Time Evolution after Double Trace Deformation

In this paper, we consider double trace deformation to single CFT${}_2$, and study time evolution after the deformation. The double trace deformation we consider is nonlocal: composed of two local operators placed at separate points. We study two types of local operators: one is usual local operator in CFT, and the other is HKLL bulk local operator, which is still operator in CFT but has properties as bulk local operator. We compute null energy and averaged null energy in the bulk in both types of deformations. We confirmed that, with the suitable choice of couplings, averaged null energies are negative. This implies causal structure is modified in the bulk, from classical background. We then calculate time evolution of entanglement entropy and entanglement Renyi entropy after double trace deformation. We find both quantities are found to show peculiar shockwave-like time evolution.


Introduction and Summary
Recent studies revealed that, introduction of non local double trace deformation can render non traversable wormholes traversable [1]. Wormholes considered in these studies are two sided black hole connecting two asymptotically AdS regions. In AdS/CFT the geometry is dual to thermofield double state on CFT L ⊗ CFT R . The non local double trace deformation added to the CFT action is where operator O L/R lives in CF T L/R . Such deformation causes null shockwave in the bulk, whose null energy can be positive or negative according to the sign of coupling g. When the averaged null energy is negative, it was found that the shock wave opens up traversable wormhole in non traversable wormhole, even if g is infinitesimally small. In this sense, wormholes in two sided black hole are barely non-traversable. Applications and generalizations of these double trace deformations result in interpretation as quantum teleportation [2], exploration into black hole interior [3] [4], and realization of eternal wormhole [5]. Traversability of wormhole is intimately related with violation of averaged null energy condition (ANEC). Averaged null energy condition is a conjecture, that states integral of null energy on null ray must be non-negative in any UV complete QFT. ANEC is proven on Minkowski spacetime and other special cases [6][7] [8] and its generalization QNEC [9][10] [11] [12]. Averaged null energy has simple intuitive interpretation, that it can measure the change of causal structure, when we perturb solution of vacuum Einstein equation by matter stress tensor. When ANE is negative, null ray in unperturbed metric becomes timelike in perturbed metric. Related to this fact, it is known that in classical general relativity, existence of traversable wormhole implies negative ANE. The results of [1] [2] are consistent with ANEC of QFT, since once one applies double trace deformation to the theory, the theory is no longer QFT nor Lorentz invariant, although it is unitary .
Interesting generalization of previous studies on double trace deformation is to consider deformation composed of HKLL bulk local operators [13], not the local operators of boundary CFT. With this deformation, one can manipulate AdS geometry directly, not indirectly through AdS boundary. Stating differently, one can create bulk local objects which are connected by quantum entanglement. Double trace deformation can be obtained effectively when we integrate out fields which are not relevant to the energy scale under consideration [14]. Study on bulk local double trace deformation may help to understand realizing traversable wormhole in pure AdS, or to understand non local effects of higher derivative interactions in string theory.
In order to understand double trace deformations in more detail, it is also important and interesting to study time evolution of field theory observables in general. In particular, it is interesting to understand the dynamics of entanglement entropy and entanglement Renyi entropy, with non local double trace deformation.
In this paper, we study time evolution of bulk AdS and boundary CFT, after double trace deformation. We focus on double trace deformation composed of boundary local operators, as well as HKLL bulk local operators. In both cases, we confirm that with appropriate choice of sign of g, we obtain negative averaged null energy. This implies time advance or time shifts, meaning originally light-like separated points become time like after incorporating the effects of the matter. Further, we confirm that HKLL bulk local double trace deformation can act as point like source of negative null energy. We then study time evolution of general CFT 2 after double trace deformation, by computing leading correction to entanglement entropy and entanglement Renyi entropy. We found the leading correction is universal, in other words it depends only on conformal dimension of double trace deformation. Their time dependence show interesting behavior will be described.
In section 2, we consider pure AdS and introduce double trace deformation which connects antipodal points in the boundary CFT. We will follow the time evolution after the quench by the double trace deformation, focusing on null energy in the bulk. We confirmed that averaged null energy on null geodesic is linear in g so in particular it can be negative.
In section 3, we consider new type of double trace deformation, composed by HKLL bulk local operators. This deformation introduces direct connection between two distinct bulk points. The motivation is to initiate the study of traversable wormholes in spacetimes with single asymptotic region using AdS/CFT. We then study the spacetime structure after the deformation, in particular null energy in the bulk. We found that, as in the case of section 2, averaged null energy on null geodesic is linear in g so in particular it can be negative.
In section 4, we consider time evolution of entanglement entropy and entanglement Renyi entropy after double trace deformation. To calculate entanglement entropy, we use known expression of modular Hamiltonian. And for Renyi entropy, we use Replica trick. In both cases, the leading corrections from the double trace deformation to vacuum are proportional to g. This means one can increase or decrease the entanglement of subsystems by double trace deformation.
In the final section, we explain implication of our results and future directions.

Double trace deformation of boundary local operators
In this section, we consider the time evolution of matter in bulk after introducing double trace deformation to boundary CFT. We consider holographic CFT dual to classical gravity on pure global AdS 3 where 0 ≤ ρ ≤ π 2 , −π < θ < π and −∞ < τ < ∞. We deform CFT Hamiltonian by double trace deformation which is designed to connect two distinct points θ = 0 and θ = π at AdS boundary. O is scalar primary operator of the CFT with conformal dimension ∆ = δ +δ. g is a deformation parameter and infinitesimally small, and f (t) is some real function that only depends on t.
Such double trace deformation does not produce non zero scalar fields at the leading order in g, but does produce nontrivial configuration of matter stress tensor in the bulk, which we will study below. From now on, we assume large central charge, so that the bulk gravity is classical Einstein gravity. We also ignore the back reaction to gravity from matter fields. However, we will treat bulk matter quantum mechanically.
We first compute perturbed bulk two point function, using HKLL bulk local operator. Two point function of HKLL bulk local operators satisfies where M 2 = ∆(∆ − 2). Let us first consider time evolution of HKLL bulk local operator. After deformation, it is given by Then the first order change of bulk two point function is given by Figure 1: The geometry of global AdS 3 . Red line is null ray, on which we compute averaged null energy. On blue plane we compute null energy, and null direction is set as that of the red line.
In the final line we used large N approximation.
Once we know the bulk two point function, we can compute expectation value of bulk stress tensor perturbatively. The bulk energy momentum tensor is so the expectation value of stress tensor can be obtained by point splitting method with coincident point divergence subtracted, as follows, where Let us assume f (t) = δ(t), so that the double trace deformation is instantaneous. Then we can obtain analytical expression for bulk stress tensors, using known expression of bulk two point function (see appendix A). In order to explore dynamics of bulk matter, we will focus on T tt and null energy in the bulk, integral of T tt on Cauchy slices, and integral of null energy on null geodesics.
Averaged null energy is defined as integral of null energy on complete null geodesic. We will consider null geodesics passing through the center of global AdS 3 , starting at t = t 0 and θ = θ 0 + π, ending at t = t 0 + π and θ = θ 0 .
for λ < 0 and for λ > 0, where λ is an affine parameter. Let's compute Null energy on those null geodesics  : Plots of null energy T λλ in the bulk on θ = 0 (and = π) plane. U is taken to be θ growing direction. We can observe non zero null energy is emitted from the two points where we inserted double trace deformation. Note that on any null geodesic incoming from θ = π and −π < θ < 0, null energy is negative, when g is positive. This result is ∆ independent. Figure 4: Plots of Averaged null energies on null geodesics when the theory is deformed by boundary double trace deformation. The null geodesic starts at boundary of AdS with θ = π and t = t 0 , then ends at AdS boundary with θ = 0 and t = t o + π. Averaged null energy is divergent and we regularized it using a defined in appendix A. The averaged null energy is proportional to g a 2∆ and the plot is describing the coefficient of g a 2∆ . Note that on any null geodesic incoming from θ = π and −π < θ < 0, null energy is negative, when g is positive. This behavior is ∆ independent.
We can compute right hand side of the equation by point splitting, Let us take f (t) = δ(t) and θ 0 = 0. We plot null energy T λλ in Figure 3 and averaged null energy dλ T λλ in Figure 4. We confirm that when the sign of g is positive, averaged null Energy is negative for −π < t 0 < 0 for any ∆. So the null ray emitted into bulk in this period gets time shift and becomes time like after incorporating stress tensor in Einstein equation. Time delay in general relativity because of null energy conditions are discussed in [16][17] [18]. Note that the averaged null energy is divergent and is proportional to g/a 2∆ , where a is lattice spacing. Naive counting of dimension of g is 1 − 2∆, so averaged null energy has naive dimension 1.

Double trace deformation by bulk local operators
In this section, we consider multi trace deformation using HKLL bulk local operators. Our bulk double trace deformations model quantum operations on bulk tensor network, bulk non local interactions, and bulk micro traversable wormhole. The motivation to study this deformation is to construct wormhole inside space with single asymptotically AdS region, unlike two sided black hole with two asymptotic regions. Time evolution of bulk after introduction of bulk local excitations was discussed in [19] [20]. In this paper, we compute time evolution of null energy and averaged null energy, and find that both of them can be negative.

Single trace deformation
To start with, we consider single trace deformation to Hamiltonian by HKLL bulk local operator. Let's assume the bulk local operator sits at the center of AdS. The deformation is where Φ bulk is HKLL bulk local operator. The vacuum expectation value of bulk local field is then given by We assume f (t) = δ(t), then we can compute expectation value of bulk energy momentum tensors.The leading term of null energy is found to be proportional to g 2 with positive coefficient as shown in Figure 5. So null energy is positive everywhere and there are no violations of ANEC in this case.

Double trace deformation
In this section, we consider double trace deformation using HKLL bulk local operators, in order to model traversable wormhole connecting bulk points. In this case the deformation operators live inside bulk, not on AdS boundary. We consider deformation of Hamiltonian where Φ bulk is HKLL bulk local operator. The computation of null energy goes in the similar way Figure 6: Plots of null energy T U U in the bulk on θ = 0 (and = π) plane. U is taken to be θ growing direction. We can observe non zero null energy is emitted from the two points where we inserted double trace deformation. Note that on any null geodesic incoming from θ = π and −π < θ < 0, null energy is negative, when g is positive. This fact is ∆ independent. The left figure is for ∆ = 1/3 case, and the right figure is for ∆ = 2/3 case.
as in the previous section. The expression for perturbed bulk two point function is which can be computed by analytical continuation of bulk two point functions [15]. Then, by using point splitting, we can compute null energy and averaged null energy in the bulk. The results are shown in Figure 6 and Figure 7. We confirm that when the sign of g is positive, averaged null Energy is negative for −3π/4 < t 0 < −π/4 for any ∆. So the null ray emitted into bulk in this period gets time shift and becomes time like after incorporating stress tensor in Einstein equation. Note that the averaged null energy is divergent and is proportional to g/a, where a is lattice spacing. Naive counting of dimension of g is 0, so averaged null energy has naive dimension 1.
One can check that when the HKLL operators approach to the boundary, then identical behavior as that of boundary local operators can be observed, as it should be. Figure 7: Plots of Averaged null energies on null geodesics when the theory is deformed by bulk double trace deformation. The null geodesic starts at boundary of AdS with θ = π and t = t 0 , then ends at AdS boundary with θ = 0 and t = t o + π. Averaged null energy is divergent and we regularized it using a defined in appendix A. The averaged null energy is proportional to g a and the plot is describing the coefficient of g a . Note that on any null geodesic incoming from θ = π and −π < θ < 0, null energy is negative, when g is positive. This behavior is ∆ independent.

CFT analysis
In this section, we consider time evolution of entanglement entropy and Renyi entropy in general two dimensional CFT on Minkowski space. Entanglement entropy has been studied from various viewpoints, including from CFT [21] [28] with non local deformation. Perturbative computation of entanglement entropy with time dependent relevant deformation is discussed in [29]. Entanglement entropies of non local theories are discussed in [30] [31]. We consider time evolution of vacuum state after acting double trace operation e −igO(0)O(L) . The non-reduced, regularized density matrix is given by where O(t − iτ ) = e H(it+τ ) Oe −H(it+τ ) , and is an infinitesimal positive constant, and N g = 0|e igOO e −2 H e −igOO |0 = 1 + O(g 2 ). We will take subregion A as [x, ∞], and consider entanglement of reduced density matrix Tr HĀ ρ(g, t).

Entanglement Entropy
Let us first begin with time evolution of entanglement entropy of subregion A. Using modular Hamiltonian H A = −logρ A , entanglement entropy can be expressed as In the case of vacuum modular Hamiltonian of half space in 2d CFT, it can be written explicitly in terms of energy momentum tensor [32], we have Figure 8: Spacetime dependence of entanglement entropy ∆S A after double trace deformation. When x > t + L or x + t < 0, ∆S A must be zero from causality.
Let us consider double trace deformation on CFT ground state in perturbation theory. Perturbation theory of entanglement entropy is discussed in [33] [34]. From the normalization condition, we have Then, the entanglement entropy is given by From conformal Ward-Takahashi identity, we have In the end, Therefore, one need to evaluate with z =x − (t ± i ) andz =x + (t ± i ). The total shift of entanglement entropy is Figure 9: When null geodesic crosses with null blue line, ANE changes by − 4πgh L 4h+1 , and 4πgh L 4h+1 with null red line. In particular, when 0 < a < 0, averaged null energy is − 4πgh L 4h+1 , and otherwise zero. Figure 10: When constant time slice crosses null blue line, the total energy changes by 4πgh L 4h+1 , and − 4πgh L 4h+1 with null red line. In particular, the total energy is always 0 after double trace deformation.
Interestingly, when t + x − L > 0 and t − x > 0, we have ∆S A = 0. The reason why∆S A = 0 in x − t − L > 0 and −x + t > 0 is because of the causality. Moreover, the correction is order g 1 , which is different from previous perturbative computation os entanglement entropy considered in [33] [34]. This is because we are using double trace and not single trace operator. Let us compare our result with energy of half space ∞ x dxT 00 (t,x). Integral of energy density is given by The averaged null energy on null geodesic (t, x) = λ, a + λ) is given by When g is positive, the average null energy is negative, which is consistent with the averaged null energy calculations in the bulk. Further, the perturbation is not UV divergent and is finite. Note that ANE is nonzero only when 0 < a < L. This is because of cancelation of positive energy flux and negative energy flux. Note that for excited states, the energy always increases. However, for our case, the total energy does not change at leading order since we are considering external perturbation.

Renyi Entanglement Entropy
Next, we consider time evolution of Renyi entropy after acting double trace deformation in two dimensional CFT. The shift of n-th Renyi entropy ∆S n A (g, t) = S n A (g, t) − S n A (0) can be computed Figure 11: Geometry of replica Σ 2 . Blue points correspond to e igOO(−(t+i )) , and red points correspond to e igOO(−(t−i )) .
using replica trick [21][35], (29) where Σ n is n-sheeted geometry, and O (i) lives on i-th sheet. We assume subregion as half line. Then the conformal transformation from Σ n to Σ 1 is Let us consider g 1 order contribution, Let us evaluate when n = 2. In the first sheet, w = −x + (t ± i ) corresponds to In the second sheet, w = −x + (t ± i ) corresponds to Figure 12: Entanglement Renyi entropy ∆S 2 A of semi infinite half line divided by coupling g. Value of Renyi entropy depends on the location of the left edge of the half line, which is specified by the coordinate (t, x). When x > t + L or x + t < 0, ∆S 2 A must be zero from causality. Left figure is for h = 0.3 and right one is for h = 0.6. In all case one can observe that Renyi entropy behaves in similar way as that of entanglement entropy.

Then we have
The results are shown in Figure 12. We can confirm that the second Renyi entropy behaves in the same way as that of entanglement entropy. In particular, we have zero Renyi entropy and zero entanglement entropy on identical regions, including regions where any change is prohibited by causality. Moreover, positive (null) energy corresponds to the growth of entanglement, and negative (null) energy corresponds to the decrease of entanglement. This is expected behavior from second law of entanglement entropy.

Outlook
In our study, we considered double trace deformation composed of boundary local operators as well as bulk local operators. We evaluated time shift caused by the deformation, by computing averaged null energy. Then we considered time evolution of CFT, by computing entanglement entropies and Renyi entropies.
In this paper, we only computed leading term in g, and entanglement introduced was order c 0 . We can consider higher order in g, as well as adding more nonlocal entanglement by e i l c l O l .Generalization of our study to higher order correction might lead to the understanding of constructing wormholes in spacetime with single boundary. Also, it is interesting to apply our bulk local double trace deformation to AdS 2 , and see if non-pertubative computation is possible, although it is not clear how to define HKLL bulk local operators when bulk metric starts fluctuating.
In our computation of dynamics of entanglement entropy and entanglement Renyi entropy, we considered instantaneous double trace deformation and leading term which is proportional to g 1 . It is interesting and tractable to generalize our work to higher order, or to triple trace deformations. Tripartite deformation can be an interesting example to examine multi partite entanglement.