information is a completely general prediction of the RT formula, applying essentially to any two regions in any state of any holographic theory. It is closely analogous to the factorization property for disconnected Wilson loops .)
The ERE for two disjoint intervals can be expressed in terms of the partition function on a certain Riemann surface of genus . For , we thus need the torus partition function, which fortunately is known for a general holographic CFT . Indeed, as we show, it exhibits a phase transition at precisely the same separation as that predicted for the EE by the RT formula. For higher values of , while the partition function is not known explicitly, we show using symmetry arguments that the ERE continues to have a phase transition at the same separation. This strongly suggests that the same will hold for , confirming this prediction of the RT formula.
The fact that we can compute the ERE explicitly only for precludes analytically continuing it to , to directly confirm or refute the full EE predicted by the RT formula. We therefore pursue a different strategy. Using the OPE, we expand the ERE, for any given , in powers of the inverse separation between the intervals. The coefficient of any given power can be computed explicitly for all using formulas for conformal blocks, and analytically continued to . We carry this out for a number of coefficients, finding that, thanks to a rather intricate pattern of cancellations, in each case the continuation to vanishes, precisely as predicted by the RT formula.
As a byproduct of our analysis of the ERE for two disjoint intervals, we find that the result for certain non-holographic CFTs with large central charges, such as large- symmetric-product theories, is precisely the same as for holographic ones. It seems that there is some form of large- universality operating here, with a large class of such CFTs having identical EREs (and therefore EEs). This possible feature of the ERE deserves further study.
We conclude in Section 5 with a list of open questions and possible generalizations of our work, and some remarks concerning our current understanding of the RT formula.
An appendix contains calculations in certain orbifold theories whose results are used in the main text.
2 Entanglement Rényi entropy: review
In subsection 2.1 we briefly motivate, define, and state (without proof) the important properties of the entanglement Rényi entropy and mutual Rényi information. In subsection 2.2, we illustrate these ideas in the simple example of two subsystems that are weakly coupled to each other. In subsection 2.3 we then briefly review the replica trick for computing the entanglement Rényi entropy, and in subsection 2.4 apply it to the simplest field theory example, a single interval in a two-dimensional conformal field theory. For more details, we refer the reader to the books [10, 11] and the review ; the latter provides a comprehensive introduction to Rényi and entanglement entropies in two-dimensional CFTs.
2.1 Basic definitions and properties
Given a density matrix and a positive real number , the Rényi entropy is defined as111Writing , we see that the definition of the Rényi entropy is similar to that of the norm of a positive function, . The difference is that, whereas in the norm we evaluate the integral with respect to a fixed measure, in the Rényi entropy we evaluate it with respect to the very density matrix whose entropy we are computing.
At the Rényi entropy is defined by taking the limit, and equals the von Neumann entropy:
Two other interesting limits are , called the Hartley entropy, where is the image of , and , called the min-entropy, where is the largest eigenvalue of . The following properties of are straightforward to prove: (1) , with equality if and only if represents a pure state; (2) is constant if and only if is proportional to the identity on , and is otherwise a decreasing function of ; (3) for it satisfies .
If the system contains a subsystem —for example, in a field theory, could be a spatial region222By region we technically mean co-dimension zero submanifold (possibly with boundary). In this paper we will not consider other types of sets, such as single points, fractals, etc.—then the Hilbert space can be expressed as the tensor product of Hilbert spaces corresponding to and to its complement : . Let be the reduced density matrix, defined in , obtained by tracing over ; this is the effective density matrix for an observer who has access only to . Its Rényi entropy is called the entanglement Rényi entropy (ERE) of , with the special case simply called the entanglement entropy (EE). It can be shown that, if the full theory is in a pure state, then .
As a special case, strong subadditivity implies the triangle inequality, namely for disjoint subsystems and ,333The ERE satisfies for classical distributions, but not in general for quantum density matrices.
The second inequality is called subadditivity, and it characterizes the EE, in the sense that any measure of entanglement that satisfies subadditivity for all subsystems and (as well as certain basic requirements such as continuity) must equal the EE [15, 16]. Subadditivity is saturated if and only if the density matrix factorizes: . Motivated partly by this fact, the mutual information (MI) is defined by
which quantifies the extent to which the degrees of freedom of and are correlated with each other, including both quantum entanglement and classical correlations. For example, the MI puts an upper bound on the connected correlator between (bounded) operators in subsystems respectively :
As a consequence of strong subadditivity, the mutual information is monotone under restriction: if then .
A natural generalization of the mutual information is to define the mutual Rènyi information (MRI):
Unlike the MI, the MRI is not necessarily positive. However, it is non-zero only when , and in this sense still quantifies the extent of correlation between and .
Another reason to study the MRI (including the MI) is that, when we are considering a field theory, it is universal, whereas the ERE (including the EE) is cutoff- or regulator-dependent. Specifically, when is a spatial region, usually suffers from an ultraviolet divergence proportional to the area of the boundary of . However, if the two regions and are disjoint and mutually disconnected, then those divergences cancel in the MRI. Since the UV regulator generally violates conformal invariance, it follows that in conformal field theories the MRI is generally conformally invariant while the ERE is not. Also, in the CFT case the ERE suffers from an infrared divergence when one of the regions is infinite in size, but this cancels in the MRI (although not when both and are infinite). We will see explicit examples of these statements throughout this paper.
2.2 Perturbative MRI
Before tackling the computation of entanglement entropies in field theories, as a warm-up we first consider the perturbative computation for subsystems that are weakly coupled to each other. We will see that in this case the MRI between the subsystems is parametrically larger than the MI, a result that foreshadows the results of Section 4 concerning holographic systems.
We begin by considering a single system, and the effect on its Rényi entropy of a small perturbation to its density matrix:
where , , and is a small parameter. We assume that the perturbation does not change the rank of the density matrix, and in particular the image of is contained in the image of . To first order in we have:
Now suppose our system is composed of two subsystems, and the unperturbed density matrix factorizes:
At zeroth order in the MRI of course vanishes. To first order we have:
As , the operators go to zero like . Hence, to first order in , the MI vanishes:
(It can be shown that the order term generically does not vanish.) This can be understood as a consequence of the fact that the MI is non-negative, since could take either sign.
2.3 Replica trick
Unfortunately, in practice there are very few known methods for computing EREs (or EEs) in field theories. One of the most useful is the replica trick, which we will review below, that allows one to compute the ERE for integer . In favorable circumstances a simple analytic form for for general real can be found which fits those data points, and from this form the EE can be read off by setting .444Throughout this paper will lie in the interval , while will be a positive integer. It is important to say at the outset that in proceeding this way we are merely presuming to have guessed the ERE correctly; firstly, nothing guarantees (in an infinite-dimensional Hilbert space) that the ERE is analytic, and, secondly, the values of a function on a countably infinite set (in this case, the integers larger than 1) are not sufficient to fix a unique analytic continuation. (There exist analytic functions, such as , that vanish for all integer but not elsewhere, including at .) Having stated this caveat, for the rest of the paper we will assume that all EREs we consider are indeed analytic functions of . We will find nothing inconsistent with this assumption.
The replica trick applies when the theory is in a state, such as the vacuum or a thermal state, whose partition function can be obtained by a path integral over some Euclidean spacetime (possibly with some operator insertions, which for the purposes of this discussion we will consider to be part of ). Let be a spatial region, and the -sheeted cover of with the sheets connected along branch cuts placed at on a constant Euclidean-time surface. Then , where is the partition function of the theory on (and, in particular, is the partition function for the original theory).555If the theory contains fermions then one needs to specify their boundary conditions across the constant Euclidean-time surface where the sheets are sewn together, which we will call . The original partition function is computed with a sign-flip on the fermionic fields across . More generally, is computed with a flip on on each sheet of , along with one on where the th sheet connects to the first sheet (but not on the other copies of ). Hence a curve that winds around a branch point times, returning to the same point on , crosses sign flips. The resulting overall flip for even is part of the definition of the twist operators and of the next paragraph. When is evaluated by passing to a coordinate system that is single-valued on , this overall flip is canceled by the branch cut in the coordinate transformation for the fermionic field. (For example, in complex coordinates if is a local coordinate on with the branch point at , and is a single-valued local coordinate on , then . For even the factor has a branch cut with a sign flip.) Hence there is no operator insertion in the new coordinate system. However, there may still be sign flips around non-contractible cycles. (See for example the case of the torus in footnote 13 below.) Hence we have (for )
To be more concrete, let us further specialize to a two-dimensional conformal field theory ,666All CFTs will be assumed compact, unitary, and modular-invariant in this paper. and let be the union of disjoint intervals , where . Then we can rewrite the expression (14) in terms of correlators of twist operators in the orbifold theory , computed on :
The correlator of twist operators is divergent, due to the singular geometry of at the branch point. It can be regularized by regularizing each twist operator separately; hence the notation and , where is a UV cutoff length, for the regularized twist operators .
2.4 Single interval in a CFT
As an example of the application of (15), the ERE for a single interval, in the vacuum, is
where is the central charge of and is a scheme-dependent quantity. Here we used the fact that the twist operators have scaling dimension
The (simplest) analytic continuation of (16) to non-integer is777It is interesting to ask what eigenvalue distribution for gives rise to the -dependence seen in (20) (neglecting the subleading and scheme-dependent quantity ). This question can be answered by defining the so-called “modular Hamiltonian” acting on . Then is related to the free energy of at the temperature : (18) (The first two equalities apply to the Rényi entropy of any system.) The density of states that gives rise to this temperature dependence for the free energy is easily found, in the saddle-point approximation, by performing a Legendre transform: (19) where is the eigenvalue of . (See  or  for the form of the full inverse Laplace transform.) It is interesting that (up to a shift of by ) has the same form as the Cardy formula for the asymptotic density of states in a CFT on a circle. Note that we have not determined which physical observable represents—it is not necessarily related to a physical energy.
which yields the EE 
Note that indeed satisfies the properties (1), (2), (3) mentioned below equation (2).
It is also possible to obtain the result (16) (and thereby derive the scaling dimension (17)) by computing and applying (14). The computation of is carried out as follows . We are in the vacuum, so the Euclidean spacetime is simply the plane, to which we add a point at infinity to make it topologically a sphere. The multi-sheeted surface is then also topologically a sphere. A Weyl transformation maps the metric on to a fiducial metric on the sphere. We then have , where is the partition function of on the sphere with the fiducial metric, and is the Liouville action:
(which depends on only through its central charge). For the metric on has a conical singularity at each branch point , so the Liouville action is divergent. The divergence can be regulated by replacing a disc of radius about each branch point with a smooth metric, which defines the regularized twist operators .
3 Holographic entanglement entropies
3.1 Ryu-Takayanagi formula
In this subsection we will provide a brief summary of Ryu and Takayanagi’s proposal for the entanglement entropy (EE) in field theories with holographic duals [4, 5], along with some of the evidence supporting it. A more complete review can be found in . We will then discuss interesting predictions it makes for the mutual information between separated regions.
The Ryu-Takayanagi (RT) conjecture is a proposed formula for the EE of a given spatial region in certain states of holographic field theories whose dual gravitational theory is classical Einstein gravity (possibly with matter). Specifically, the proposal concerns states that admit a description as static classical solutions in the dual theory, such as the vacuum and thermal states.888Possible generalizations to time-dependent states were proposed in . We work in a fixed constant-time (i.e. timelike-Killing-field orthogonal) slice of the bulk. The conjecture states that
where is the minimal-area surface in the bulk that is homologous to , i.e. such that there exists a region with . (As we will see, this topological condition plays a crucial role in several checks of the proposal.) The area is evaluated with respect to the Einstein-frame metric.
An interesting question, assuming the RT formula is valid, is how it gets corrected by quantum effects and by higher-derivative (e.g. ) corrections to the classical action in the bulk. Quantum effects presumably lead to corrections to (23) (starting at order ), although a specific form has not been proposed. On the other hand, in the presence of higher-derivative corrections to the classical bulk action, it is expected that the EE is given by minimizing a corrected geometrical functional; based on consistency with black-hole entropy (discussed below), the functional should coincide with Wald’s black-hole entropy formula  when evaluated on a horizon.
The RT proposal passes several basic checks. For example, if is the entire boundary, then should simply be the statistical entropy of the state. Indeed, according to the RT proposal we should take to be the minimal surface in the bulk that is homologous to the boundary; this will generally be the horizon, if there is one, giving agreement with the Bekenstein-Hawking entropy. If there is no horizon, then the boundary is homologically trivial in the bulk (i.e. the topological boundary of the bulk is precisely the boundary where the field theory lives); hence the minimal surface is the empty set, giving .999Since we are considering static spacetimes, any black holes in the spacetime should be eternal, so the maximally extended spacetime may include other, topologically disconnected boundaries. Consider, for example, the maximally extended spacetime of the AdS-Schwarzschild black hole, whose boundary has two connected components. It is believed  that the field theory defined on both boundaries represents the thermofield double of the field theory defined on only one boundary. In this picture, the black hole spacetime, which represents a thermal and therefore mixed state in the original field theory, represents a pure state in the thermofield double. This result is faithfully reproduced by the RT prescription; the full boundary (including both components) is homologically trivial in the bulk, giving . (Again, this is the order entropy—the RT formula does not capture the entropy due for example to a gas of gravitons in thermal AdS, which is of order .) Furthermore, when the total entropy is zero (or of order ), then if we instead take to be a subset of the boundary, we expect from (4) that . Indeed, in this case the entire boundary is homologically trivial in the bulk, so and are homologous, implying .
Another important check on the RT proposal is that it satisfies the strong subadditivity (SSA) property (3) for any regions and , as can be shown by a simple geometrical argument . (Interestingly, the proof of SSA based on the RT formula is far simpler than the general proof.) Since, as mentioned in subsection 2.1, subadditivity, which is implied by SSA, characterizes the EE, this is quite strong evidence in favor of the RT formula. However, it is not sufficient to prove its correctness, since it only shows that (3) is satisfied for subsystems corresponding to geometrical regions, whereas for the characterization proof one needs it to hold for all subsystems. The proof of SSA extends trivially to the inclusion of higher-derivative corrections, as long as they are extensive.
As a final check, let us see how the RT formula reproduces the EE (21) of a single interval , in the vacuum of a two-dimensional CFT. The vacuum is described holographically by AdS, whose metric on a constant-time slice is
here is the coordinate along the boundary and is the radial coordinate, with the boundary being at . We employ a simple UV cutoff in which we shift the boundary to . The minimal surface is a geodesic connecting the points on the boundary , which is an arc of a circle (almost a semi-circle) with center . Applying (23) and using the standard holographic relation , one finds [4, 5]
matching (21). (In this scheme, the finite part vanishes.) In higher dimensional CFTs, although one does not have exact formulas for the EEs even of simple regions, the leading UV divergence is known and matches that predicted by the RT formula [4, 5].
3.1.3 Application to disconnected regions
For a time it was believed that the RT formula should only apply to connected regions. (See for example the paper .) The reason was that, when applied to the union of two intervals (a case that will be considered in detail in the next section), it disagreed with a calculation by Calabrese and Cardy  which (like the formula (21) for a single interval) was supposed to be valid in any two-dimensional CFT. However, those same authors have since shown that their original calculation was incorrect. At present, there is no reason to believe that the RT formula, if it is valid at all, would not apply equally well to connected and to disconnected regions. For example, all the checks discussed above apply to both cases (including the last one, which can be considered a computation of the EE of the disconnected region ).
When applied to a disconnected region, the RT formula makes a fascinating prediction for the mutual information (MI) between its components, similar to the phase transition for disconnected Wilson loops found by Gross and Ooguri . For simplicity, let us consider two disjoint and mutually disconnected regions , . Each has a corresponding minimal surface , and region , . (We assume the generic situation that , are disjoint and mutually disconnected.) When we consider the region , the disconnected surface is topologically allowed and locally minimal. Assuming that the full bulk spacetime is itself connected, surfaces will also exist that connect and . However, if the separation between and is sufficiently large compared to their sizes (and any other scales defined in the theory or state), then will necessarily be the globally minimal surface. Then we have , so the MI vanishes. More precisely, is of order , rather than .101010A closely related phenomenon, in which the and are held fixed but the bulk spacetime is deformed, was discussed in the paper . This implies that, from a quantum information point of view, the two regions are approximately decoupled from each other. (See for example the bound (6) on correlators between and . Note however that this bound does not directly give us information about correlators of local operators, which are generally not bounded operators.) If we then imagine bringing and closer to each other, then it may happen that, at some critical separation, the minimal surface will switch from to one that connects and (see for example figure 1). In this case, the MI will (in the thermodynamic/classical limit ) undergo a first-order phase transition; it will become non-zero, with a continuous value but discontinuous first derivative as a function of the separation between and . Section 4 will be devoted to a detailed study of these phenomena in the simplest example, namely two intervals in the vacuum of a two-dimensional CFT.
3.2 Fursaev’s ERE calculation
In the paper , Fursaev gave a derivation, based on the replica trick, of the RT formula. In this subsection, we will briefly summarize his argument, and then point out a flaw that results in an incorrect value for the entanglement Rényi entropy (ERE).
In our sketch of Fursaev’s argument, for simplicity we will take the bulk action to be pure Einstein gravity; matter fields and higher-derivative (e.g. ) corrections are straightforwardly incorporated, as he discusses. We will also assume that the ultraviolet divergence in the field theory is cut off in some manner whose details will not concern us. Fursaev’s starting point is (14), where is the partition function on the -sheeted Euclidean spacetime . Recall that, if is the spatial region whose EE we are computing, then the sheets of are connected by a branch cut along on a constant-time slice. In a holographic theory, this partition function is given by the gravitational path integral over Euclidean geometries whose conformal boundary is . In the classical limit, this path integral goes over to its saddle-point approximation , where is the minimal value of the Euclidean Einstein-Hilbert action among extrema obeying the boundary conditions. Fursaev constructs a set of geometries with boundary , then minimizes the Euclidean action within that set. He takes as given the bulk Euclidean spacetime representing the original state of the system; its boundary is and its Euclidean action is . He takes copies of and connects them along a branch cut , which is a spatial region in lying in the same constant-time slice as . In order for this -sheeted bulk geometry to have boundary , the part of the boundary of that lies in must coincide with (i.e. ); apart from this condition, the choice of is at this point arbitrary. The branch “point” is , the rest of the boundary of ( and ). He now evaluates the Euclidean Einstein-Hilbert action for this geometry. There are two contributions. First, the geometry is made up of copies of , so there is a contribution , which is independent of the choice of . In addition, the Ricci scalar has a delta function along the branch “point” , which is co-dimension 2 and hosts a conical singularity with excess angle . It therefore contributes a term to the action. Minimizing this action over all possible choices of , he obtains the minimal surface , and (from (14)) the ERE
Since there is no -dependence, the analytic continuation is particularly simple: . Finally, setting , he obtains the RT formula (23).
The problem with this derivation is that the action has been extremized only with respect to a subset of the degrees of freedom in the metric, namely the choice of . The resulting field configuration is therefore not guaranteed to be a true saddle-point, and in fact it does not solve the Einstein equation: the Einstein tensor has a delta function supported on due to the conical singularity, with no corresponding source.
We can confirm that the ERE (26) is incorrect by comparing it to the exact result in a case where the latter is known. For example, when is a single interval in the vacuum of a two-dimensional CFT, the exact ERE (16) depends on (by the factor ), whereas the Fursaev result (26) is independent of . What is the true saddle point in this case? The Euclidean space is a plane, and the corresponding bulk geometry is hyperbolic 3-space (a.k.a. Euclidean AdS). The saddle point corresponding to the -sheeted cover is also . The easiest way to see this is to add a point at infinity to to make it a sphere; then its -sheeted cover is also a sphere, so the corresponding bulk geometry is . This geometry is smooth, in contrast to Fursaev’s, which is copies of glued together in such a way as to create a conical singularity along the geodesic connecting the endpoints of . Given that the bulk geometry is for all , why does its action depend on ? The bulk action is divergent due to the infinite volume near the boundary; while the full bulk geometry is for any , the cutoff geometry depends on . (The full geometry depends only on the Weyl class of the boundary metric, which is the same for all , since the sphere admits a unique Weyl class. On the other hand, the cutoff geometry is sensitive to the actual boundary metric. This is the holographic manifestation of the Weyl anomaly .) The -dependence can most easily be calculated by performing a Weyl transformation to put the metric on into a standard form and taking into the account the resulting change in the partition function due to the Liouville action, as described at the end of subsection 2.4, or equivalently by the holographic renormalization procedure .
It is worth noting that the true saddle point with boundary can be obtained topologically by gluing copies of together in precisely the manner described by Fursaev. This will continue to be the case in the more complicated examples we will study in the next section, suggesting that, while it carries the wrong metric, Fursaev’s construction may be topologically correct in general. This would explain why the topological condition on the minimal surface that he suggested—that should be homologous to —appears to be correct.
Finally, it is intriguing that, while Fursaev’s value (26) for the ERE is incorrect for , it somehow manages to give the right answer for the EE (), assuming that the RT conjecture holds. We can only speculate that, if there is some sense in which the spacetimes and their bulk duals can be defined for non-integer values of , then his construction may be correct “at linear order” in a neighborhood of .
4 Mutual Rényi information between two intervals
As we saw in subsection 2.4, the entanglement entropy for a single interval in the vacuum of a two-dimensional CFT depends only on the theory’s central charge. The fact that the Ryu-Takayanagi formula correctly reproduces this entropy, as reviewed in subsection 3.1.1, is an important check on the proposal, but does not give us any new information. The next simplest configuration we can consider in such a theory consists of two disjoint intervals. As suggested by the fact that the Rényi entropies (15) depend in this case on four-point rather than two-point functions of twist operators, we would expect the EE to depend on the full operator content of the theory, rather than simply its central charge. As we will see in this section, the RT formula can give us significant new physical information in this case. The new predictions in turn give us the opportunity to subject the formula to new and highly non-trivial quantitative tests.
We begin by reviewing the necessary formulas and setting up the basic properties of the ERE for two intervals.
4.1 General properties
We consider two separated intervals , () in the vacuum of a conformal field theory with central charge . As discussed in subsection 2.1, it is convenient to consider the mutual Rényi information (MRI) between the two intervals,
which measures the extent to which the degrees of freedom of the two intervals are entangled with each other (including both classical correlations and quantum entanglement).
We first consider the integer case . Using (15), the MRI is given in terms of a finite ratio of four-point and two-point functions of twist operators in the orbifold theory :
where we’ve defined the renormalized twist operators:
This is an example of the UV divergences in the EREs, which occur at the endpoints of the intervals, cancelling in the MRI, as discussed at the end of subsection 2.1.
Since the twist operators are primaries, the transformation law for the four- and two-point functions implies that is conformally invariant, and therefore depends only on the cross-ratio
which lies in the interval . By a conformal transformation, the four points , , , can be brought to 0, , 1, respectively, so we have:
where . (The scaling dimensions of the twist operators are given by (17).) Notice that, like the UV divergence, the IR divergence in the ERE cancels in the MRI. The four-point function, and therefore , is an analytic function of in the interval .
It is useful to note that the four-point function in (32) can be expanded as a power series in , where the powers are the dimensions of operators in the orbifold theory,111111Throughout the paper, we use primes on untwisted operators of , to distinguish them from operators of . and the coefficients are given in terms of OPE coefficients:
Note that only untwisted operators contribute to the sum. Assuming we are dealing with a unitary theory, the operator with lowest scaling dimension is the unit operator, for which (by the normalization of the twist operators) the OPE coefficients are 1. Hence goes to 0 as , as we would expect on physical grounds. For example, if we fix the sizes of the intervals and take their separation to infinity, we would expect all correlations between them to go to zero. We will study the higher-order terms in the expansion (33) in subsections 4.4 and 4.6, and in the appendix.
A final important property of the MRI, implied by the invariance of the four-point function in (32) under , is
At the level of the definition (27) of the MRI, this relation is due to the fact that, in a pure state (in this case, the vacuum), , so .
We have listed five general properties that the MRI satisfies for integer , but for the reasons given we either know or expect each to hold for general values of :
UV finiteness, and IR finiteness when one of the intervals is semi-infinite;
conformal invariance, implying
for all ,
analyticity of as a function of .
So far we have not assumed anything about the theory (other than unitary and compactness). In the rest of this section, we will study the function in holographic CFTs, as well as certain other theories with large central charge.
4.2 Prediction from Ryu-Takayanagi formula
As in the holographic derivation of the EE for a single interval, reviewed in subsection 3.1.2, we use the fact that the holographic dual of the vacuum is AdS, with , and we cut off integrals near the boundary at radial coordinate value .
The RT formula is straightforward to apply to the union of two intervals . There are two locally minimal surfaces in the bulk that are homologous to this boundary region, as shown in figure 1. The first is the union of the minimal surfaces for the two intervals separately, (similarly for the corresponding bulk region ). This has “area” (i.e. length)
(see (25)). The other locally minimal surface connects to and to : . (The corresponding bulk region is a semi-annulus connecting the two intervals: .) Its area is
It is easy to see that is the globally minimal surface when , and otherwise (see (31)), so
Of the five properties of the MI listed at the end of the previous subsection, this formula obeys the first four. It does not obey the last—analyticity—as it has a discontinuous first derivative at and vanishes for . These two features were anticipated in the discussion in subsection 3.1.3. The discontinuity in the first derivative occurs because the global minimum switches between the local minima as we vary , and is reminiscent of phase transitions due to competing saddle points of the Euclidean action, such as the Hawking-Page transition. As in that case, the transition is presumably sharp only in the classical limit in the bulk, which corresponds to the thermodynamic () limit of the CFT, and gets smoothed out by finite- effects. Similarly, the vanishing of the MI for is presumably true only at order ; if the MI vanished exactly for , then the reduced density matrix for the two intervals would factorize, , implying that the two intervals are completely decoupled from each other; in particular, it would imply that all connected correlators vanish, which is certainly not the case. Thus we should expect both perturbative and non-perturbative corrections to (42) in , with the first perturbative correction at order . Nonetheless, since the MI is apparently parametrically small for —smaller than the EE for either interval separately or for their union, and smaller than the MI for —it appears that the density matrix factorizes approximately.
Unlike quantum corrections, we do not expect higher-derivative (e.g. ) corrections to the classical bulk action to change the result (42), for the following reason. As discussed in subsection 3.1.1, such corrections are believed to correct the area functional appearing in the RT formula without changing the basic prescription of minimizing over topologically allowed surfaces. The symmetries of AdS guarantee that the minimal surfaces shown in figure 1 remain uncorrected; furthermore, the corrected “area” of each curve is unchanged when written as a function of , since we know that the EE is always given by (21). In fact, this argument applies for any bulk gravitational action, not just Einstein-Hilbert with small higher-derivative corrections.
4.3 Universality in the large- limit?
Given any family of CFTs that admit a large- limit, such as holographic ones, we can consider the expansion of the MRI in powers of . Since the number of degrees of freedom is of order , the leading term will be at most of that order, so we have
In particular, we focus our attention on the leading function . In the previous subsection, we used the RT formula to compute, for holographic theories,
and argued that this result should hold no matter what the bulk gravitational theory is. As discussed, (44) has two striking qualitative features, namely its discontinuous first derivative at and the fact that it vanishes for . In the rest of this section, we will study using the replica trick, and find independent evidence for both phenomena. In the next subsection, we will compute in holographic theories, and show that the result is independent of the details of the bulk theory (e.g. the presence of higher-curvature corrections), and applies also to symmetric-product theories , even though their large- limit is not described by classical gravity. Like (44), the result will have a phase transition at . In subsection 4.5 we will argue that this phase transition occurs also in for , at least in holographic theories. Then in the last subsection we will use CFT techniques to study the expansion of in powers of for general , and find evidence that every coefficient in this expansion goes to 0 in the limit . That analysis will assume very little about the CFT , essentially just that the number of operators below any given dimension stays finite as goes to infinity, a condition that holds for both holographic and symmetric-product theories (but not, for example, in the power theory without the orbifold).
These results not only give strong quantitative support to the RT formula, but point to a broader picture, namely that a large class of large- CFTs—including holographic and symmetric-product theories—share the same leading MRI , as a function of both and . Although we do not know the explicit form of this function except for , we can deduce that it is analytic in except at , where it has a discontinuous first derivative, and satisfies the following properties:
Based on these considerations, it appears that in the range the MRI is parametrically larger for (where it is of order ) than for (where it is of order ). This is similar to what we found in the perturbative calculation of subsection 2.2. It would be interesting to find a simple toy model of a system with degrees of freedom, in which the MRI between two subsystems is of order , but the MI is only of order .
4.4 MRI for
In this subsection we will begin by expressing the mutual Rényi information in a general CFT in terms of its torus partition function. Using this expression, we will calculate the order- part in a general holographic CFT, finding that—like the RT prediction (44) for —it is analytic except at , where it has a discontinuous first derivative. We will then show that is precisely the same function in large- symmetric-product theories, supporting the idea of universality (i.e. theory-independence in the large- limit) proposed in the previous subsection.
4.4.1 General CFTs
We begin by applying the formula (32) for . In the orbifold theory, there is a unique twist operator . Lunin and Mathur  showed that its four-point function is given by121212In terms of Lunin and Mathur’s variables, and . The four-point function of twist fields was computed in  in the case where the underlying CFT is a free scalar field.
where is the partition function for on a flat rectangular torus131313The fermion sign flips explained in footnote 5 imply that fermionic fields should have antiperiodic (NS) boundary conditions on both cycles of the torus. The reason is that, on the double-sheeted plane, in going around either cycle one encounters two sign flips, so no overall flip. The coordinate transformation to the torus introduces a flip, just as when passing from the plane to the cylinder. Hence the partition function is not invariant under the full modular group, but it is invariant under . with modular parameter ; and are related by
As goes from 0 to 1, goes from to 0, with corresponding to (see figure 2). Since
the invariance of the four-point function (47), and hence of the ERE, can be traced to the modular invariance of the torus partition function, . The reason for the appearance of the torus partition function of is that the four-point function of twist operators is the (renormalized) zero-point function on the two-sheeted Riemann surface with a branch cut connecting to and another one connecting to , which is a torus with complex structure . The Weyl transformation that flattens it, when plugged into the Liouville action, leads to the prefactor .
The first term in (4.4.1) is ( times) the free energy of on a circle of unit circumference at temperature . (Note however that the basic cycles of this torus, which we interpret as space and Euclidean time directions when we speak of the free energy, are not the same as the space and time directions of the Euclidean plane where the theory was originally defined, whose double cover is . Rather, the spatial circle of the torus encircles the points and , staying on one sheet, while its Euclidean time circle encircles and , crossing each branch cut once.)
Let us consider the expansion of (4.4.1) for small values of , where .141414Defining , the expansions of the theta and eta functions for small are as follows: , Hence