A General Proof of the Quantum Null Energy Condition
We prove a conjectured lower bound on in any state of a relativistic QFT dubbed the Quantum Null Energy Condition (QNEC). The bound is given by the second order shape deformation, in the null direction, of the geometric entanglement entropy of an entangling cut passing through . Our proof involves a combination of the two independent methods that were used recently to prove the weaker Averaged Null Energy Condition (ANEC). In particular the properties of modular Hamiltonians under shape deformations for the state play an important role, as do causality considerations. We study the two point function of a “probe” operator in the state and use a lightcone limit to evaluate this correlator. Instead of causality in time we consider causality in modular time for the modular evolved probe operators, which we constrain using Tomita-Takesaki theory as well as certain generalizations pertaining to the theory of modular inclusions. The QNEC follows from very similar considerations to the derivation of the chaos bound and the causality sum rule. We use a kind of defect Operator Product Expansion to apply the replica trick to these modular flow computations, and the displacement operator plays an important role. Our approach was inspired by the AdS/CFT proof of the QNEC which follows from properties of the Ryu-Takayanagi (RT) surface near the boundary of AdS, combined with the requirement of entanglement wedge nesting. Our methods were, as such, designed as a precise probe of the RT surface close to the boundary of a putative gravitational/stringy dual of any QFT with an interacting UV fixed point. We also prove a higher spin version of the QNEC.
- I Introduction and summary
- II Background
- III Replica trick for the modular Hamiltonian
- IV Evaluation of the modular Hamiltonian
- V Modular flow
- VI General properties of
- VII Loose ends
- VIII Discussion
- A Half-sided modular inclusions
- B More details on defect operator spectrum
- C Multi-replica operators
- D Modular flow from the replica trick
- E Double modular flow from single modular flow
- F Higher spin and
I Introduction and summary
Bounds on the stress tensor of a QFT have important consequences for the semi-classical limit of gravity - using these bounds we can rule out pathological spacetimes that might arise when coupling gravity to matter in the form of this QFT. Typically these pathologies have their root in some form of causality violation of the resulting spacetime. However naive bounds that apply to classical field theory, like the local Null Energy Condition (NEC) , are violated quantum mechanically. The NEC was central to the classical proofs of the black hole area law Hawking (1971), singularity theorems Penrose (1965), topological cencorship Friedman et al. (1993), etc Hawking and Ellis (1973). In order to generalize these proofs to the quantum regime several new energy conditions on have been conjectured with various degrees of non-locality Borde (1987); Klinkhammer (1991); Wald and Yurtsever (1991); Verch (2000); Ford and Roman (1995); Fewster (2012); Wall (2010, 2013, 2012). Despite their origin in gravitational physics these generalizations often have a limit which applies directly to the QFT in curved space, and it is furthermore interesting to study their validity and consequences even in flat Minkowski space Hofman and Maldacena (2008); Hofman (2009). Perhaps most excitingly their validity almost always relates to bounds on the behavior and manipulation of quantum information in the QFT Casini (2008); Blanco and Casini (2013), further strengthening the important connection between gravity and quantum information.
Recently two different proofs of the Averaged Null Energy Condition (ANEC) in any UV complete QFT have appeared in the literature Faulkner et al. (2016); Hartman et al. (2016). In Minkowski space this is the positivity constraint:
The proof Faulkner et al. (2016) works in Minkowski space and for null integrals along a complete null geodesic generator of the horizon of a static black hole. This is sufficient to rule out using these black holes as traversable wormholes Gao et al. (2016); Graham and Olum (2007). The proof of the ANEC in Hartman et al. (2016) was based on causality considerations applied to the two point function of a probe operator evaluated in the state . In Faulkner et al. (2016) the proof was based on monotonicity of relative entropy under shape deformations of an entangling region which is taken to be a null deformed cut of the Rindler horizon . This in turn imposed a condition on the negativity of the shape deformations of vacuum modular Hamiltonians for these entangling cuts which was then proven to be related to the ANEC operator in (1). Indeed it does not take much more work to use these modular Hamiltonians combined with monotonicity of relative entropy to prove the quantum half-ANEC:
which is then an important ingredient in the semi-classical proof of the Generalized Second Law for black hole mechanics Wall (2012). It turns out however that we need an even more local constraint in order to generalize other classical gravitational theorems, such as the Bousso/covariant entropy bound Bousso (1999), to the semi-classical regime. One such constraint is the Quantum Null Energy Condition (QNEC) Bousso et al. (2016a, b) which is logically more general than the ANEC and the half-ANEC, implying both of these if it is true. For certain special cases the QNEC is the functional derivative , or shape deformation, of (2). Since we don’t expect a second order shape deformation of relative entropy to be constrained in sign for a general quantum system, if we are to prove the QNEC it will involve an essentially new ingredient beyond monotonicity. As we will see the QNEC follows from a more fine grained notion of causality compared to the results in Hartman et al. (2016), where we make use of the probe operator , but at the same time add the action of modular Hamiltonians into the mix.
We will prove a (slight) generalization of the QNEC. This is essentially an integrated version:
where and are two spatial regions of a fixed Cauchy slice. They should satisfy the inclusion property where is the domain of dependence of . The parameterizes a null line passing from the entangling surfaces to located at fixed and locally labels the coordinates along the entangling surface with labeling null coordinates transverse to the surface. We will require the surfaces to be locally stationary at the point , which means the extrinsic curvature in one of the null directions vanishes at as well as a sufficient number of its derivatives. Other than this we only require that the domains of dependences and are non timelike separated, so for example could have multiple disconnected components with non trivial topology etc.
The essential idea for the proof is to study the following correlator:111If the state of interest is not pure, then represents the purifcation of the state in a doubled Hilbert space.
where the two probe operators are inserted in the region and respectively. We then act on these operators with modular flow using the (full) modular Hamiltonians defined for the sub regions respectively and for the state . The modular Hamiltonian can be defined abstractly (with some technical assumptions on the state ) with respect to the algebra of bounded operators within the region, and Tomita-Takesaki theory Takesaki (1970); Haag (2012) guarantees that the modular flowed operators for real are still contained within the algebra of operators of that region. More constructively the modular Hamiltonian is related to the reduced density matrix of restricted to and is sometimes referred to as the entanglement Hamiltonian, and modular flow simply involves time evolution using this Hamiltonian. In this paper we will mostly be interested in the full modular Hamiltonian which is , and it is important to note that for the defining state.
For some special cases modular evolution can be local, as for example the case where is a half-space/Rindler cut in Minkowski space and for the vacuum Bisognano and Wichmann (1976). The action of is then a boost holding fixed . This is our definition of the term in the denominator of (4) where and are half space cuts such that and are parallel to and at respectively. Later in the paper we will slightly refine this definition of the denominator to allow for to be null cuts of the Rindler horizon agreeing locally with . In this case the denominator can be constrained using the so called theory of half-sided modular inclusions Wiesbrock (1993); Borchers (1996); Araki and Zsidó (2005); Borchers (1992, 1995) - the computation of which has some overlap with the recent paper Casini et al. (2017).
Since the probe operators are initially spacelike separated they commute, and since and are spacelike seperated the modular evolved operators will also commute:
for real . This fact, pertaining to causality, translates into a statement about analyticity of in the complex plane. Indeed a generalization of Tomita-Takesaki’s modular theory Buchholz et al. (1990); Wiesbrock (1993); Borchers (1995); Araki and Zsidó (2005) establishes the analytic extension of in the strip .
We will use this analyticity to prove the QNEC. Roughly speaking, if the QNEC were violated the modular evolved operators could exit their respective causal domains and cause a branch cut along giving a non-zero commutator (5). Since we are using modular evolved operators this is a subtle violation of causality, but one which makes sense in the context of AdS/CFT. Indeed it was recently shown that modular evolved operators give a way to reconstruct bulk operators localized within the entanglement wedge associated to some boundary sub region such as Jafferis et al. (2015); Faulkner and Lewkowycz (2017). The entanglement wedge is believed to be the largest region containing information reconstructible using operators acting on the sub Hilbert space in the QFT Czech et al. (2012); Headrick et al. (2014); Almheiri et al. (2015); Dong et al. (2016), and so these bulk regions are causally constrained by the boundary theory. Additionally the QNEC was proven for theories with an AdS/CFT dual using exactly this causality requirement Koeller and Leichenauer (2016); Akers et al. (2016); Fu et al. (2017a) - more specifically the entanglement wedge nesting (EWN) requirement. As we will explain there is a very precise sense in which this paper can be thought of as studying subtle QFT causality requirements via the causality properties of a gravitational dual with an emergent radial direction Maldacena (1999). Most QFTs do not have a classical gravity dual, but in some sense since we will only be studying properties of the gravitational system close to its boundary, the real stringy/strongly interacting nature of the dual gravitational system is suppressed. Our results thus identify (5) with the QFT equivalent of EWN.
Of course having set up the problem it may seem hard to compute in any useful way, that is, retaining full generality over the state as well as the generality of the entanglement cuts and . This is because are complicated non-local operators. We will manage to make progress here using a lightcone limit for the operators , as pictured in the setup of Figure 1, where the operators are separated in the direction by an amount and as is held fixed. This is a very similar limit to that considered in the causality ANEC proof Hartman et al. (2016) although now in the presence of entanglement cuts through points collinear with the operators. In this limit we can use the replica trick to compute properties of the general modular Hamiltonians , coupled with a defect lightcone OPE argument. The defect is the non-local co-dimension 2 twist operator of the -replicated theory and a large part of our computations involve controlling the spectrum of local operators on the defect (referred to as defect operators) in the limit.
For large (but not too large as to move us out of the lightcone limit) we find the small but growing correction term:
where is the QNEC object in (3). Note that we have introduced a quantity we call via its usual relation to in holographic theories (101) in units where . There is no need for to be small. We have also defined in terms of the kinematics of the operator insertions which exactly plays the role of the radial coordinate in an emergent AdS. Here is the coordinate distance between and and must be true. All we have to do now, taking inspiration from the chaos bound Maldacena et al. (2016) and causality bound Hartman et al. (2015) stories, is prove that along the lines in the complex -strip. Analyticity then allows us to extract from (6) as an integral over along these same lines which is then constrained to be positive thus proving the QNEC.
Let us add one more comment on the meaning of . Note that the vacuum two point function, in the denominator of (4), can be computed using the fact that are simple boosts and one finds for large :
The leading correction to in (6) can be understood as resulting from a shift:
in the appearing in (7). This is suggestive of a tendency towards shifting the branch cut singularity in the two point function onto the real axis if for large enough . This is not a precise argument since the shift is only important when the small correction in competes with , but that’s okay since the precise argument was given above. However it allows us to identify the gravitational time delay/advance that we should look for in the bulk, and we will identify this by (very slightly) generalizing the arguments of Koeller and Leichenauer (2016) which proved the QNEC for holographic theories using entanglement wedge nesting (EWN) of the RTRyu and Takayanagi (2006)/HRTHubeny et al. (2007)/quantum extremalEngelhardt and Wall (2015) surface near the boundary.
Many of the properties of the function are the same as the function defined in Maldacena et al. (2016) for studying chaos using an out of time order four point function for two different operators in a thermal state. The analogy is strengthened by taking the thermal state to be that of the Rindler state, and time to be generated by boosts using the Rindler Hamiltonian. This is then the same setup as Hartman et al. (2016) for proving the ANEC using causality - although the equivalent function is constrained in different kinematic regimes - determined by how large is and whether the operators are in a lightcone limit for the causality bound (large ) or the Regge limit (even larger , but not as large as the scrambling time ) for the chaos bound. The analog of our setup would evolve not with the Rindler Hamiltonian but with the complicated modular Hamiltonian of the state and now reduced to two different entangling regions. From this point of view our paper represents a generalization of the chaos and causality bound setup, however we have not yet explored the extent to which we can apply this setup usefully to non-relativistic quantum systems and generic perturbed thermal states. We also do not have much to say about the equivalent “even larger” regime of in a large- theory analogous to the Regge limit leaving these fascinating generalizations for future work.
The paper is organized as follows. In Section II we start with background on various known results that will be useful to us, including a discussion of the holographic proof of the QNEC as well as a discussion of geometric modular Hamiltonians and their action on local operators. We note an interesting relation to the well studied theory of half-sided modular inclusions. In Section III we discuss the use of the replica trick to compute properties of general modular Hamiltonians. We then consider the defect OPE which is necessary to carry out the replica trick computation. This includes a discussion of possible local defect operators that arise when . In Section IV we compute the matrix elements of the modular Hamiltonian in the state excited by in the lightcone limit. In Section V we use this result to find the action of modular flow in a perturbative expansion with respect to the lightcone limit which gives the result (6). In Section VI we detail the general properties of which lead to the QNEC. In Section VII we discuss some loose ends, including an understanding of local geometric contributions to entanglement entropy that can contaminate the QNEC quantity and thus invalidate the bound for non-stationary entanglement cuts. We also discuss a higher spin version of the QNEC, generalizing the higher spin version of the ANEC proven in Hartman et al. (2016). We conclude in Section VIII with several possible extensions. Some computations and details are relegated to the Appendices.
In this section we collect some known results from the literature that we will make use of throughout the paper. We will bring our own perspective to these results relevant to our discussion. Let us set the stage by setting up the problem we wish to study more precisely than in the introduction.
ii.1 Setup and conventions
We take the metric to be flat:
where we use and for null coordinates adapted to an entangling surface which passes through the point (we have set relative to the introduction!). We use both and to maximize our variable options. Wick rotation is given by such that and .
The first entangling surfaces will be defined close to via
such that is a space like region ending on to the “left” - roughly speaking within the wedge close to . The other entangling surface is displaced in the direction at :
where again is a space-like region to the “left” of this cut. This description could break down far from the null line passing through both and at , but the details far away will not play a role in our computations. For now we will be agnostic to the exact shape of the entangling regions, except to require that the . We will later discover that some further local conditions are required in order to claim the QNEC bound - these are similar conditions to those discussed in Koeller and Leichenauer (2016), that locally the entangling cuts should be stationary with for sufficiently many derivatives. These further conditions contain the special case where and are general null cuts of the Rindler horizon such that and are left arbitrary (except for the inclusion requirement for all .) However our results are much more general than this.
To lighten the notation we will use the following:
where and and we have made explicit which state the modular Hamiltonian refers to. For example where is the CFT vacuum. We will often suppress the label on the operator insertions. We also suppress which should be understood from the superscript label. For most of the paper, except in Section VII and below, we will take the undeformed regions to be flat Rindler cuts that agree with and at . We turn now to a description of the modular Hamiltonians for these regions in vacuum.
ii.2 Vacuum modular Hamiltonians and modular inclusions
We start by describing a special class of modular Hamiltonians, these are the so called local modular Hamiltonians which apply for relativistic vacuum states and for simple flat Rindler cuts. Modular flow in this case is just a local boost around the entangling surface and the modular Hamiltonians for two flat cuts of the same Rindler horizon form an algebra which is the one that naturally arrises in the theory of half-sided modular inclusions Wiesbrock (1993). This case applies to the modular Hamiltonians for the two uniform Rindler cuts in vacuum which are important for evaluating in the lightcone limit. The action is simple:
and for :
Furthermore these satisfy an algebra:
where is the translation operator in the direction . This algebra is 2 dimensional and isomorphic to the algebra associated with the affine group . For the pattern of modular flow in the correlator we find:
where the generates a translation in the null direction . Note that is clearly a positive operator via vacuum stability, which it must have been due to the negativity constraint on modular Hamiltonians under shape deformations Blanco and Casini (2013).
In this paper we will work in a limit where the modular Hamiltonians are well approximated by these modular Hamiltonians plus computable corrections. In Section VII we will find in order to account for some of these corrections it is useful to consider a more general class of vacuum modular Hamiltonians, the form of which was only recently elucidated Faulkner et al. (2016); Koeller et al. (2017); Casini et al. (2017).222These modular Hamiltonians for general QFTs are consistent with those of free theories which were worked out by A. Wall Wall (2012) based on light front quantization. This class derives from arbitrarily shaped null cuts of the Rindler horizon in vacuum. An important result now comes from the theory of half-sided modular inclusions which can be used to prove that the algebra (17), suitably generalized, continues to apply in this more general case.
One proceeds in two steps, the details of which are given in Appendix A. Firstly we recall that half sided modular inclusions apply to the case where is an arbitrary null cut of the Rindler horizon satisfying where is a uniform Rindler cut ending on with an associated local modular Hamiltonian. The region then has the nesting property that
These conditions are enough to prove the result that the algebra defined in (17) continues to hold with the replacements:
where the notation defines the modular Hamiltonian as a functional of the specific entangling cut of the Rindler horizon (recall that and for now.) Intriguingly one way to prove this is by studying a very similar correlation function to that which appears in namely:
Applying the nesting property (19) and positivity properties of Wiesbrock (1993) argued for an analytic extension that is periodic and holomorphic in the thermal strip: with . Similarly is necessarily bounded in this strip and the only way to satisfy all these conditions is if is a constant independent of . Expanding about one derives the algebra in (20). 333One approach that we tried in order to prove the QNEC was to study in the case where not all the conditions above are satisfied - in particular the nesting property of boosted regions (19) is generically going to fail for non-local modular Hamiltonians associated to a non vacuum state . We did not get this approach to work and instead settled on the modular flow pattern in .
Continuing on, this algebra allows us to find an expression for the null deformed modular Hamiltonian in terms of an integral of the stress tensor:
This was recently shown in Casini et al. (2017) and we will give a slightly different proof of this in Appendix A. The main ingredients in our proof are the algebra (20) as well as the recent computation of linearized shape deformations to the Rindler modular Hamiltonian Faulkner et al. (2016) which allows us to fix the modular Hamiltonian for small . This result proves the conjectured answer in Faulkner et al. (2016) that the higher order corrections in the expansion are essentially trivial.
Note that these new modular Hamiltonians are not local in the sense that they do not generate local flows. With the result (20) in hand one can then just go and calculate the algebra when and are both deformed null cuts (see Appendix A and Casini et al. (2017)):
While the action of these modular Hamiltonians is not local, it will become local when acting on operators in the lightcone limit (close to the Rindler horizon.) This should allow us to compute the action of the vacuum modular Hamiltonian perturbatively in the lightcone limit, which goes into computing . As we will see the details of this computation will not be important, excepting that they satisfy the modular inclusion algebra (23).
We make a final point returning to the simple uniform null cuts. The nested boosts relevant to that we computed in (18) tells that for large we simply have a null translation by a small amount . This means we can take large without the operators exploring too much of the spacetime and this will be important for us to claim the more general QNEC results. We can also understand what happens if we move the two entangling cuts away from each other slightly in the direction by an amount . The inclusion property is now only true if . These modular Hamiltonians are not constrained by the algebra of half sided modular Hamiltonians, but since here they are simple boosts we can just explicitly compute the modular flow. Consider the flow:
which for large still gives an operator shifted in the null direction, but the operator is now moving to large . If we plug this into the vacuum correlator in the denominator of and expand for small we have:
So unless we set we find a small but growing term which should be compared to (6) and (8). Without making this later expansion the two operators will eventually become time-like separated from each other if . Since in this case the two domains of dependence and are not causally disconnected there is no issue with the necessary appearance of a branch cut in along . However this gives us some intuition for the growing QNEC term we are claiming for more general modular Hamiltonians and states. Consider a holographic theory. If the QNEC is violated then as one moves slightly inwards in the holographic direction the two bulk entanglement wedges for and will come into causal contact. Since the JLMS Jafferis et al. (2015) result tells us that boundary modular flow equals bulk modular flow, a similar algebra for modular Hamiltonians should now apply except in the bulk and now determined via the relative position of the RT surface (elucidated further in the next subsection). Near the boundary we can approximate the cut with a bulk Rindler cut except slightly deformed due to the movement of the RT surface in the direction as we move inwards from the boundary. From this consideration we expect to find the same growing term that we found by shifting on the boundary. In particular the wrong sign which applies when is an indication that the entanglement wedges are coming into causal contact. We now turn to a holographic calculation demonstrating that indeed this bulk causality consideration is determined by the sign of the QNEC quantity .
ii.3 Holographic proof and EWN
Let us attempt to identify the gravitational time delay/advance directly in the bulk. In this section we assume our CFT has a description in terms of a weakly coupled classical Einstein gravity theory. This is only true for a small class of theories, but these theories allow us to develop intuition for the general case. The results here are not new and were originally worked out in Koeller and Leichenauer (2016), relating the QNEC in holographic theories to entanglement wedge nesting (EWN.) The EWN property states that if two boundary regions satisfy then the dual entanglement wedges must satisfy the same condition. The entanglement wedge of a region is the domain of dependence of the spacelike region located between on the boundary of AdS and the RT surface . This requirement can be understood as being basic to the program of entanglement wedge reconstruction Akers et al. (2016).
We will work out a slight generalization for the integrated version of the QNEC in (3). For simplicity we will ignore many complications due to extrinsic curvature effects and effects arising due to a relevant deformation which takes us from the UV CFT to a more general QFT. These more complicated effects were discussed carefully in Koeller and Leichenauer (2016).
The metric solving Einstein’s equations near the boundary of AdS has a Fefferman-Graham expansion:
Similarly the two RT entangling surfaces parameterized via have an expansion:
where we find this form by solving the extremal surface condition close to the boundary. Here are not fixed by the asymptotic boundary conditions. They are state () dependent and can be related to the CFT stress tensor and the shape variation of the holographic EE respectively:
The later relation may be less familiar to the reader, but can be thought of as the usual Hamilton-Jacobi relation between conjugate coordinates in the sense where time and the area of the RT surface or is like the action holding fixed the boundary value : . We will take for simplicity to suppress additional leading terms that would arise in the expansion of multiplying various local extrinsic curvature invariants. We also remind the reader that and .
Now we consider a high energy particle moving near the boundary of AdS in the direction along a null geodesic with approximately fixed and paramaterized by the coordinate . To leading order we only need to track the small change . This particle will be analogous to our probe. To see if the two entangling wedges are causally disconnected we consider this null geodesic to pass through the point at and fixed. The particle then picks up a delay in the direction as it propagates to :
Comparing this new coordinate to the position of the RT surface we find this is determined by the QNEC quantity:
The lightcone limit allows us to study particles propagating near the boundary of AdS and weakly interacting via graviton exchange with the state . This turns out to be a useful picture in any interacting CFT Fitzpatrick et al. (2014, 2013); Komargodski and Zhiboedov (2013), and this was the original motivation for studying the lightcone limit in this context. The delay one extracts from this picture is the total delay of the particle integrated over all boundary times - causality then imposes the ANEC constraint. The boundary theory ANEC is in this way related to the Gao-Wald causality condition on the bulk Kelly and Wall (2014), that the fastest path in the full spacetime between two null separated points on the boundary is a null line on the boundary. By studying causal curves that reach into the bulk and are sensitive to the boundary theory stress tensor term in the metric the authors Kelly and Wall (2014) used Gao-Wald to prove the ANEC. This does not usefully constrain a more local version of the NEC because propogating the particle from the boundary into a fixed causes a large delay which swamps the delay/advance due to the term in the metric. This can only be removed by taking the two points on the boundary to be infinitely separated in the null direction. Entanglement wedge nesting is a much more fine grained version of causality that allows us to directly study the gravitational delay/advance at a fixed coordinate via the introduction of the entangling surfaces. And it turns out the way to extract this from the boundary theory is with the correlator in .
ii.4 Tomita-Takesaki theory
In order to prove various (non perturbative) properties of we will need to have a better understanding of modular flow for a more general class of states than for the vacuum of a QFT. For now we will present (a brutalized version of) the abstract algebraic discussion of Tomita-Takesaki theory, see for example Haag (2012). This will pertain to the action of a single modular flow - the double modular flow will be discussed later in Section VI. The idea is to consider a von Neumann algebra of bounded operators associated with some local region in spacetime say . If we additionally have a state on the total Hilbert space that is cyclic and separating for - meaning that is dense in the total Hilbert space for all operators and that cannot annihilate , then one can define the following modular operators:
where is anti-unitary and is positive and Hermitian, but generally unbounded. To make contact with the (full) modular Hamiltonian one writes where now will not be a positive operator. One can then show that:
where is the commutant which is then the bounded operators associated to the region .
Physically, cyclic and separating just means that the state has a large amount of entanglement between and and we expect that all reasonable QFT states one might consider have this property. For the case of the vacuum the Reeh-Schlieder theorem Schlieder (1965) rigorously establishes this fact. In a quantum system with a finite dimensional Hilbert space this condition would be equivalent to the statement that the reduced density matrix (for a finite quantum system ) has full rank and so is invertible Papadodimas and Raju (2014), however it will be important to acknowledge the fact that in an infinite quantum system, since is unbounded we have to carefully specify the domain on which it acts. For example it is known that is generally in the domain of for and is generally in the domain of for Haag (2012).
An important consequence of this structure is an abstract version of the KMS condition. To understand this we consider the correlator (which is a baby version of ):
which can be analytically continued into complex in the strip . On the upper/lower edge we have:
where . The difference across the cut, which arises in the strip after we identify at , is just the commutator where . Analyticity along is simply related to the fact that the original operators and commute.
We can give a less rigorous discussion of these results by appealing to the analogy with thermal systems. For example, if the subspace had a trace, we can then replace the correlators as:
where we have set . This expression demonstrates where the strip comes from. In an infinite dimensional system the sum over intermediate eigenstates of is not guaranteed to converge outside of this range. In our case there is no trace, however we could regulate things around the entangling surface with a hard wall cutoff in order to introduce a trace.
Moving forward we want to study the situation where there are now two algebras with a common cyclic and separating state and the inclusion property . This is harder to study but we can use various results from the literature. We will explain these in a later section.
Iii Replica trick for the modular Hamiltonian
Our computation of will now begin in earnest. Our first task is to compute matrix elements of sandwiched between excited by the operator insertions. To do this we will need to use the replica trick.
Previous discussion of using the replica trick to compute the modular energy of excited states has appeared in Lashkari (2016) (also Srosi and Ugajin (2016); Ruggiero and Calabrese (2017)). This was then used to study the modular energy in 2d CFTs. While we will take a very similar approach there will be an important difference. We would like to write the answer in terms of twist operators in the orbifold theory . It is not totally clear this is possible since, as noted in Lashkari (2016), the replica trick in this case explicitly breaks the symmetry which cycles through the replicas. For this reason the results in Lashkari (2016) are left in the form of correlation functions on n-sheeted branched coverings without the symmetry. On the other hand the orbifold theory is much more under control since we can use standard results about defect CFTs Bill et al. (2016); Gliozzi et al. (2015); Gaiotto et al. (2014); Bill et al. (2013); Liendo et al. (2013) in order to make progress with computations. This will be the main technical difficulty that we have to overcome here.
iii.1 Replica trick
The replica trick is a way of computing properties of the operator using the limit:
This is useful because it is sometimes possible to compute traces over for integer using a path integral. The limit is then only achievable once an analytic extension is found from integer to complex . While this is usually subtle the replica trick has yielded many powerful results relating to entanglement entropy in QFT Callan and Wilczek (1994); Calabrese and Cardy (2004); Casini and Huerta (2009); Lewkowycz and Maldacena (2013).
We will firstly be interested in simply evaluating the half modular Hamiltonian: , thought of as an operator on the total Hilbert space, between matrix elements of the defining state excited by local operator insertions. This is not a totally well defined object in the continuum and so will only be an intermediate step towards computing the full version: which is well defined. We will not be completely explicit about how we regulate to define it, but we will assume this regulator allows us to define a trace over the various tensor factors in the Hilbert space.
which we can write as a trace over the Hilbert space
The trace in (42) can be computed using a path integral. We first write a path integral representation of by integrating over Euclidean space with a branch cut running along and different boundary conditions above and below used to represent the density matrix. To be concrete let us take the state to be defined via local operator insertions which we will also denote as (perhaps smeared appropriately). We place two operators and on the Euclidean section above and below the Cauchy slice on each replica The details of this state and the Euclidean path integral used to construct these states will not matter, except to note that for now we take to be a pure state. We will extend the proof to the case of mixed states in Section VII.
We now write a path integral representation for which differs from by the additional operator insertions within the path integral. We imagine slicing open the path integral along radial lines emanating outwards from and integrating forward in a clockwise angular direction444We work clockwise because the entangling region starts on the left of the cut. This results in some funny minus signs, such as the Euclidean holomorphic coordinates close to the entangling surface satisfies where increases in the clockwise direction and is the radius with specifying the region. We wick rotate as . - so the ordering of operator insertions (including the operators that create the state) in the Hilbert space language is always angular ordering.
Putting the various density matrices together and tracing we can write the answer as a correlation function on a non-trivial manifold which consists of copies/replicas of the dimensional Euclidean space which are cut and joined cyclicly along . Sometimes we will refer to this space as a branched manifold. The state operator insertions and both arise on each replica and live on the same single replica. Then:
where means insert the operator symmetrically on each replica. This is not yet an orbifold correlation function. The branched manifold can be alternatively represented by using a co-dimension 2 (non-local) twist defect operator living on :
where the orbifold/gauging of by the discrete cyclic permutation symmetry is necessary in order to remove the existence of extra conserved stress energy tensors from the new replicas - thus allowing us to apply standard CFT considerations to the orbifold theory on the original (unbranched) manifold but now in the presence of a co-dimension twist operator.555Orbifolds of 2d CFTs are well studied Polchinski (1998). The higher dimensional versions have received less attention, see Belin et al. (2017) for a recent discussion which however is complicated by non-trivial topology. We can literally view the resulting theory as a discrete gauging of the replica symmetry, by coupling the theory to a continuum version of a discrete gauge theory as reviewed in Banks and Seiberg (2011). Indeed the state operator insertions are clearly symmetric under the symmetry so they are genuine orbifold operators. To unclutter the discussion moving forward we will often suppress the existence of these operators and consider them part of the definition of the twist operator where the later replacement is for further decluttering purposes.
Unfortunately the operators and are quite clearly not orbifold operators, so (44) is not yet well defined. We cannot simply symmetrize each individual or operator over the action of the group since the two operators are necessarily inserted on the same replica. Thus we consider to be a bi-local operator with a non-local string attached whose sole job is to keep track of the relative position of the operator on the different replicas, say when we move one of the operators around the twist defect relative to the other.We can then -symmetrize this bi/non-local operator by summing this composite over the different replicas. We take this as our definition of (44) which we rewrite as:
where the superscript notation specifies which replica the operator descends from on . In Section VII we will discuss a more precise definition of this bi-local operator where the string attached is actually a sum over Wilson lines for the orbifold gauge group . For now we note that, due to the non-local nature of this operator, we must pick where we place the branch cuts in the definition of in order to define which local operator lives on which replica (and thus define what we mean by ). Excepting the effective string that remains attached between and and moves past the twist operator interesecting the region , the choice of exactly where we place the branch cut goes away upon moving to the orbifold theory - as it must.
Now we would like to compute by doing a similar replica trick to compute . Notice that the difference here is the positioning of the branch cut. However in the orbifold theory, by definition, there is no knowledge of the position of the branch cut so one might conclude incorrectly that the answer, upon subtraction, is . The reason we find a non-zero answer can be understood since moving the position of the branch cut from yields a different ordering for the bi-local operator . The conclusion is that we can compute the full modular Hamiltonian as:
where for the later non/bi-local operator we have moved the relative to around the twist operator once. That is:
At this point we have now set up the problem. Computing any of these correlation functions seems difficult. We aim to make progress by bringing the operators close to the twist defect and using a defect operator product expansion (dOPE.) We turn to this now.
iii.2 Defect OPE
If we take the pair of operators close to the defect, say at a point along the defect, we can imagine zooming out and replacing these with a sum over local defect operators on . That is . Note that we might have done this in two steps, first replacing the pair of operators by a sum of ambient local orbifold operators using the regular OPE, then bringing these operators close to the defect. This would look roughly like:
One might even think that there is another way to do this - first bring one of the operators close to the defect and expanding this in terms of defect operators. However this later method is not possible because individually is not an orbifold operator. Thus there is really only one channel we can evaluate this correlator in.666In a more familiar setting, if the later channel had been allowed, the equality between these two expansions would be akin to a bootstrap constraint on the defect CFT spectrum. See Liendo et al. (2013); Gliozzi et al. (2015); Bill et al. (2016) for recent work on this bootstrap problem. Additionally we will choose to ignore the intermediate step of the ambient OPE involving the sum over in (48) - mostly because it turns out in the limit we can directly compute the dOPE coefficients for this full replacement without having to sum over an infinite set of intermediate operators.
We can compute the dOPE coefficients as follows. Firstly note that since the replacement is done locally we could have done the same replacement on a twist defect within a totally different setup but using the same replacement coefficient .777We are lying a little here. It turns out that is sensitive to the local extrinsic curvature of the defect at , in analogy to regular OPE coefficients being sensitive to local curvature invariance of the metric if we make an OPE expansion of two local CFT operators in curved space Hollands and Wald (2010). We will fix this lie in Section VII. Thus let us consider a flat/planar twist defect defined in the vacuum of a CFT and living along . To extract a particular OPE coefficient we must also insert some other defect operator far away from at a point . All of this still in the presence of the bi-local . Now in this new setup take the ’s close to the defect simultaneously and make the same replacement we did above:
where in the notation we established above