Entanglement Wedge Cross Section from CFT: Dynamics of Local Operator Quench
YYITP-19-82
Entanglement Wedge Cross Section from CFT:Dynamics of Local Operator Quench
Yuya Kusuki, Kotaro Tamaoka
Center for Gravitational Physics,Yukawa Institute for Theoretical Physics (YITP), Kyoto University,Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan.
Abstract
We derive dynamics of the entanglement wedge cross section from the reflected en-tropy for local operator quench states in the holographic CFT. By comparing between thereflected entropy and the mutual information in this dynamical setup, we argue that (1)the reflected entropy can diagnose a new perspective of the chaotic nature for given mixedstates and (2) it can also characterize classical correlations in the subregion / subregion du-ality. Moreover, we point out that we must improve the bulk interpretation of a heavystate even in the case of well-studied entanglement entropy. Finally, we show that wecan derive the same results from the odd entanglement entropy. The present paper is anextended version of our earlier report arXiv:1907.06646 and includes many new results:non-perturbative quantum correction to the reflected / odd entropy, detailed analysis in bothCFT and bulk sides, many technical aspects of replica trick for reflected entropy whichturn out to be important for general setup, and explicit forms of multi-point semi-classicalconformal blocks under consideration. a r X i v : . [ h e p - t h ] S e p ontents A and B . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Quench inside Region A and B . . . . . . . . . . . . . . . . . . . . . . . . . . 22
10 Discussion 49A Semiclassical Fusion and Monodromy Matrix 52B Semiclassical 5-point Block 53
B.1 Proof of (2.33) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53B.2 Proof of (2.43) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55B.3 Proof of (5.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
C Heavy-Heavy-Light OPE Coe ffi cient 56 Introduction & Summary
The non-equilibrium dynamics in a given strongly coupled system attracts a lot of attention inthe physics community. One useful tool to capture this dynamical process is entanglement en-tropy , which measures entanglement between subsystem A and its complement ¯ A . This quantityis defined by S ( A ) = − tr ρ A log ρ A , (1.1)where ρ A is a reduced density matrix for a subsystem A , obtained by tracing out its complement.The Renyi entropy is a generalization of the entanglement entropy, which is defined as S ( n ) ( A ) = − n log tr ρ nA , (1.2)and the limit n → S ( A ). For thismeasure, a large number of works have been done to characterize the dynamics, for example,after joining quench [1], global quench [2, 3], splitting quench [4] and double quench [5–7].In particular, our interest in this paper is to study a local operator quench state [8, 9], which iscreated by acting a local operator O ( x ) on the vacuum in a given CFT at t = | Ψ ( t ) (cid:105) = √N e − (cid:15) H − iHt O ( x ) | (cid:105) , (1.3)where x represents the position of insertion of the operator, (cid:15) is an UV regularization of thelocal operator and N is a normalization factor so that (cid:104) Ψ ( t ) | Ψ ( t ) (cid:105) = dynamics of correlations between two dis-joint intervals. A natural challenge for this purpose is to investigate the dynamics of somequench state by utilizing correlation measures. One progress in this direction had already donein [10, 11], which studied universal features of dynamics after a global quench by using theentanglement entropy for two disjoint intervals, or equivalently, the mutual information andshowed that entanglement spreads as if correlations were carried by free quasiparticles after aglobal quench. And also it was shown that this quasiparticle picture breaks down in the holo-graphic CFT [12]. It suggests that the mutual information is very useful to probe the universalfeature of correlation dynamics in a given CFT class. (see also [13], which studied the dy-namics of the mutual information after a joining quench.) However, what we have to mentionis that our interest is the correlation between two disjoint intervals, which are not necessarilycomplementary to each other, therefore, the state cannot be described by pure state. For mixedstates, we do not have the unique measure for the bi-partite correlation. For this reason, we arealso interested in other correlation measures. For example, one of other interesting correlationmeasures is negativity [14, 15] and in [16, 17], the time-dependence of the correlation betweentwo disjoint intervals is studied by using the negativity.In this paper, we will make use of reflected entropy [18] as a tool to probe dynamics ofcorrelations between two intervals. The definition is as follows. We consider the followingmixed state, ρ AB = (cid:88) n p n ρ ( n ) AB , (1.4)2here each ρ ( n ) AB represents a pure state as ρ ( n ) AB = (cid:88) i , j (cid:113) λ in λ jn | i n (cid:105) A | i n (cid:105) B (cid:104) j n | A (cid:104) j n | B , (1.5)where | i n (cid:105) A ∈ H A , | i n (cid:105) B ∈ H B and λ in is a positive number such that (cid:80) i λ in =
1. The real number p n is the corresponding probability associated with its appearance in the ensemble. For thismixed state, we can provide the simplest purification for this mixed state as | √ ρ AB (cid:105) = (cid:88) i , j , n (cid:113) p n λ in λ jn | i n (cid:105) A | i n (cid:105) B | j n (cid:105) A ∗ | j n (cid:105) B ∗ , (1.6)where | i n (cid:105) A ∗ ∈ H ∗ A and | i n (cid:105) B ∗ ∈ H ∗ B are just copies of H A and H B . Then, the reflected entropy isdefined by S R ( A : B ) ≡ − tr ρ AA ∗ log ρ AA ∗ , (1.7)where ρ AA ∗ is the reduced density matrix of ρ AA ∗ BB ∗ = | √ ρ AB (cid:105) (cid:104) √ ρ AB | after tracing over H B ⊗H ∗ B . We have to emphasize that this quantity measures not only quantum correlations but alsoclassical correlations, like mutual information. Actually, these two quantities for the vacuumare very similar, however, we will give quite di ff erences by considering dynamical setups. Interestingly, if we restrict ourselves to two-dimensional CFTs, we can analytically evaluatethis quantity in the path integral formalism, like entanglement entropy. For this reason, weconsider a 2D CFT in this paper.An important point is that this quantity has a simple holographic dual interpretation, so-called entanglement wedge cross section , S R ( A : B ) = E W ( A : B ) , (1.8)where E W ( A : B ) is entanglement wedge cross section defined as the area of the minimal surfacebipartitioning the entanglement wedge region, first introduced in [19, 20]. (See also [21–47] forfurther developments in this direction.) That is, the reflected entropy is computable both in bulkside and CFT side and also meaningful in quantum information theory, in a similar manner tothe RT and HRT formula [48–50]. Thus this is a very good useful to investigate the quantumgravity in the context of the AdS / CFT, however, there is little understanding of its propertyfor now. In particular, there is no understanding on the non-equilibrium properties of the re-flected entropy even in the holographic CFT. This naturally motivates us to study the dynamicsof the reflected entropy. This study might give new insights into the relation of dynamics ofcorrelations between in the holographic CFT and in the quantum gravity.On this background, in this paper, we will study the time-dependence of the reflected entropyafter a local quench as a first step to understand the dynamics of the reflected entropy. Wewould like to point out the advantage of considering the local operator quench. Technically,the reflected entropy after a local quench is calculated by the
Regge limit of n -point conformal Here, we mean the vacuum by the mixed state ρ AB which comes form the vacuum for the whole system bytracing over H AB . odd entanglement entropy . The odd entanglement entropy is definedby S O ( A : B ) ≡ lim n O → − n O (cid:34) tr (cid:16) ρ T B AB (cid:17) n O − (cid:35) , (1.9)where ρ AB is a reduced density matrix for subsystems A and B , obtained by tracing out itscomplement. The limit n O → T B is thepartial transposition with respect to the subsystem B . (Note that it is equivalent to act T A insteadof T B .) Interestingly, it is conjectured that this quantity has a simple bulk interpretation as S O ( A : B ) − S ( A : B ) = E W ( A : B ) , (1.10)where S ( A : B ) is the entanglement entropy for the subsystems A and B . This is verified forthe vacuum and thermal state in the 2D holographic CFT [37], however, it is nontrivial that thisrelation also holds in other setups. For this reason, we will also study this quantity in the samesetup and investigate whether the relation can also be applied to nontrivial states or not. Wewould like to mention that this quantity can be calculated in the same way as negativity [14,15].More precisely, this is given by the analytic continuation of an odd integer of the same replicapartition function as negativity. Here we briefly summarize our results. • CFT vs. Gravity (in Section 2 and 9)It has been argued that the entanglement entropy after a local quench state is realized bygeometries with a falling particle [52]. From this observation, it is naturally expected thatthe reflected entropy for a locally excited state would be also the dual to the entanglementwedge cross section in that geometry. In this paper, we calculate the reflected entropy forsuch a dynamical state and compare it to the dynamics of the entanglement wedge crosssection. As a result, we find the perfect agreement. This is a new support of the dynamicalgeneralization of the reflected entropy / entanglement wedge cross section conjecture. • Technical aspects of replica trick for reflected entropy (in Section 2)When we use the replica trick, we should use the conformal blocks not for original theory(Virasoro conformal blocks) but for orbifold theory. In the case of the entanglemententropy (and odd one) we can justify the use of former blocks. However, this turns out tobe not the case for the reflected entropy. We clarify many technical aspects of the replica4 ②③ Figure 1: Three setups considered in this paper. We fist study the setup (0 < u < − v < − u < v ), second, (0 < u < v < u < v ) , and finally, (0 < v < u < v < − u ). In any setups, weexcite the vacuum by acting an local operator on x = t = • Dynamics of reflected entropy (and entanglement of purification) vs. mutual information(in Section 4)One motivation is to understand the dynamics of the reflected entropy. In this paper,we will consider three patterns of a local operator quench as shown in Figure 1. Firstobservation for the reflected entropy is that the time-dependence is captured by the quasi-particle picture [2, 10] as seen in the mutual information and the negativity. For example,if we consider a setup (cid:13) in Figure 1, we find that the reflected entropy becomes non-zeroonly in the time region t ∈ [ u , v ]. However, the time-dependence in the non-zero regionis very complicated, therefore, it cannot be completely explained by the quasi-particlepicture.We compare our results for the reflected entropy to the dynamics of the mutual infor-mation in the same setup and find both similarities and di ff erences. For example, thetime dependence of the reflected entropy is discontinuous, unlike the mutual information.Moreover, we give a natural explanation that the reflected entropy probes more classicalcorrelations than the mutual information from our dynamical setup . This implies the quantities dual to the entanglement wedge cross section can not be any axiomatic measures
5s a comment, our physical interpretation in this section can be also applied to entangle-ment of purification because in the holographic CFT, the reflected entropy reduces to theentanglement of purification. • What is dual to a heavy state? (in Section 5)Another interest is to understand what is the holographic dual to a heavy state in CFT.The first study has been done in [53] by making use of the entanglement entropy. Theresult suggests that the entanglement entropy for a heavy state can be approximated theholographic entanglement entropy in the BTZ background. In this paper, we considerreflect entropy for a heavy state to make it clear. This approach is quite natural becausereflected entropy is more refined tool than entanglement entropy. Consequently, we finda contradiction between their bulk interpretation and the entanglement entropy whichcomes from the pure state limit of the reflected entropy. To resolve this problem, wegive an improved bulk interpretation of the heavy state and then we obtain the perfectagreement between our bulk interpretation and the reflected entropy in the heavy state. • Quantum correction (in Section 6)We can evaluate some quantum corrections to the reflected entropy, which is consistentwith a naive expectation from the physical viewpoint. And also the reflected entropy withsome quantum corrections also satisfies some important inequalities of the holographicreflected entropy. • Dynamics in other CFTs (in Section 7)If one wants to characterize the holographic CFT by the reflected entropy, it is necessaryto find out a unique feature of the holographic reflected entropy. For this purpose, wefirst tried to compare the holographic result to that in rational CFTs (RCFTs). As a result,we show that the time-dependence for these two CFTs are quite di ff erent. From thisobservation, we could argue that the dynamics of the reflected entropy is very sensitive towhether a given CFT is chaotic or not. In other words, we can make use of the reflectedentropy as a probe of the chaotic nature of a given CFT (see also [54]). • Agreement with odd entanglement entropy (in Section 8)We can show that the odd entanglement entropy also reproduces the entanglement wedgecross section in our dynamical setup. Actually, the similarity between the holographicodd entanglement entropy and the holographic reflected entropy can be explained by aspecial property of the linearized conformal block. Therefore, instead of providing thedetailed calculations, we show how the odd entanglement entropy reduces to the reflectedentropy in the holographic CFT. of the quantum entanglement. This conjecture has been proven recently by [44]. Reflected Entropy of Local Operator from CFT
The reflected entropy can be evaluated in the path integral formalism [18]. For example, theRenyi reflected entropy in the vacuum can be computed by a path integral on m × n copies asshown in Figure 2. Here, we would view this manifold as a correlator with twist operators as inthe lower of Figure 2, where we define the twist operators σ g A and σ g B . Here, we focus on thefollowing mixed state, ρ AB = tr AB | Ψ ( t ) (cid:105) (cid:104) Ψ ( t ) | , (2.1)where Ψ ( t ) is a time-dependent pure state as | Ψ ( t ) (cid:105) = √N e − (cid:15) H − iHt O (0) | (cid:105) . Then, in a similarmanner to the method in [8], the replica partition function in this state can be obtained by acorrelator as 11 − n log Z n , m (cid:0) Z , m (cid:1) n , (2.2)and Z n , m ≡ (cid:68) σ g A ( u ) σ g − A ( v ) O ⊗ mn ( w , ¯ w ) O ⊗ mn † ( w , ¯ w ) σ g B ( u ) σ g − B ( v ) (cid:69) CFT ⊗ mn , (2.3)where we abbreviate V ( z , ¯ z ) ≡ V ( z ) if z ∈ R and the operators O are inserted at w = t + i (cid:15), ¯ w = − t + i (cid:15), w = t − i (cid:15), ¯ w = − t − i (cid:15). (2.4)Here O ⊗ N ≡ O ⊗ O ⊗ · · · ⊗ O is an abbreviation of the operator on N copies of CFT (CFT ⊗ N ) .To avoid unnecessary technicalities, here we do not show the precise definition of the twistoperators σ g A and σ g B (which can be found in [18]) because in many parts of this paper, we onlyuse the following properties of the twist operators, h σ gA = h σ g − A = h σ gB = h σ g − B = cn (cid:32) m − m (cid:33) ( = nh m ) ,σ g − A g B = σ g n ⊗ σ g − n , (2.5)where the twist operator σ g n is just the usual twist operator σ n based on the n -cyclic permutationgroup, which has the conformal dimension h σ gn = c (cid:16) n − n (cid:17) ( ≡ h n ). Note that the secondproperty is a naive expression, which will be explained more explicitly in Section 2.1.1.The reflected entropy is defined by the von-Neumann limit of this partition function,lim n , m → − n log Z n , m (cid:0) Z , m (cid:1) n , (2.6)where the analytic continuation m → m . For simplicity, we always omit the transposition of operators on the reflected sheets. B
21 4 433 6 5 785 6 9 8792 1 1010 1112 𝑚2𝑚2 (cid:3250)(cid:3250)(cid:3127)(cid:3117)(cid:3251)(cid:3251)(cid:3127)(cid:3117)
Figure 2: The path integral representation of the Renyi reflected entropy. Edges labeled with thesame number get glued together. We can instead view it as a correlator with four twist operators (cid:68) σ g A ( u ) σ g − A ( v ) σ g B ( u ) σ g − B ( v ) (cid:69) CFT ⊗ mn .It is hard to calculate the numerator in (2.2) in general. Fortunately, in the case of interest,i.e., holographic CFTs, this 6-point function can be approximated by a single conformal blockas in [53], for example, if we set 0 < (cid:15) (cid:28) t < u < − v < − u < v then the correlation functionis approximated by( C n , m ) (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3041)(cid:3034) (cid:3251) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3041) × (anti-holomorphic part) , (2.7)8here C n , m is the OPE coe ffi cient (cid:104) σ g − A | σ g B (1) | σ g B g − A (cid:105) . This coe ffi cient can be calculated by themethod developed in [55], and the result is C n , m = (2 m ) − h n . (2.8)The details of this derivation can be found in Appendix C of [18]. As explained in [53], wehave many choices of the single block approximation aside from (2.7) because we can also de-compose the 6-point correlator in terms of the conformal block transformed by the monodromytransformation . The correct result is obtained by the maximal single conformal block approx-imation. More detailed explanations and its explicit calculations are shown in the followingsubsections. Before moving on to the calculation, we discuss two technical complications due to the unusualreplica trick for the reflected entropy: (1) We have an analytic continuation of an even integer m related to preparing a canonically purified state, but eventually take m → n related to the Renyiindex. Since we finally take the m , n → In general, we cannot approximate the conformal block of the orbifold theory (“orbifold block”)appeared in (2.7) by the Virasoro conformal block. This is because there is the current associatedwith the replica symmetry. However, in the limits n , m → c → ∞ , these two blocks canbe related. Indeed, the orbifold block in this limit can be related to a “square” of the Virasoroconformal block. This “squaring” (or “doubling”) essentially comes from the doubling of thepurified Hilbert space. Interestingly, this doubling also explains the origin of the double ofentanglement wedge cross section for reflected entropy in holographic CFTs. Therefore, let usfirst explain why this works in our case.Since we analytically continue an even integer m to the real number, replica sheets labelledby m = , . . . , m − m , . . . , m − ( k , l ) Operator on ( k , l )-sheet. ( k = , . . . , m − l = , . . . , n − O ⊗ n ( k ) (cid:78) n − l = O ( k , l ) σ (0) n O ( k , l ) (e π i z ) σ (0) n (0) = O ( k , l + ( z ) σ (0) n (0), (if k = O ( k , l ) (e π i z ) σ (0) n (0) = O ( k , l ) ( z ) σ (0) n (0), (otherwise). σ ( m / n O ( k , l ) (e π i z ) σ ( m / n (0) = O ( k , l + ( z ) σ ( m / n (0), (if k = m ), O ( k , l ) (e π i z ) σ ( m / n (0) = O ( k , l ) ( z ) σ ( m / n (0), (otherwise). σ ⊗ nm O ( k , l ) (e π i z ) σ ⊗ nm (0) = O ( k + , l ) ( z ) σ ⊗ nm (0).¯ σ ⊗ nm O ( k , l ) (e π i z ) ¯ σ ⊗ nm (0) = O ( k − , l ) ( z ) ¯ σ ⊗ nm (0). σ (cid:48)⊗ nm O ( k , l ) (e π i z ) σ (cid:48)⊗ nm (0) = O ( k + , l + ( z ) σ (cid:48)⊗ nm (0), (if k = , m ), O ( k , l ) (e π i z ) σ (cid:48)⊗ nm (0) = O ( k + , l ) ( z ) σ (cid:48)⊗ nm (0), (otherwise) .¯ σ (cid:48)⊗ nm O ( k , l ) (e π i z ) ¯ σ (cid:48)⊗ nm (0) = O ( k − , l − ( z ) ¯ σ (cid:48)⊗ nm (0), (if k = , m ), O ( k , l ) (e π i z ) ¯ σ (cid:48)⊗ nm (0) = O ( k − , l ) ( z ) ¯ σ (cid:48)⊗ nm (0), (otherwise) .Then, the operator O ⊗ mn can be written as O ⊗ mn = O ⊗ n (0) ⊗ · · · ⊗ O ⊗ n ( m / ⊗ · · · . (2.9)Throughout this paper, we suppress the transposition acting on the operators on second halfsheets concerning to m . We have to emphasize that in the analytic continuation of even m , theoperator O ⊗ mn does NOT reduce to O but the “square” of O aslim m ∈ even → O ⊗ mn → O ⊗ n (0) ⊗ O ⊗ n (1 / . (2.10)One can also find the same decoupling in the original paper [18], where the analytic continuationleads to lim m ∈ even → σ g − A g B → σ (0) n ⊗ ¯ σ (1 / n . (2.11)It means that this tricky analytic continuation provides two decoupled sheets labeled by 0 and1 / σ g B = σ ⊗ nm , σ g − B = ¯ σ ⊗ nm , σ g A = σ (cid:48)⊗ nm , σ g − A = ¯ σ (cid:48)⊗ nm , σ g − A g B = σ (0) n ⊗ ¯ σ ( m / n , (2.12)and the conformal block can be re-expressed by 𝑂 ((cid:2868))⊗(cid:3041) ⊗ 𝑂 ((cid:3040)/(cid:2870))⊗(cid:3041) ⊗ ⋯ (cid:3040)⊗(cid:3041) 𝜎 (cid:3041)(cid:2868) ⊗ 𝜎(cid:3364) (cid:3041)(cid:3040)/(cid:2870) 𝜎 (cid:3041)(cid:2868) ⊗ 𝜎(cid:3364) (cid:3041)(cid:3040)/(cid:2870) (cid:3040)⊗(cid:3041) (cid:3040)⊗(cid:3041) (cid:3040)⊗(cid:3041) 𝑂 (cid:2868)⊗(cid:3041) ⊗ 𝑂 (cid:3040)/(cid:2870)⊗(cid:3041) ⊗ ⋯ (cid:2993) , (2.13)10here · · · means the rest of O ⊗ mn , that is, (cid:78) l = ˆ0 , , ,..., ˆ m ,..., n − O ⊗ n ( l ) , which is not important becauseit disappears in the limit m → { O ⊗ n (0) , σ (0) n } do not interact with { O ⊗ n ( m / , σ ( m / n } , therefore, the component ofthe conformal block (i.e., three point block) is decoupled into two parts, for example, (cid:104) σ (0) n ⊗ ¯ σ ( m / n | O ⊗ n (0) ⊗ O ⊗ n ( m / | σ (0) n ⊗ ¯ σ ( m / n (cid:105) = (cid:104) σ (0) n | O ⊗ n (0) | σ (0) n (cid:105) (cid:104) σ ( m / n | O ⊗ n ( m / | σ ( m / n (cid:105) . (2.14)Let us highlight this decoupling by 𝑂 ((cid:3040)/(cid:2870))⊗(cid:3041) (cid:3040)⊗(cid:3041)(cid:3040)⊗(cid:3041) (cid:3040)⊗(cid:3041) (cid:3040)⊗(cid:3041) 𝑂 (cid:2868)⊗(cid:3041) (cid:2993) 𝑂 (cid:3040)/(cid:2870)⊗(cid:3041) (cid:2993) 𝑂 ((cid:3040)/(cid:2870))⊗(cid:3041) 𝜎 (cid:3041)(cid:2868) 𝜎 (cid:3041)(cid:2868) 𝜎(cid:3364) (cid:3041)(cid:3040)/(cid:2870) 𝜎(cid:3364) (cid:3041)(cid:3040)/(cid:2870) . (2.15)Roughly, each decoupled contribution can be regarded as the independent Virasoro conformalblocks up to the universal contributions from external operators . We will see each decoupledblock provides the entanglement wedge cross section, thus we obtain the double of the entan-glement wedge cross section in total. Note that these blocks are quite similar to the one for theodd entanglement entropy. This is the main reason why it also reproduces the cross section.Having this doubling in mind, we will often suppress the above lengthy doubling expression(2.15) and instead double the conformal dimension for internal operators.The analytic continuation of the even integer m gives rise to another subtle issue. In order toobtain the correct normalization for the density matrix, Z , m should not be regarded as the naivetr ρ mAB , namely the Renyi entropy after a local quench, Z , m (cid:44) (cid:68) σ g m ( u ) σ g − m ( v ) O ⊗ m ( w , ¯ w ) O ⊗ m † ( w , ¯ w ) σ g m ( u ) σ g − m ( v ) (cid:69) CFT ⊗ m , (2.16)where the twist operator σ g m is just the usual twist operator σ m . This is just because the naivetr ρ mAB is (strictly speaking) di ff erent from the normalization of the purified state. In other words,the naive one cannot take into account the above squaring e ff ect. As a result of this squaring,the analytic continuation of the denominator in (2.2) is given by the square of the two-pointfunction, lim m ∈ even → (cid:0) Z , m (cid:1) n = (cid:68) O ( w , ¯ w ) O † ( w , ¯ w ) (cid:69) n = (2 i (cid:15) ) − nh O , (2.17) To be precise, the decoupled conformal block in (2.15) is still not the Virasoro block because there is thecurrent associated with the Z n symmetry (see [12], which discusses this problem). However, in the large c limit,this type of blocks with twist operators can be related to the Virasoro block. In fact, this assumption is often usedin the calculation of the entanglement entropy and it is verified in the holographic CFT by comparing with thegravity calculation [53]. Moreover, this is also verified by comparing with a completely independent calculationwithout relying on twist operators [51]. h O is the conformal dimension of the operator O . It would be worth noting that we canconfirm the necessity of this squaring from the pure state limit of ρ AB , where our reduced densitymatrix ρ AA ∗ becomes “square” of ρ A . Second one is physically more important—the two limits m → n → c limit. We should first take the limit n →
1. There is a physical rea-son: in order to obtain the correct cross section of the entanglement wedge, we should preparethe precise entanglement wedge at first. In terms of the single conformal block approximation,it means that we have to choose the maximal channel in the limit n → m (seeFigure 3).However, in the following, we calculate the reflected entropy by taking first the limit m → n → h n (cid:28) nh m , instead of first taking n → m →
1. Let usstress that this is just for the simplification of calculation and presentation. Indeed, as we showin the following, our result from this procedure perfectly reproduces the bulk calculation. Wecan also show this validity in another way. The reason why two limits m → n → c limit could changeif the order is reversed. And in fact, we use the assumption 2 h n (cid:28) nh m only to specify thedominant channel. That is, after identifying the dominant channel, the order of the two limits isnot important. Therefore, we can calculate the correct reflected entropy by taking first the limit m → n → h n (cid:28) nh m .It would be interesting to comment that the non-commutativity of n → m → Actually, if we take first the limit m →
1, then the limit n → S R ( A : B )[ O ] ≥ I ( A : B )[ O ]. (cid:3034) (cid:3250) 𝑂 ⊗(cid:3040)(cid:3041) (cid:3041) (cid:3034) (cid:3251) 𝜎 (cid:3034) (cid:3251)(cid:3127)(cid:3117) 𝜎 (cid:3034) (cid:3250)(cid:3127)(cid:3117) 𝑂 ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3041) 𝜎 (cid:3034) (cid:3250) 𝑂 ⊗(cid:3040)(cid:3041) (cid:3041) (cid:3034) (cid:3251) 𝜎 (cid:3034) (cid:3251)(cid:3127)(cid:3117) 𝜎 (cid:3034) (cid:3250)(cid:3127)(cid:3117) 𝑂 ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3041) minimizemaximize maximizeCFT description minimize Figure 3: To reproduce the entanglement wedge cross section, we first take the large c limitand approximate the correlator by the maximal single conformal block. However, we haveto take care of the fact that this maximization is done by two maximization processes. First,we maximize the propagations between external operators (lines colored by blue) and second,we maximize the internal line (colored by red). This order of processes corresponds to theminimizations in bulk side as shown in the upper of this figure. As mentioned in the maintext, this order of maximizations can be accomplished by the large c limit under the assumption2 h n (cid:28) nh m . A and B We first consider the setup, 0 < (cid:15) (cid:28) u < − v < − u < v and we assume the connectedcondition < ( v − u )( u − v )( v − v )( u − u ) < , (2.18)which means that in the bulk side, the entanglement wedge for two intervals A = [ u , v ] and B = [ u , v ] is connected. In this article, we only focus on this connected case because the reflected In the CFT side, the transition between connected and disconnected entanglement wedge can be interpreted asa change of the dominant conformal block as shown in [62]. One can show this connected condition from the CFTside by calculating the entanglement entropy for two intervals A = [ u , v ] and B = [ u , v ] after a local quench. < t < u ), the (cid:15) → (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3041)(cid:3034) (cid:3251) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3041) −−−→ (cid:15) → (2 i (cid:15) ) − mnh O × (cid:3034) (cid:3250) (cid:3041)(cid:3034) (cid:3251) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) , (2.19)which means Z n , m (cid:0) Z , m (cid:1) n −−−→ (cid:15) → (cid:68) σ g A ( u ) σ g − A ( v ) σ g B ( u ) σ g − B ( v ) (cid:69) CFT ⊗ mn , (2.20)Therefore, the reflected entropy for the excited state in the early time is just given by that forthe vacuum, like the entanglement entropy after a local quench [53]. Note that the explicit formof the vacuum reflected entropy is c + (cid:113) ( − u + v )( v − u )( − u + u )( − v + v ) − (cid:113) ( − u + v )( v − u )( − u + u )( − v + v ) , (2.21)which exactly matches the entanglement wedge cross section in pure AdS [19]. This can beimmediately shown by using the asymptotic form of the Virasoro block (see (A.2)), (cid:3034) (cid:3250) (cid:3041)(cid:3034) (cid:3251) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) → h n + (cid:113) ( − u + v )( v − u )( − u + u )( − v + v ) − (cid:113) ( − u + v )( v − u )( − u + u )( − v + v ) − h n , (2.22)where we take first the limit c → ∞ , second m →
1, and finally n → u < t < − v , only the holomorphic part of the OPE between O ⊗ mn and O ⊗ mn † crosses a branch cut on the real axis from u to v , which means that the limit (cid:15) → Regge limit [63, 64]. Before evaluating the 6-point correlatorfor u < t < − v , we should take care of the fact that there are other choices of the conformalblock expansion and a single block approximation besides (2.7). The point is that the correlatoris invariant under a monodromy transformation, which moves the operators O ⊗ mn , O ⊗ mn † aroundthe twist operators. On the other hand, each individual conformal block is not invariant. Thus,we have other choices of the single conformal block approximation and the correct choice ismaximal one under the assumption 2 h n (cid:28) nh m .Fortunately, we find that the correct choice for u < t < √− v u is just the following channelwithout monodromy tranformations, In fact, this calculation cannot be found in previous works but we can calculate it by the method developed in thissection (which is explained later in Section 4). In the following, we will abbreviate σ n ⊗ ¯ σ n by 2 h n . C n , m ) (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3040) (cid:3041) × (anti-holomorphic part) , (2.23)For u < t < − v (in particular, u < t < √− v u ) , the e ff ect of crossing the branch cut can beillustrated for the holomorphic part by (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3041)(cid:3040) , (2.24)which is a conformal block mapped by a monodromy transformation, which moves the operator O ⊗ mn † clockwise around the twist operator σ g B (i.e., ( w − u ) → e − π i ( w − u )). In general,the e ff ect of the monodromy transformation is encapsulated in the monodromy matrix , whichdoes not depend on a given CFT data and, therefore, can be evaluated exactly. For the Virasoroblock, the monodromy matrix is usually expressed by [65] (the notation is as in [51, 61]) (cid:3036)(cid:3037)(cid:3038)(cid:3039) = (cid:90) S d α p M ( − )0 ,α p (cid:34) α j α i α k α l (cid:35) × (cid:3036)(cid:3037)(cid:3038) (cid:3043)(cid:3039) , (2.25)where we introduce the following Liouville notation, c = + Q , Q = b + b , (2.26)and the Liouville momentum, α i ( Q − α i ) = h i . (2.27)The contours run from Q to Q + i ∞ and also runs clockwise around α p = α i + α j + lb < Q and α p = α k + α l + lb < Q ( l ∈ Z ≥ ). The superscript ( − ) of the matrix M means the clockwise monodromy. It is worth to note that this monodromy matrix only depends on thefour external operators { i , j , k , l } and the internal operator p , that is, it is independent of otheroperators described by {· · · } in (2.25). If one is interested in the details of these transformations,one can refer to [51].Like the Virasoro block, the orbifold block (2.24) can also be expressed in terms of a certainmonodromy matrix as (cid:90) S d α p ˜ M ( − )0 ,α p (cid:34) α m α m α O α O (cid:35) × (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3040) (cid:3041) , (2.28) Here we choose the convention 0 < b <
1, which is possible if c >
25, in particular, large c . M and define the Liouville momentum, α m ( Q − α m ) = h m (cid:32) = c (cid:32) m − m (cid:33)(cid:33) , α O ( Q − α O ) = h O , ¯ α O ( Q − ¯ α O ) = ¯ h O . (2.29)Although the explicit form of ˜ M is unknown, that appearing in our calculation can be related tothe Virasoto monodromy matrix from the fact (2.15). We will explain it in more details when˜ M appears in the calculation of the reflected entropy.The anti-holomorphic part does not change in this time region. As a result, the approximated6-point function with the monodromy e ff ect (2.24) for u < t < √− v u can be shown as( C n , m ) (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3041)(cid:3040) × (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3040) (cid:3041) , (2.30)where the overline means the anti-holomorphic part. To proceed further, we consider the Reggelimit, which comes from the limit (cid:15) →
0. In fact, the Regge limit of the block is universal[51] because the integral in (2.28) is dominated by a Liouville momentum α min such thatthe corresponding conformal dimension h min = α min ( Q − α min ) is minimal in the set { h | h = α ( Q − α ) s . t . α ∈ S } . In our case, this saddle point contribution comes from the clockwiseintegral around α min . For this reason, we introduce the following notation, M ( − )0 ,α min ≡ Res (cid:16) − π i M ( − )0 ,α p ; α p = α min (cid:17) . (2.31)We comment on a trivial property of any monodromy matrix,˜ M ( − )0 ,α min (cid:34) α αβ β (cid:35) −−−→ α → . (2.32)By using this fact, we obtain (see Appendix B in more details) (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3041)(cid:3040) −−−→ (cid:15) → h n (2 i (cid:15) ) − nh O + (cid:113) ( − u + v )( v − t )( − u + t )( − v + v ) − (cid:113) ( − u + v )( v − t )( − u + t )( − v + v ) − h n , (2.33)where we take first the limit c → ∞ , second m →
1, third n →
1, and finally (cid:15) →
0. Notethat the OPE limit between O ⊗ mn and O ⊗ mn † in the limit m ∈ even → m →
1, the contribution from the monodromy matrix becomes trivial. The Regge limit of the Virasoro block had first studied in [66]. And from the observations [67, 68], it wanshown that the singularity of the Virasoro block is closely related to the fusion matrix [61, 69] and consequently,the explicit form of the Regge limit is obtained by using the monodromy matrix, which can be rexepressed by thefusion matrix [51].
16n what follows, we will not display the trivial ones under this limit. On the other hand, theanti-holomorphic part is just given by the OPE limit, (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3040) (cid:3041) −−−→ (cid:15) → h n (2 i (cid:15) ) − n ¯ h O + (cid:113) ( − u + v )( v − u )( − u + u )( − v + v ) − (cid:113) ( − u + v )( v − u )( − u + u )( − v + v ) − h n . (2.34)Substituting these holomorphic part (2.33) and anti-holomorphic part (2.34), and (2.17), (2.8)into (2.2), we obtain the reflected entropy at u < t < √− v u as c + (cid:113) ( − u + v )( v − t )( − u + t )( − v + v ) − (cid:113) ( − u + v )( v − t )( − u + t )( − v + v ) + c + (cid:113) ( − u + v )( v − u )( − u + u )( − v + v ) − (cid:113) ( − u + v )( v − u )( − u + u )( − v + v ) . (2.35)For √− v u < t < − v , the 6-point conformal block is NOT dominated by the usual block,but the block illustrated by ( C n , m ) (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3041)(cid:3040) × (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3041)(cid:3040) . (2.36)The e ff ect of crossing the brunch cut is the same as that for u < t < √− v u . This e ff ectcancels the monodromy illustrated in (2.36), therefore, the approximated 6-point function at √− v u < t < − v results in( C n , m ) (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3040) (cid:3041) × (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3041)(cid:3040) . (2.37)Each holomorphic and anti-holomorphic conformal block is the same as (2.34) and (2.33),consequently, we obtain c + (cid:113) ( − u + v )( v + t )( − u − t )( − v + v ) − (cid:113) ( − u + v )( v + t )( − u − t )( − v + v ) + c + (cid:113) ( − u + v )( v − u )( − u + u )( − v + v ) − (cid:113) ( − u + v )( v − u )( − u + u )( − v + v ) . (2.38)For − v < t < − u , the dominant channel is given by( C n , m ) (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3041)(cid:3040) × (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3041)(cid:3040) . (2.39) In the anti-holomorphic ¯ z plane, the imaginary direction is flipped, therefore, the arrow of the monodromytransformation is also flipped.
17n a similar way as (2.24), the holomorphic part of the block is a ff ected by crossing the brunchcut as (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3041)(cid:3040) , (2.40)In the time region − v < t < − u , the anti-holomorphic part of the OPE between O ⊗ mn and O ⊗ mn † also crosses a branch cut on the real axis from − u to − v . This a ff ects the anti-holomorphicblock as (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3040) (cid:3041) . (2.41)Taking account of the e ff ects (2.40) and (2.41), the approximated 6-point function for − v < t < − u can be illustrated by( C n , m ) (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3040) (cid:3041) × (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3040) (cid:3041) . (2.42)Let us evaluate the Regge limit of this approximated 6-point function. By our result [51] again,we obtain (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3040) (cid:3041) −−−→ (cid:15) → ( − i (cid:15) ) h n − nh O (cid:32) ( u − v )( u − v )( t + u )( t + u )( t + v )( t + v ) (cid:33) h n M n [ O ] , (2.43)where we take first the limit c → ∞ , second m →
1, third n →
1, and finally (cid:15) →
0. Here M n [ O ] is a constant, given by the monodromy matrix, and the asymptotic expression in theselimits is M n [ O ] → (cid:32) ˜ M ( − )0 , α n (cid:34) α n α n α O α O (cid:35)(cid:33) = (cid:32) i ¯ γ sinh π ¯ γ (cid:33) − h n , (2.44)where α n is given by α n ( Q − α n ) = h n and we define ¯ γ = (cid:113) c ¯ h O −
1. The square comes from thedecoupling of the orbifold block into two Virasoro blocks as explained in (2.15). More detailedcalculation can be found in Appendix B. Thus, we obtain the reflected entropy as c (cid:34) t + u )( t + u )( t + v )( t + v ) (cid:15) ( u − v )( u − v ) (cid:32) sinh π ¯ γ ¯ γ (cid:33) (cid:35) + c + (cid:113) ( − u + v )( v − u )( − u + u )( − v + v ) − (cid:113) ( − u + v )( v − u )( − u + u )( − v + v ) , if − v < t < − u . (2.45)18owever, as explained later (in Section 5), there is another possibility to dominate the 6-point correlater by the following channel,( C n , m ) ( C n , O ) (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993)(cid:3034) (cid:3251) (cid:3041) (cid:3041) × (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993)(cid:3034) (cid:3251) (cid:3041) (cid:3041) . (2.46)The constant C n , O is the OPE coe ffi cient with O ⊗ mn . The intermediate state p corresponds to thedominant contribution to the correlator. In the m , n → p is given by O ⊗ in the CFTof interest [70]. In the bulk side, this channel corresponds to the disconnected entanglementwedge cross section which ends at the block hole horizon (which is discussed more in Section5). The e ff ect of crossing the branch cut is illustrated by (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993)(cid:3034) (cid:3251) (cid:3041) (cid:3041) × (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993)(cid:3034) (cid:3251) (cid:3041) (cid:3041) . (2.47)Each conformal block is approximated in the von-Neumann limit as (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993)(cid:3034) (cid:3251) (cid:3041) (cid:3041) −−−→ (cid:15) → (2 i (cid:15) ) h n − nh O (cid:32) ( u − v )( u − v )( t − u )( t − u )( t − v )( t − v ) (cid:33) h n , (2.48)and (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993)(cid:3034) (cid:3251) (cid:3041) (cid:3041) −−−→ (cid:15) → ( − i (cid:15) ) h n − nh O (cid:32) ( u − v )( u − v )( t + u )( t + u )( t + v )( t + v ) (cid:33) h n . (2.49)Thus, we obtain the reflected entropy as c (cid:34) ( t − u )( t − u )( t − v )( t − v ) (cid:15) γ ( u − v )( u − v ) (cid:35) + c (cid:34) ( t + u )( t + u )( t + v )( t + v ) (cid:15) ¯ γ ( u − v )( u − v ) (cid:35) + (const.) (2.50)We can immediately find that the (cid:15) -singularity of this result is much larger than 2.45), therefore,we can neglect this possibility.The calculation of the reflected entropy for − u < t is almost the same as the derivation of192.35), therefore, we can summarize our results as S R ( A : B )[ O ] = c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if t < u , c log + (cid:113) ( − u + v v − t )( − u + t )( − v + v − (cid:113) ( − u + v v − t )( − u + t )( − v + v + c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if u < t < √− v u , c log + (cid:113) ( − u + v v + t )( − u − t )( − v + v − (cid:113) ( − u + v v + t )( − u − t )( − v + v + c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if √− v u < t < − v , c log (cid:34) t + u )( t + u )( t + v )( t + v ) (cid:15) ( u − v )( u − v ) (cid:16) sinh π ¯ γ ¯ γ (cid:17) (cid:35) + c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if − v < t < − u , c log + (cid:113) ( − u + v − t − u − u + u − v − t ) − (cid:113) ( − u + v − t − u − u + u − v − t ) + c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if − u < t < √− u v , c log + (cid:113) ( − u + v t − u − u + u − v + t ) − (cid:113) ( − u + v t − u − u + u − v + t ) + c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if √− u v < t < v , c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if v < t . (2.51)We can also consider the case 0 < (cid:15) (cid:28) u < v < u < v . The di ff erent monodromy e ff ectfrom the above case can happen when t > v . For v < t < u , we find the dominant channel( C n , m ) (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3041)(cid:3034) (cid:3251) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3041) × (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3041)(cid:3034) (cid:3251) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3041) . (2.52)The monodromy e ff ect can be illustrated as (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3041)(cid:3034) (cid:3251) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3041) × (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3041)(cid:3034) (cid:3251) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3041) . (2.53)Combining (2.52) with (2.53), we find that the reflected entropy can be evaluated by the follow-ing approximated correlator, 20 C n , m ) (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3041)(cid:3034) (cid:3251) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3041) × (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3041)(cid:3034) (cid:3251) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3041) . (2.54)Note that one of the e ff ects of crossing branch cut cancels the monodromy around z = u displayed in (2.52). We can apply the same technique as (2.33) to calculate each these left andlight blocks and then we obtain S R ( A : B )[ O ] = c + (cid:113) ( − u + v )( t − u )( − u + u )( − v + t ) − (cid:113) ( − u + v )( t − u )( − u + u )( − v + t ) + c + (cid:113) ( − u + v )( v + t )( − u − t )( − v + v ) − (cid:113) ( − u + v )( v + t )( − u − t )( − v + v ) , if v < t < u . (2.55)In a similar manner, we can also evaluate the reflected entropy for the other time region andthus we obtain S R ( A : B )[ O ] = c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if t < u , c log + (cid:113) ( − u + v v − t )( − u + t )( − v + v − (cid:113) ( − u + v v − t )( − u + t )( − v + v + c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if u < t < v , c log + (cid:113) ( − u + v t − u − u + u − v + t ) − (cid:113) ( − u + v t − u − u + u − v + t ) + c log + (cid:113) ( − u + v v + t )( − u − t )( − v + v − (cid:113) ( − u + v v + t )( − u − t )( − v + v , if v < t < u , c log + (cid:113) ( − u + t )( v − u − u + u − t + v − (cid:113) ( − u + t )( v − u − u + u − t + v + c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if u < t < v , c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if v < t . (2.56)However, this is not complete. In this setup, we have to take disconnected channel intoaccount. The entanglement wedge for two sybsystems after a local quench was studiedin [13]. From this result, we can find that in the case 0 < (cid:15) (cid:28) u < v < u < v , there is apossibility that the disconnected entanglement wedge is chosen as the minimal RT surface fortwo sybsystems. If we set | u − v | = | u − v | = l and | u − v | = d for simplicity, then thetransition time between connected and disconnected entanglement wedge is given by¯ v ≡ u + l + d (2 l + d ) d + dl − l , ¯ u ≡ u − l d + dl − l . (2.57) Here, we mean disconnected by the disconnected entanglement wedge, which generally leads to vanishing ofmutual information. This is NOT the disconnected entanglement wedge cross section, like the right upper sketchof Figure 6. S R ( A : B )[ O ] = c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if t < u , c log + (cid:113) ( − u + v v − t )( − u + t )( − v + v − (cid:113) ( − u + v v − t )( − u + t )( − v + v + c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if u < t < ¯ v , , if ¯ v < t < ¯ u , c log + (cid:113) ( − u + t )( v − u − u + u − t + v − (cid:113) ( − u + t )( v − u − u + u − t + v + c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if ¯ u < t < v , c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if v < t . (2.58) A and B In this subsection, we consider a local excitation inside the interval A . We can accomplish theevaluation for this state by using the same 6-point correlator (2.2) with 0 < (cid:15) (cid:28) v < u < v < − u . The early time reflected entropy is again that for the vacuum due to the same reason as inthe calculation of (2.20). For v < t < u , the dominant channel is( C n , m ) (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3041)(cid:3040) × (anti-holomorphic part) , (2.59)and the monodromy e ff ect by crossing the branch cut is (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3040) (cid:3041) × (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3041)(cid:3040) . (2.60)Therefore, the Regge limit of the 6-point function for v < t < u is approximated by( C n , m ) (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3040) (cid:3041) × (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3041)(cid:3040) . (2.61) Here, we choose the principle sheet on where the operators O ⊗ mn and O ⊗ mn † are inserted at t = S R ( A : B )[ O ] = c + (cid:113) ( u − t )( v − u )( − u + u )( t − v ) − (cid:113) ( u − t )( v − u )( − u + u )( t − v ) + c + (cid:113) ( u − v )( v − u )( − u + u )( v − v ) − (cid:113) ( u − v )( v − u )( − u + u )( v − v ) , if v < t < u . (2.62)This is a similar result to (2.35), because both of them is based on almost the same monodromytrajectory. On the other hand, the monodromy e ff ect at u < t < v is quite di ff erent from thecase discussed in the subsection 2.2.For u < t < v , the dominant channel is again( C n , m ) (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3041)(cid:3040) × (anti-holomorphic part) , (2.63)and the e ff ect of crossing the branch cut is (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3040) (cid:3041) × (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3040) (cid:3041) . (2.64)Combining these monodromy, we obtain a similar result to (2.45), c (cid:34) t − u )( t − u )( t − v )( t − v ) (cid:15) ( u − v )( u − v ) (cid:32) sinh π ¯ γ ¯ γ (cid:33) (cid:35) + c + (cid:113) ( − u + v )( v − u )( − u + u )( − v + v ) − (cid:113) ( − u + v )( v − u )( − u + u )( − v + v ) , if u < t < v . (2.65)The other choice (2.46) can be neglected for the same reason.We do not show the calculation of the reflected entropy for t > v because what we need todo is just to repeat the above. The result is as follows,23 R ( A : B )[ O ] = c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if t < v , c log + (cid:113) ( − u + t )( v − u − u + u − t + v − (cid:113) ( − u + t )( v − u − u + u − t + v + c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if v < t < u , c log (cid:34) t − u )( t − u )( t − v )( t − v ) (cid:15) ( u − v )( u − v ) (cid:16) sinh π ¯ γ ¯ γ (cid:17) (cid:35) + c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if u < t < v , c log + (cid:113) ( − t + v v − u − t + u − v + v − (cid:113) ( − t + v v − u − t + u − v + v + c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if v < t < √− u v , c log + (cid:113) ( t + v v − u t + u − v + v − (cid:113) ( t + v v − u t + u − v + v + c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if √− u v < t < − u , c log + (cid:113) ( − u + v v − u − u + u − v + v − (cid:113) ( − u + v v − u − u + u − v + v , if − u < t . (2.66) The reflected entropy measures correlations between A and B . In particular, if we restrict our-selves to a pure state (e.g., ρ = | Ψ ( t ) (cid:105) (cid:104) Ψ ( t ) | , studied in Section 2) and set B = ¯ A , then one canfind that this measure reduces to entanglement entropy. In general, the reflected entropy has thefollowing property, S R ( A : B ) = S ( A ) , if ρ AB is a pure state. (3.1)We should check that our result is consistent with this property.We consider a local excitation in an interval A = [ l , l ] with 0 < l < l . The entanglemententropy for this locally excited state had studied in [53, 71] (non-perturbatively in [51, 61, 66])and the result is S ( A )[ O ] = c log (cid:16) l − l µ (cid:17) , if t < l , c log (cid:16) ( l − t )( t − l ) (cid:15) ( l − l ) sinh( π ¯ γ )¯ γ (cid:17) + c log (cid:16) l − l µ (cid:17) , if l < t < l , c log (cid:16) l − l µ (cid:17) , if t > l , (3.2)where a positive constant µ is a UV cuto ff to regulate the twist operators. To compare theentanglement entropy with the reflected entropy (2.65), we take the pure state limit by setting, v = l − µ, u = l + µ, v = l − µ, u = l + µ, (3.3)24nd then take the limit µ →
0. As a result, we obtain S R ( A : B )[ O ] −−−−−−−−−→ pure state limit c (cid:32) ( l − t )( t − l )( l − l ) (cid:15)µ sinh( π ¯ γ )¯ γ (cid:33) , if l < t < l , (3.4)which perfectly reproduces the entanglement entropy (3.2). Note that reflected entropy can beused as a natural regulator for entanglement entropy in QFT [18] and here one can find that thereflected entropy plays a role as a regulator of the entanglement entropy after a local quench. In this section, we would like to understand how dynamics of the correlation measures is char-acterized. To this end, we will show various plots of the reflected entropy and read o ff importantnature of its dynamics. It is important to emphasize that there is another useful correlation mea-sure, mutual information . Therefore, it is very interesting to discuss similarities and di ff erencesbetween dynamics of reflected entropy and mutual information. To simplify the comparisonwith mutual information, we show the explicit form of the mutual information after a localquench in the following, • Quench outside intervals ( 0 < (cid:15) (cid:28) u < − v < − u < v and O is acted on x = t = I ( A : B )[ O ] = S ( u , v ) + S ( u , v ) − S ( v , u ) − S ( u , v ) , if t < u , S ( u , v , t ) + S ( u , v ) − S ( v , u , t ) − S ( u , v ) , if u < t < √− u v , S ( u , v , t ) + S ( u , v ) − S ( v , u , − t ) − S ( u , v ) , if √− u v < t < − v , S ( u , v , t ) + S ( u , v , − t ) − S ( v , u ) − S ( u , v ) , if − v < t < − u , S ( u , v , t ) + S ( u , v ) − S ( v , u ) − S ( u , v , − t ) , if − u < t < √− u v , S ( u , v , t ) + S ( u , v ) − S ( v , u ) − S ( u , v , t ) , if √− u v < t < v , S ( u , v ) + S ( u , v ) − S ( v , u ) − S ( u , v ) , if v < t , (4.1) • Quench inside intervals (0 < (cid:15) (cid:28) v < u < v < − u and √− v u < v and O is acted on x = t = ( A : B )[ O ] = S ( u , v ) + S ( u , v ) − S ( v , u ) − S ( u , v ) , if t < v , S ( u , v ) + S ( u , v , t ) − S ( v , u , t ) − S ( u , v ) , if v < t < u , S ( u , v , t ) + S ( u , v , t ) − S ( v , u ) − S ( u , v ) , if u < t < √− u v , S ( u , v , t ) + S ( u , v , − t ) − S ( v , u ) − S ( u , v ) , if √− u v < t < v , S ( u , v ) + S ( u , v , − t ) − S ( v , u ) − S ( u , v , t ) , if v < t < √− u v , S ( u , v ) + S ( u , v , − t ) − S ( v , u ) − S ( u , v , − t ) , if √− u v < t < − u , S ( u , v ) + S ( u , v ) − S ( v , u ) − S ( u , v ) , if − u < t , (4.2)where we define S ( x , y ) = c y − x µ , S ( x , y , t ) = c | ( y − x )( y − t )( t − x ) | (cid:15)µ sinh πγγ , (4.3)and we assume γ = ¯ γ = (cid:113) c h O − It means that the falling particle bulk interpretation [52] of a local quench statecan be applied not only to the single interval entanglement entropy but also to more refinedcorrelation measures, mutual information and reflected entropy.Note that in [13], the holographic mutual information is compared to not the local operatorquench state but the joining quench state. Therefore, they find the di ff erence between the bulkresult and the CFT result. Particularly, the remarkable di ff erence is that the long range entangle-ment is found only in the CFT side (which can be also found for negativity [17]). However, ourapproach shows that the local operator quench state perfectly reproduces the holographic mu-tual information and then we cannot find such a long range entanglement. It means that the longrange entanglement is a particular feature of the joining quench state. We expect that this longrange e ff ect can be completely understood by the recent development of the bulk interpretationof the joining quench state [4, 6]. A similar CFT calculation can be found in [72], but it might not be rigorous because their calculation of the6-point Virasoro block is based on a wrong assumption, even in the Regge limit, σ n × ¯ σ n = I + · · · . (4.4)As shown in [51], the Regge limit of this OPE is dominated by a NON-vacuum state. Actually, in a specialcase, this assumption somehow gives a correct result and their final expression becomes consistent with the bulkcomputation. However, in general, this assumption leads to a wrong estimate. On the other hand, our methodintroduced in Section 2 can be applied to any situations.
10 15 20 t Reflected Entropy vs. Mutual Information S R ( A:B ) I ( A:B ) t Reflected Entropy vs. Mutual Information S R ( A:B ) I ( A:B ) t Growth of Reflected Entropy vs. Mutual Information Δ S R ( A:B ) Δ I ( A:B ) t Growth of Reflected Entropy vs. Mutual Information Δ S R ( A:B ) Δ I ( A:B ) Figure 4: Reflected entropy (blue) and mutual information (yellow) for a state locally quenchedoutside two intervals. Here we have set ( u , v , u , v ) = ( − , − , , (cid:15) = − , γ = c . We check that this parameter set satisfies the connected condition0 < ( v − u )( u − v )( v − v )( u − u ) < . Each blue dot shows a transition of itself or its first derivative.In Figure 4, we show the time-dependence of reflected entropy and mutual information inthe setup ( 0 < (cid:15) (cid:28) u < − v < − u < v ). A first observation of this graph is that the reflectedentropy is always larger than the mutual information. In fact, as shown in [18], the reflectedentropy is bounded by the mutual information as S R ( A : B ) ≥ I ( A : B ) . (4.5)That is, our result is perfectly consistent with this lower bound. An important di ff erence be-tween mutual information and reflected entropy can be found at t = √− u v , √− u v , themutual information is continuous, on the other hand, the reflected entropy is discontinuous. Tomake it clear, we zoom into early time region in the right of the figure. In the lower two plots,we show the di ff erence between the local quench state and the vacuum state, ∆ S R ( A : B ) = S R ( A : B )[ O ] − S R ( A : B )[ I ] , ∆ I ( A : B ) = I ( A : B )[ O ] − I ( A : B )[ I ] , (4.6)which measure a growth of correlations after a local quench. In fact, they behave very similarly,but interestingly, we find the following inequalities for the mutual information and reflectedentropy, ∆ S R ( A : B ) ≥ ∆ I ( A : B ) , if t (cid:60) [ − v , − u ] , ∆ S R ( A : B ) ≤ ∆ I ( A : B ) , if t ∈ [ − v , − u ] . (4.7) This does not contradict with S R ( A : B ) ≥ I ( A : B ) because this is just a di ff erence between the excited stateand the vacuum state. S R ( A : B ) ≥ I ( A : B ) has already shown in Figure 4.
27t implies that the reflected entropy measure the dynamics of the correlations in a quite di ff erentway from the mutual information. And this inequalities might be a key to understanding whatcorrelations are measured by reflected entropy from the physical view point. Possibly, it mightbe interpreted in the following. The growth in t ∈ [ − v , − u ] is strongly caused by the quantumcorrelations, on the other hand, it would be expected that in t (cid:60) [ − v , − u ], the excitationchanges both quantum correlations and classical correlations in a similar manner. The point isthat in the holographic CFT, the mutual information probes quantum correlations more purelythan the reflected entropy. Therefore, the quantum correlations in t ∈ [ − v , − u ] comparedwith the classical correlations result in the large growth of the mutual information, thus weobtain ∆ S R ( A : B ) ≤ ∆ I ( A : B ), while in t (cid:60) [ − v , − u ], the change of the quantum correlationsare not larger than the classical correlations enough to satisfy ∆ S R ( A : B ) ≤ ∆ I ( A : B ).Note that if we take two intervals A = [ −∞ , v ] and B = [ u , ∞ ] and focus on the late timelimit t (cid:29) (cid:15) , then these two quantities approach ∆ S R ( A : B ) ∼ ∆ I ( A : B ) ∼ ∆ S ( A ) + ∆ S ( B ) , (4.9)where ∆ S ( A ) is the growth of the entanglement entropy for the interval A after a local quench, ∆ S ( A ) ∼ c t (cid:15) . (4.10)This would be natural because in the late time limit, quasi particles do not interact with eachother.We have to comment that the reflected entropy is expected to be non-zero only in the timeregion t ∈ [ u , v ] from the quasi particle picture [2, 10] and our result is perfectly consistentwith this expectation. However, the behavior in the time-dependent region cannot be capturedby the quasiparticle picture, which is one of the characteristics of the holographic CFT. It wouldbe worth mentioning that in the nontrivial time region t ∈ [ u , v ], there are two phases as shownin the figure. The remarkable features in each phase is as follows: • t ∈ [ u , − v ] ∪ [ − u , v ]The reflected entropy is independent of the conformal dimension h O and does not includehigh energy scale (the UV cuto ff parameter (cid:15) ). • t ∈ [ − v , − u ]The reflected entropy depends on the conformal dimension h O and includes high energyscale. This intuition comes from the inequality S R ( A : B ) ≥ I ( A : B ). Moreover, the holographic mutual informationsatisfies the monogamy relation, while the holographic reflected entropy only satisfies the strong superadditivity,which is a weaker version of the monogamy relation. We do not have a further explanation for the reflected entropy,however, we can give a clearer explanation for the entanglement of purification by the fallowing inequality for anyseparable state, E P ( A : B ) ≥ I ( A : B )2 > I ( A : B )2 . (4.8)
10 15 20 t Reflected Entropy vs. Mutual Information S R ( A:B ) I ( A:B ) t Reflected Entropy vs. Mutual Information S R ( A:B ) I ( A:B ) t Growth of Reflected Entropy vs. Mutual Information Δ S R ( A:B ) Δ I ( A:B ) t Growth of Reflected Entropy vs. Mutual Information Δ S R ( A:B ) Δ I ( A:B ) Figure 5: Reflected entropy (blue) and mutual information (yellow) for a state locally quenchedinside two intervals. Here we have set ( u , v , u , v ) = ( − , , , (cid:15) = − , γ = c . We check that this parameter set satisfies the connected condition0 < ( v − u )( u − v )( v − v )( u − u ) < . Each blue dot shows a transition of itself or its first derivative.It means that when the left or right moving excitation enters one interval, the excitation a ff ectsthe reflected entropy but its e ff ect is not so strong, on the other hand, if both left and rightmoving excitations enter two intervals, then the reflected entropy becomes much larger thanthat for the vacuum. This strong e ff ect comes from the entanglement between two intervals,which is created by the excitation. However, we do not have any clear explanation of the smalle ff ect found in t (cid:60) [ − v , − u ]. Note that this small e ff ect does not appear in RCFTs (see Section7). In Figure 5, we show the reflected entropy and the mutual information in the di ff erent setup(0 < (cid:15) (cid:28) v < u < v < − u and √− v u < v ). The main di ff erence is that there isan additional transition for the mutual information, in that, the first derivative of the mutualinformation is discontinuous at t = √− u v , which can not be observed for the reflected entropy.Moreover, we can find the inequalities (4.7) and the transition of the reflected entropy at t = √− u v as seen in Figure 5. And also we find the agreement with the quasi particle picture in t (cid:60) [ v , − u ].Finally, we would like to comment that our interpretation by comparing between the re-flected entropy and the mutual information can be also applied to the entanglement of purifica-tion. This is because these two quantities reduces to the same entanglement wedge cross sectionin the holographic CFT. 29 Reflected Entropy in Heavy State from CFT
We consider a CFT on a circle with length L . Then, the reflected entropy for a heavy state canbe calculated from 11 − n log Z n , m (cid:0) Z , m (cid:1) n , (5.1)where Z n , m = (cid:68) O ⊗ mn (cid:12)(cid:12)(cid:12)(cid:12) σ g A ( − u ) σ g − A ( − v ) σ g B ( u ) σ g − B ( v ) (cid:12)(cid:12)(cid:12)(cid:12) O ⊗ mn (cid:69) CFT ⊗ mn . (5.2)Here, this correlator is defined on a cylinder. This can be mapped to the plane ( z , ¯ z ) by z = e π iwL , ¯ z = e − π iwL . (5.3)In this coordinates, we can also evaluate it by a single block approximation as in Section 2.In this setup, we are very interested in a question, whether we can reproduce the transitionof the entanglement wedge cross section or not. It is known that the entanglement wedgecross section has a transition as shown in the upper of Figure 6. That is, it is possible thatthe minimal cross section is given by the disconnected codimension-2 surfaces which haveendpoints on the black hole horizon, instead of the connected surface. Actually, there is noreason to reproduce this transition from the reflected entropy (5.1) because our heavy state(i.e., ρ = | O (cid:105) (cid:104) O | ) is “pure” but the BTZ microstate is “mixed”. Nevertheless, it might bepossible to find this transition from the CFT side, because the reduced density matrix could beapproximated by that for a microstate of BTZ in the large c limit. Naively, we can expect thatthe transition in the bulk side can be translated into a change of the dominant channel as shownin the lower of Figure 6. If this naive expectation is true, then the disconnected cross sectionshould be reproduced from the following single block approximation,( C n , m ) ( C n , O ) (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993)(cid:3034) (cid:3251) (cid:3041) (cid:3041) × (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993)(cid:3034) (cid:3251) (cid:3041) (cid:3041) . (5.4)The intermediate state p is dominated by O ⊗ as explained below (2.46). The constant C n , O isthe OPE coe ffi cient between O ⊗ mn and σ g − B g A and its asymptotics in the limits c → ∞ , n → C n , O → γ h n ¯ γ h n , (5.5)with γ = (cid:113) c h O − γ = (cid:113) c ¯ h O −
1. This is justified in the holographic CFT [73], whichis explained in Appedix C. Notice that we have no exponential suppression from the OPE coef-ficients. Moreover, the degeneracy of the primary fields should be also 1 becasue we are takingthe OPE including twist operators.The limit m → (cid:104) O ⊗ | O ⊗ (cid:105) n = which means that the LLHHLL block isdecomposed into two HHLL blocks as (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993)(cid:3034) (cid:3251) (cid:3041) (cid:3041) −−−−→ HHLL (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993)(cid:3034) (cid:3251) (cid:3041) × (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:2993) (cid:3041) . (5.7)Note that this expression is precise only under the m , n → S R ( A : B ) = c (cid:32) coth πγ ( v + u )2 L (cid:33) + c (cid:32) coth π ¯ γ ( v + u )2 L (cid:33) . (5.8)The detailed calculation is shown in Appendix B.3. CFT description 𝜎 (cid:3034) (cid:3250) 𝑂 ⊗(cid:3040)(cid:3041) (cid:3041) (cid:3034) (cid:3251) 𝜎 (cid:3034) (cid:3251)(cid:3127)(cid:3117) 𝜎 (cid:3034) (cid:3250)(cid:3127)(cid:3117) 𝑂 ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3041) 𝜎 (cid:3034) (cid:3250) 𝑂 ⊗(cid:3040)(cid:3041) 𝜎 (cid:3034) (cid:3251)(cid:3127)(cid:3117) 𝜎 (cid:3034) (cid:3250)(cid:3127)(cid:3117) 𝑂 ⊗(cid:3040)(cid:3041)(cid:2993) 𝑝 𝜎 (cid:3034) (cid:3251) (cid:3041) (cid:3041) transitiontransition Figure 6: The non-trivial entanglement wedge cross section in the BTZ background has twocandidates. One is the connected codimension-2 surface and the other is the disconnectedcodimension-2 surfaces which have endpoints on the black hole horizon. The correct choiceis the minimal one. If we could observe this transition in the CFT side, it should come from achange of the dominant channel in the large c limit as shown in the lower of this figure. We can show this fact by using the Virasoro algebra as in Appendix E of [74] and this is also justified by themonodromy method [70]. c limit [75–79] can also be found in the reflectedentropy. Our result also answers the interesting physics question, what is the bulk dual of ourquench state. We show that the surface ends at the horizon of the black hole. This can beexplained by considering the horizon as an end of the world brane [80–83]. In this case, thesurface can end at the horizon even if we consider a pure state black hole. We have to mentionthat this idea should be also applied to the entanglement entropy in a heavy state because thereflecte entropy (5.8) should reproduce the entanglement entropy by the relation (3.1) in thepure state limit. Note that the entanglement entropy from the pure state limit of the reflectedentropy does not match the result in [53]. This is because their derivation assumes that thechange of the dominant channel (i.e., the transition shown in Figure 6) does not happen. Thiswas because we expected the OPE coe ffi cients in another channel is suppressed exponentiallyunder the large c limit. However, as we have seen here, this is actually not the case at leastunder the n → In the calculation of entanglement entropy in the holographic CFT, the large c limit commuteswith the von-Neumann limit in usual setups (vacuum state, local quench state, etc.). However,we have to calculate the reflected entropy by taking first the limit c → ∞ even if we considerthe vacuum state. We discuss this problem in this section.To calculate reflected entropy or entanglement entropy for two intervals A and B in thevacuum state, we start with the semiclassical block (A.2) (and its anti-holomorphic block), F LLHH ( h p | z ) = (1 − z ) h L ( δ − (cid:32) − (1 − z ) δ δ (cid:33) h p − h L + (1 − z ) δ − h p , (6.1)with δ = (cid:113) − c h H and then we obtain the entanglement entropy [62] by setting h p to be zeroand h H = h L = c (cid:16) n − n (cid:17) , S ( A : B ) = c z µ , (6.2)and the reflected entropy (2.21) by setting h H , h L = h p = c (cid:16) n − n (cid:17) , S R ( A : B ) = c + √ − z − √ − z . (6.3)32ere, we focus on the nontrivial case where the entanglement wedge is disconnected. The crossratio is related to the coordinated as z = ( v − u )( u − v )( v − v )( u − u ) , (6.4)and the connected condition can be expressed in terms of the cross ratio as 0 < z < .If one wants to take first the von-Neumann limit, one cannot use the semiclassical block be-cause this block is defined in the limit c → ∞ with h p c , h L c , h H c fixed. Actually, the von-Neumannlimit also simplifies evaluating the block, for example, the entanglement entropy calculated bythe following simplification at any c > F LLLL (0 | z ) −−−−→ h L → − h L log z + O ( h L ) , (6.5)which perfectly reproduces (6.2). On the other hand, the reflected entropy is calculated by F LLLL ( h p | z ) −−−−→ h p → + h p log z √ − z + O ( h p ) , (6.6)where we first take the limit h L →
0. The result is˜ S R ( A : B ) = c √ − zz , (6.7)which is quite di ff erent from (6.3). We have to mention that we take the large c limit after thevon-Neumann limit to approximate the correlator by a single block. The motivation to reversethere two limits, c → ∞ and the von-Neumann limit, is to understand non-perturbative e ff ectsto the reflected entropy.The discrepancy between (6.7) and (6.3) means that the two limits c → ∞ and m , n → ff ects in the reflectedentropy, which cannot be found in the entanglement entropy because c → ∞ and n → S R ( A : B ) as the reflected entropyincluding quantum corrections . We can immediately find that the inequality ˜ S R ( A : B ) ≥ I ( A : B ) is satisfied from the left of Figure 7 and also show the two monotonicity inequalities of theholographic reflected entropy, S R ( A : BC ) ≥ I ( A : B ) + I ( A : C ) , S R ( A : BC ) ≥ S R ( A : B ) . (6.8)We plot the di ff erence between ˜ S R ( A : B ) and S R ( A : B ) in the right of Figure 7. From this, wecan find that the quantum correction is always negative. This is natural because the quantumcorrection should smooth the transition of the reflected entropy at z = , therefore, the quantum More precisely, in the calculations of entanglement entropy and reflected entropy, we assume that our CFThas c > h L , h p → S R ( A : B ) in order to connect twodisconnected lines at z = as sketched in Figure 8. Note that other quantum corrections comefrom sub-leading conformal blocks. This e ff ect can also be calculated in the same way and itexpected to be negative. This is one of interesting directions for future research.The non-perturbative e ff ect for a local quench state can be also evaluated in the same way.In Figure 9, we show the time-dependence of the non-perturbative e ff ect in the same setup asin Figure 4. One can find that the non-perturbative e ff ect after the transition at t = √− u v becomes very small. It is natural because this transition at t = √− u v is attributed not bythe large c limit but by the (cid:15) → z Reflected Entropy vs. Mutual Information S ˜ R ( A:B ) I ( A:B ) z - - - - - - S ˜ R ( A:B )- S R ( A:B ) Figure 7: (Left) This shows the z -dependence of the non-perturbative reflected entropy and themutual information. We can check the inequality ˜ S R ( A : B ) ≥ I ( A : B ). (Right) The di ff erencebetween ˜ S R ( A : B ) and S R ( A : B ). We can find that the quantum correction is always negative.It might be natural because the quantum corrections should smooth the transition at z = inthe left figure, in other words, the corrections should decrease the classical reflected entropy S R ( A : B ). It is very interesting to compare our result to the dynamics of the reflected entropy in otherCFTs, in particular, integrable CFTs. There are many works to study entanglement entropyafter a local quench in various setups [4, 5, 52, 53, 71, 86–91]. Their motivation is to characterizeCFT classes by the dynamics of entanglement. And from those results, this quantity is expectedto capture the chaotic natures of CFTs. On this background, it is naturally expected that byusing a refined tool, reflected entropy, we can obtain more information to classify CFTs. In thissection, we will briefly discuss how the reflected entropy grows after a local quench in RCFTsand investigate whether the RCFT reflected entropy has a di ff erent growth from the holographicreflected entropy or not. 34 o smooth the transition,quantum corrections decrease the classical (cid:3019) ?Classical (cid:3019)(cid:3019) with full quantum corrections ? Figure 8: Sketch of the e ff ect of quantum corrections. It is naturally expected for the quantumcorrections to decrease the classical reflected entropy to smooth the transition. t - - - - - - - S ˜ R ( A:B )- S R ( A:B ) Figure 9: The di ff erence between ˜ S R ( A : B ) and S R ( A : B ) for a local quench state. Here theparameters are set to be ( u , v , u , v ) = ( − , − , , (cid:15) = − , γ = ¯ γ = c is removed.An important di ff erence between the holographic CFT and RCFTs is that in the former,the OPE in the Regge limit does not contain the vacuum state, whereas in the later, the vac-uum state can propagate even in the Regge limit. As a result, the time-dependence cannot befound in RCFTs. We will briefly explain this mechanism of the vanishing time-dependence byconsidering an analogy of (2.33) (see also (B.1)) in RCFTs.In our CFT, the Regge limit of this block is obtained by the monodormy matrix as (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3041)(cid:3040) −−−→ (cid:15) → ˜ M ( − )0 , α m (cid:34) α m α m α O α O (cid:35) × (2 i (cid:15) ) h α m − nmh O (cid:3034) (cid:3250) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) (cid:3034) (cid:3251) (cid:3041)(cid:3040)(cid:2870)(cid:3080) (cid:3288) , (7.1) In the analytic continuation m →
1, the exponent is replaced by 2 mnh O → nh O by the squaring rule (2.10). h a = α ( Q − α ). We would like to mention that the time-dependence is encapsulated inthe position of the external operator h α m . On the other hand, if we consider the Regge limit inRCFTs, (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3041)(cid:3040) −−−→ (cid:15) → M ( − )0 , (cid:34) α m α m α O α O (cid:35) × (2 i (cid:15) ) − nmh O (cid:3034) (cid:3250) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) (cid:3034) (cid:3251) (cid:3041)(cid:3040) . (7.2)The key point is that the operator h α m is replaced by the identity, therefore, this 5-point blockreduces a 4-point block, (cid:3034) (cid:3250) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) (cid:3034) (cid:3251) (cid:3041)(cid:3040) = (cid:3034) (cid:3250) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) (cid:3034) (cid:3251) (cid:3041) . (7.3)This means that the time-dependence disappears in this single block approximation. The way tocalculate the reflected entropy in RCFTs is just repeating the calculation in Section 2 replacing(7.1) by (7.2). As a result, if we consider the setup ( 0 < (cid:15) (cid:28) u < − v < − u < v and O isacted on x = t = ∆ S R ( A : B )[ O ] = , if t < − v , d O , if − v < t < − u , , if − u < t , (7.4)where d O is a constant, so-called quantum dimension , which is re-expressed in terms of themodular S matrix as [86, 87] d O = S O S . (7.5)We would like to comment that this result is consistent with the relation (3.1). Namely, inthe pure state limit ( ¯ A = B ), the reflected entropy reduces to the entanglement entropy, whichimplies ∆ S R ( A : B )[ O ] = ∆ S ( A )[ O ] = d O . (7.6)This is consistent with the previous result ∆ S ( A )[ O ] = log d O in [86]. We will show the de-tailed calculation in a future paper about reflected entropy in finite c CFTs. Consequently, wecan conclude that the reflected entropy in RCFTs cannot grow as in the holographic CFT andin fact, the dynamics can be fully captured by the quasi-particle picture. More concretely, ifthe quasi-particle enters a interval, then entanglement is created between the interval and its36omplement. In terms of the reflected entropy, this phenomena can be observed as a non-zeroconstant characterized by the quantum dimension, like entanglement entropy [86, 87]. In otherwords, the RCFT reflected entropy after a local quench is characterized by a step function,which is quite di ff erent from the holographic case. We show the comparison of the reflectedentropy between holographic CFT and Ising model in Figure 10. One can find two significantdi ff erences from this Figure, • The small e ff ect in t ∈ [ u , − v ] ∪ [ − u , v ] does not appear in RCFTs, unlike the holo-graphic CFT (see also Section 4). • The holographic CFT shows the logarithmic growth in t ∈ [ − v , − u ], on the other hand,the growth of RCFT approaches a finite constant.This di ff erence between RCFT and holographic CFT means that the reflected entropy might bealso related to a nature of chaos in a given CFT, therefore, we expect that by making use ofthe reflected entropy, we can also study the information scrambling [12, 61], which might bea interesting direction for future work. It would be interesting to note that this growth pattern(7.4) is exactly the same as that of the mutual information. This is a stronger version of thedecoupling relation (4.9), which is quite natural because in RCFTs, the quasi particle picturecan be applied in any time region.These properties are quite di ff erent from the holographic case as show in (4.7), therefore,we could classify CFTs by studying whether the growth of reflected entropy and mutual in-formation are di ff erent or not. Further studies in this direction shed light on what correlationsare measured by reflected entropy. We would also like to mention that the quantum dimensioncan be interpreted as an e ff ective degrees of freedom included in the operator O and our resultsuggests that the reflected entropy captures this degrees of freedom, like entanglement entropy. As mentioned in the introduction, the odd entanglement entropy in holographic CFTs alsomatches the reflected entropy (the entanglement wedge cross section) in our dynamical setup.These agreement can be understood from a similarity of the methods to calculate the odd entan-glement entropy and the reflected entropy especially in the holographic CFTs. (Interestingly,this agreement is also the case for RCFTs. ) In this section, we will sketch the proof of thiscoincidence. An interesting point is that this quantity is not based on the purification, therefore,it is nontrivial in this sense that this quantity also reproduces the entanglement wedge crosssection, like the reflected entropy.Following the definition (1.9), the odd entanglement entropy in our setup can be obtainedfrom the following correlation function,tr (cid:16) ρ T B AB (cid:17) n = (cid:68) σ n ( u ) ¯ σ n ( v ) O ⊗ n ( w , ¯ w ) O ⊗ n † ( w , ¯ w ) ¯ σ n ( u ) σ n ( v ) (cid:69) CFT ⊗ n (cid:10) O ( w , ¯ w ) O † ( w , ¯ w ) (cid:11) n , (8.1) Here, we mean not S R ( A : B ) = I ( A : B ) but ∆ S R ( A : B ) = ∆ I ( A : B ).
10 15 20 t Δ S R ( A:B ) Growth of Reflected Entropy in holographic vs. RCFT holographicRCFT
Figure 10: The growth of reflected entropy in holographic CFT (blue) and Ising model (yellow). ∆ S R means the di ff erence between the excited state and the vacuum state. Here ( u , v , u , v ) = ( − , , , (cid:15) = − and we divide them by c . We choose γ = O = σ in Ising model. Each blue dot shows a transition of itself or its first derivative.where σ n and ¯ σ n correspond to the usual twist operators with twist number ± n is theanalytic continuation of an odd integer. If one assumes an even integer analytic continuation,the (8.1) is nothing but the one for the negativity. Note that for the odd entanglement entropythe complications from the decoupling e ff ect (as like reflected entropy and negativity) do notappear. This is just because we take here the analytic continuation of an odd integer, thus nodecoupling of the replica sheet happens [14, 15]. Therefore, we can safely use the Virasoroconformal blocks for the calculation of the odd entanglement entropy as like the entanglemententropy in holographic CFTs.If one evaluates the (8.1) in the holographic CFTs, one can again approximate it as a sin-gle semiclassical conformal block. The semiclassical conformal block (more precisely, thelinearized semicalssical block [92]) has the following form,log F ( z i ) ∼ h f ( z i ) + h p f p ( z i ) + O ( h , h p ) , (8.2)where external dimensions h and internal dimensions h p are given by the form, h ∼ h p ∼ σ c with σ (cid:28) , (8.3)and the functions f and f p are of order one. The Landau symbol O ( x , y ) stands for variousquantities vanishing as x n y m → n + m ≥
2. Here the entanglement entropy is obtained by f ( z i ) because the corresponding correlator is dominated by the vacuum block (i.e., h p =
0) [62].Let us recall the case of the reflected entropy. After all the reflected entropy came from this f p ( z i ) because we take the limit h → m → h n aslog F ( z i ) = < denominator in (2 . > + h n f p ( z i ) + O (cid:16) (1 − n ) (cid:17) . (8.4)38 inimizeEWCS + EE Odd Entanglement EntropyEntanglement Wedge Cross Section
MinimizeEWCS
Figure 11: The reflected entropy is given by minimizing the red line in the left, on the otherhand, the odd entanglement entropy is given by minimizing the red lines in the right, which isthe sum of two RT surfaces and the entanglement wedge cross section.Here the first term is compensated by the denominator and the second term 2 f p corresponds tothe value of reflected entropy. Remind that the factor 2 of 2 f p comes from “doubling of Virasoroblock” due to the doubling of the Hilbert space (namely, the even integer analytic continuation).Thus, we can immediately show that S O ( A : B )[ O ] − S ( A : B )[ O ] = c f p ( z i ) = S R ( A : B )[ O ] , (8.5)where we used the fact that the conformal block related to the odd entanglement entropy hasthe intermediate dimension h p = h n as shown in [37]. It means that the calculation of the oddentanglement entropy is just a repetition of that in section 2.Strictly speaking, it might happen to find the disagreement between reflected entropy andodd entanglement entropy, because reflected entropy is based on the minimal of the entan-glement wedge cross section, on the other hand, the odd entanglement entropy computes theminimal of the sum of two RT surfaces and the “entanglement wedge cross section” (see Figure11) . This could cause a change of the dominant channel of the single block approximations.However, we can easily check the agreement between the reflected entropy and the odd en-tanglement entropy (up to prefactor 2) by assuming µ (cid:28) (cid:15) (cid:28)
1. We expect that these twominimizing problems provide the same result .Since the reflected entropy for RCFT in section 7 relies on the single conformal block ap- We abused the word “entanglement wedge cross section” (precisely, the minimal surface which ends at twoRT surfaces). It is not necessary that this corresponds to the (minimal) entanglement wedge cross section. In the regime µ ∼ (cid:15) , the area of the “entanglement wedge cross section” could be comparable to area of thetwo RT surfaces. In such regimes, we potentially have this deviation. Clarifying such possibilities in more generaldynamical setup might be an interesting future direction. ∆ S O ( A : B )[ O ] − ∆ S ( A : B )[ O ] = ∆ S R ( A : B )[ O ] (for RCFT) . (8.6)Therefore, we can use S O ( A : B ) as a signature of the chaos as like the reflected entropy.However, we suspect that the “bare values”, S O ( A : B )[ O ] − S ( A : B )[ O ] and S R ( A : B )[ O ] forRCFT, should behave quite di ff erently. In this section, we consider the entanglement wedge cross section in the Poincare AdS geom-etry, d s = d z − d t + d x z , (9.1)with a falling particle whose trajectory is given by z − t = (cid:15) , x = . (9.2)Here (cid:15) corresponds not to the cuto ff for radial direction (UV cuto ff in CFT side) but to the sizeof the particle. We will define the cuto ff for radial direction by µ . We also set AdS radius (cid:96) AdS ≡ x , t ) = (0 ,
0) in the holographic CFT [52].Since the falling particle gets boosted under the time evolution, we must take into accountthe back-reaction due to the boosted particle. By using the global coordinates, one can putthe falling particle always on the center and represent the back-reacted geometry outside of theparticle [52, 93] as d s = − ( r + − M )d t + d r r + − M + r d θ , (9.3)where M characterizes the mass of the particle. For M <
1, this metric describes the geometrywith a conical deficit located at r =
0. For M ≥
1, it gives rise to the static BTZ geometrywith mass M −
1. In particular, we are interested in the latter BTZ setup. To this end, onecan analytically continue the former results to the latter ones √ − M → i √ M − ≡ i γ . Notethat one can identify the present γ = √ M − γ = (cid:113) c h O −
1. The static BTZ corresponds to γ = ¯ γ . In section 9.3, we will briefly discussthe γ (cid:44) ¯ γ case, dual to the rotating BTZ blackhole.Since the above geometries are locally AdS , it is very useful to write them by using theembedding coordinates in R , :d s = η AB d X A d X B = − d X − d X + d X + d X , (9.4)with X = − , (9.5)40here we defined X · Y ≡ η AB X A Y B . (9.6)Then the geometry (9.2) is given by X = tz , (9.7a) X = (cid:15) + (cid:15) − ( z + x − t )2 z , (9.7b) X = xz , (9.7c) X = − (cid:15) + (cid:15) − ( z + x − t )2 z . (9.7d)On the other hand, one can describe the back-reacted geometry in global coordinates as thefollowing coordinates: X = (cid:114) r + − M − M sin (cid:16) √ − M τ (cid:17) , (9.8a) X = (cid:114) r + − M − M cos (cid:16) √ − M τ (cid:17) , (9.8b) X = r √ − M sin (cid:16) √ − M θ (cid:17) , (9.8c) X = r √ − M cos (cid:16) √ − M θ (cid:17) , (9.8d)where we chose τ ∈ [0 , π ] ( τ ∈ [ − π, t ≥ t ≤
0) and θ ∈ [0 , π ] ( θ ∈ [ − π, x ≥ x ≤ θ ∼ θ + π whichwill become important for later analysis. Having this identification in mind, we can easily relatethese two geometries by using the above embedding coordinates. First, we derive the geodesic distance between two geodesics anchored on the boundary points.This will be very useful to obtain the entanglement wedge cross section of our interests. In theembedding coordinates, the length of the geodesics ending on the bulk points X i and X j is givenby σ ( X i , X j ) = log( ξ − i j + (cid:113) ξ − i j − (cid:113) ξ − i j +
1) (9.9a) ξ − i j = − X i · X j . (9.9b)On the other hand, the spacelike geodesics γ i j anchored on two bulk points X i and X j is givenby X Ai j ( λ ) = m A e − λ + n A e λ , (9.10)41here m = n = , m · n = − . (9.11)If we have X ( λ i ) = X i , X ( λ j ) = X j , (9.12)as a boundary condition and if both X i and X j are su ffi ciently close to the boundary, we canwrite X Ai j ( λ ) = X Ai e − λ + X Aj e λ (cid:112) − X i · X j , (9.13)where e − λ i = e λ j = (cid:112) − X i · X j ( ≡ (cid:113) ξ − i j ) . (9.14)We would like to find the pair of parameters ( λ, λ (cid:48) ) = ( λ ∗ , λ (cid:48)∗ ) which minimizes (extremizes) thelength of geodesics σ ( λ, λ (cid:48) ) ≡ σ ( X ( λ ) , X ( λ (cid:48) )). As a result, we find λ ∗ =
14 log (cid:34) ( ξ − )( ξ − )( ξ − )( ξ − ) (cid:35) , λ (cid:48)∗ =
14 log (cid:34) ( ξ − )( ξ − )( ξ − )( ξ − ) (cid:35) , (9.15)and ξ − i j ( λ ∗ , λ (cid:48)∗ ) = √ v (1 + √ u ) . (9.16)Here u and v are given by, u = ξ − ξ − ξ − ξ − , v = ξ − ξ − ξ − ξ − , (9.17)and reduce to the standard cross ratio in the CFT side. Therefore, we have obtained E W = G σ ( λ ∗ , λ (cid:48)∗ ) = G log + √ u + (cid:113) (1 + √ u ) − v √ v , (9.18)which has e ff ectively the same form as the AdS one in the embedding coordinates [23]. Herewe introduced the Newton constant by G , which is related to the central charge as c = G by the AdS / CFT dictionary [94]. We will apply the above formula (9.18) in order to obtain theentanglement wedge cross section in the falling particle geometry. Notice that, however, we hadthe identification θ ∼ θ + π along the angular direction. Therefore, we have multiple solutions,most of which correspond to the solutions with non-trivial winding around the deficit angle(or the blackhole). What we need to pick up is the one which reproduces the correct minimalsurfaces (namely, the correct entanglement wedge) and gives the minimal cross section of theentanglement wedge. 42igure 12: Left: our setup in the Poincare coordinates. Black curves ending on the boundary areminimal surfaces and the shaded region corresponds to the (time slice of) entanglement wedge.Another solid curve anchored on the minimal surfaces represents the minimal cross section ofthe entanglement wedge. Right: The back-reacted geometry in the global coordinates. To beprecise, each “boundary” points map to the di ff erent time and radial slices, thus the right panelis quite schematic. For each figure, the black-colored circle represents the black hole. Here we illustrate an example of the holographic local quench. In section 9.2.1, we will see theperfect agreement with the CFT analysis. Quite similar analysis show the agreement even inother setups. To avoid redundancy, we will not present other examples here. In section 9.2.2,we also comment on the non-zero size case.
Let us consider the bulk dual of a local (heavy) operator quench outside region between A and B . Namely, we assume A = [ u , v ] and B = [ u , v ] where 0 < u < − v < − u < v (seeFigure 12). To make life simpler and for comparison with the CFT results, we focus on thesmall particle limit (cid:15) →
0. Without this assumption, we will observe many transitions betweenthe three phases (see Figure 13). We comment on these transitions briefly in the upcomingsubsection. At the first time, 0 < (cid:15) (cid:28) t < u , the falling particle is outside of the entanglementwedge and does not a ff ect any back-reaction to its inside. Indeed we can compute geodesics in43igure 13: Three possibilities for entanglement wedge (shaded regions) and its cross section(dotted lines): disconnected (left), connected (center) and splitting cross sections (right). In thesmall particle limit (cid:15) →
0, we can fix our phase either disconnected (left) or connected (center)for every time regions.global coordinates and then back to the original metric by using the following relation:( τ u , θ u , r u ) = (cid:32) t (cid:15) u − t , π − u (cid:15) u − t , | u − t | µ(cid:15) (cid:33) , (9.19)( τ v , θ v , r v ) = (cid:32) t (cid:15) v − t , π − v (cid:15) v − t , | v − t | µ(cid:15) (cid:33) , (9.20)( τ u , θ u , r u ) = (cid:32) t (cid:15) u − t , π − u (cid:15) u − t , | u − t | µ(cid:15) (cid:33) , (9.21)( τ v , θ v , r v ) = (cid:32) t (cid:15) v − t , π − v (cid:15) v − t , | v − t | µ(cid:15) (cid:33) . (9.22)Thus, we obtain E W = c + (cid:113) ( v − u )( v − u )( u − u )( v − v ) − (cid:113) ( v − u )( v − u )( u − u )( v − v ) , (if 0 < t < u ) , (9.23)at the leading order of (cid:15) expansion. Notice that this is just the same cross section as one forPoincare AdS .In the regime u < t < √− u v , the falling particle is getting closer to the entanglementwedge, but still outside of the entanglement wedge. Since the coordinates across the singularityon u , the relation between two coordinates changes slightly,( τ u , θ u , r u ) = (cid:32) t (cid:15) u − t , π − u (cid:15) u − t , | u − t | µ(cid:15) (cid:33) → (cid:32) π + t (cid:15) u − t , − u (cid:15) u − t , | u − t | µ(cid:15) (cid:33) , (9.24)whereas that for other coordinates ( u , v , v ) does not change. From the CFT viewpoint, thise ff ect can be seen as the monodromy transformation in (2.24) although here we have no distinc-tion between the left and right moving. We will take the same replacement for each coordinate44igure 14: The manipulation in order to obtain the correct entanglement wedge. In the rightpanel, dotted lines describe the non-minimal surfaces and “cross section” obtained naively from(9.18). After shifting θ u → θ u + π , we achieve the left panel which describes the correctentanglement wedge and its cross section.( u , v , v ) when the time t exceeds each (absolute) value. In this regime, the back-reaction tothe minimal surfaces becomes visible, so the entanglement wedge cross section does, E W = c
12 log 1 + (cid:113) ( v − t )( v − u )( t − u )( v − v ) − (cid:113) ( v − t )( v − u )( t − u )( v − v ) + c
12 log 1 + (cid:113) ( v − u )( v − u )( u − u )( v − v ) − (cid:113) ( v − u )( v − u )( u − u )( v − v ) , (if u < t < √− u v ) . (9.25)When the particle enters the entanglement wedge ( √− u v < t < − v ), we cannot use the for-mula naively. This is because the original one captures the non-minimal surfaces (see left panelof Figure 14). Thus, we should utilize the identification so that we have correct entanglementwedge. This can be achieved by shifting the θ u → θ u + π , which is the same manipulationwhen one computes the holographic entanglement entropy (see right panel of Figure 14).Then we get E W = c
12 log 1 + (cid:113) ( t + v )( v − u )( − t − u )( v − u ) − (cid:113) ( t + v )( v − u )( − t − u )( v − u ) + c
12 log 1 + (cid:113) ( v − u )( v − u )( u − u )( v − v ) − (cid:113) ( v − u )( v − u )( u − u )( v − v ) , (if √− u v < t < − v ) . (9.26)Note that the aforementioned manipulation corresponds to choosing the unusual conformalblock in (2.36) which is compensated by the monodromy in the holomorphic part.When the particle is falling near the center of the entanglement wedge ( − v < t < − u ), thecorresponding minimal cross section acquires the significant e ff ects on the back-reaction. Theminimal one can be obtained from the (9.18) by shifting θ v → θ v + π and θ u → θ u + π (see45igure 15: The manipulation in order to obtain the minimal cross section of the entanglementwedge.Figure 15), E W = c γπγ(cid:15) (cid:115) ( t + u )( t + u )( t + v )( t + v )( u − v )( u − v ) + c
12 log 1 + (cid:113) ( v − u )( v − u )( u − u )( v − v ) − (cid:113) ( v − u )( v − u )( u − u )( v − v ) , (if − v < t < − u ) . (9.27)For − u < t , we can repeat the similar analysis. In summary, we have obtained, E W = c log + (cid:113) ( v − u v − u u − u v − v − (cid:113) ( v − u v − u u − u v − v (if 0 < t < u ) c log + (cid:113) ( v − t )( v − u t − u v − v − (cid:113) ( v − t )( v − u t − u v − v + c log + (cid:113) ( v − u v − u u − u v − v − (cid:113) ( v − u v − u u − u v − v (if u < t < √− u v ) c log + (cid:113) ( t + v v − u − t − u v − u − (cid:113) ( t + v v − u − t − u v − u + c log + (cid:113) ( v − u v − u u − u v − v − (cid:113) ( v − u v − u u − u v − v (if √− u v < t < − v ) c log γπγ(cid:15) (cid:113) ( t + u )( t + u )( t + v )( t + v )( u − v )( u − v ) + c log + (cid:113) ( v − u v − u u − u v − v − (cid:113) ( v − u v − u u − u v − v (if − v < t < − u ) c log + (cid:113) ( t + u u − v t + v u − u − (cid:113) ( t + u u − v t + v u − u + c log + (cid:113) ( v − u v − u u − u v − v − (cid:113) ( v − u v − u u − u v − v (if − u < t < √− u v ) c log + (cid:113) ( t − u u − v t − v u − u − (cid:113) ( t − u u − v t − v u − u + c log + (cid:113) ( v − u v − u u − u v − v − (cid:113) ( v − u v − u u − u v − v (if √− u v < t < v ) c log + (cid:113) ( v − u v − u u − u v − v − (cid:113) ( v − u v − u u − u v − v (if v < t ) (9.28)46igure 16: Two dotted lines ( Σ u v and Σ v u ) show another possibility for the entanglementwedge cross section which ends on the horizon. This happens only when the particle is fallinginside the entanglement wedge.These results perfectly agree with the CFT results (2.51). There is another interesting possibility — the entanglement wedge cross section splits into twopieces and each of them ends on the falling particle (see Figure 16). Although it will neverbecome dominant contribution for the (cid:15) → (cid:15) has a comparable length scale with each interval ( A , B anddistance between them). Even in this case, one can use (9.9) for each segment ( Σ u v and Σ v u )in the Figure 16 and find the “minimal” ones . After all, we obtain E W = G ( σ ( Σ u v ) + σ ( Σ v u )) , (9.29) σ ( Σ ) = log r ∗ + (cid:112) r ∗ − ( M − √ M − , (9.30)where r ∗ corresponds to the “turning point” in the geodesics anchored on the boundary points.One can see the r ∗ in the literature [52] (see also appendix B of [13]): r ∗ = (cid:113) − A − B (1 − M ) + (cid:112) (1 − A − B (1 − M )) + B (1 − M ) √ B . (9.31) As discussed in the below, we must minimize not the segments of cross sections but the minimal surfaces,otherwise what we compute is no longer the minimal cross section of the entanglement wedge. �������
60 70 80 90 t normal cross section vs. splitting cross section normal cross sectionsplitting cross section Figure 17: This plot shows the time dependence of the splitting cross section (Figure 16) andthe normal cross section discussed in the previous section (middle panel of Figure 13). Here weset − u = v = , − v = u = , γ = , and (cid:15) = .
1. In this setup, the splitting cross sectionbecomes a minimum one.Here we defined A = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin( √ − M ∆ τ ∞ )sin( √ − M ∆ θ ∞ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (9.32) B = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cos( √ − M ∆ τ ∞ ) − cos( √ − M ∆ θ ∞ ) √ − M sin( √ − M ∆ θ ∞ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (9.33)where ( ∆ τ ∞ , ∆ θ ∞ ) = ( τ v − τ u , θ v − θ u ) or ( τ u − τ v , θ u − θ v ). We also have the possibilities( ∆ τ ∞ , ∆ θ ∞ ) → ( ∆ τ ∞ , π − ∆ θ ∞ ). Note that the minimum value of the σ ( Σ ) does not alwayscorrespond to the correct entanglement wedge. We must carefully choose the one which mini-mizes the area of the minimal surfaces . For example, in the small size limit (cid:15) →
0, the analyticexpression in − v < t < − u is given by E W = c (cid:113) ( t − u )( t − v )( u − t )( v − t ) γ (cid:15) ( u − v )( u − v ) , (if − v < t < − u ) . (9.34)Obviously, this possibility is excluded from the (cid:15) dependence. However, this is what we haveseen in (2.50) as a (non-dominant) conformal block. Rather interestingly, we can confirm large c conformal blocks nicely tell us the each possibility for each phase. Moreover, this splittingcross section can be a dominant one if (cid:15) becomes non-zero (see Figure 17 as an example). We can easily extend the previous calculations to the rotating BTZ black hole with angularmomentum J . In the CFT side, we let the local operator have the scaling dimension h O (cid:44) ¯ h O .48he embedding coordinates in the rotating case are given by X = (cid:115) r − r + r + − r − sin ( r + τ − r − θ ) , (9.35a) X = (cid:115) r − r + r + − r − cos ( r + τ − r − θ ) , (9.35b) X = (cid:115) r − r − r + − r − sin ( r + θ − r − τ ) , (9.35c) X = (cid:115) r − r − r + − r − cos ( r + θ − r − τ ) , (9.35d)where r + ( r − ) correspond to the radius of the outer (inner) horizon, r + − r − = √ M − − J ≡ γ, (9.36) r + + r − = √ M − + J ≡ ¯ γ. (9.37)Note that the above coordinates cover only the region r > r + . Here γ and ¯ γ are the same onein the CFT. In the previous subsections, we assumed J =
0, hence γ = ¯ γ . By using the abovecoordinates, one can check that the local heavy operator with h O (cid:44) ¯ h O consistently reproducesthe rotating BTZ results.
10 Discussion
We will propose some remaining questions and interesting future works at the end of this paper: • information spreadingOne of our basic questions is how question information spreads in a strongly coupledsystem. A useful tool to probe how information spreads is mutual information as studiedin [12]. It is natural to expect that reflected entropy provides new information about thisproblem. What we need to calculate reflected entropy in their setup is the light conesingularities of the 6-point conformal blocks. Nevertheless there are currently no explicitforms of the light-cone singularity of Virasoro blocks, they are recently investigated bynumerically [66,67] and analytically [68] in large c and [61,69] in general c >
1. Now thatwe have all tools to accomplish this task, it would be very interesting to study informationspreading by making use of reflected entropy. (In the bulk side, a first step in this directionhas already been taken in [54]) • monotonicityThe entanglement of purification has some useful properties and the holographic reflectedentropy satisfies all these inequalities. However, if we leave the holographic CFT, some49f them break down. It would be interesting to clarify how the quantum corrections breakdown them. In particular, there is little knowledge about the monotonicity for reflectedentropy, S R ( A : BC ) ≥ S R ( A : B ) . (10.1)Our approach developed in this paper can be applied non-perturbatively to non-trivial state, therefore, we believe that our approach makes it clear before long. • relation to negativityThere is an interesting proposal for the relation between entanglement wedge cross sec-tion and negativity in [30, 43]. The negativity can also be calculated in CFT by the replicatrick. It would be very interesting to compare the reflected entropy and the negativity fora local quench state. This trial should reveal di ff erences and similarities of them. Webelieve that our approach developed in this paper is useful to calculate the negativity aftera local quench and it will bring about a deep understanding of this relation. • Renyi reflected entropyAs shown in this paper, the reflected entropy in the holographic CFT is approximated bythe thermal reflected entropy. However, it is not trivial for the Renyi reflected entropy toalso show this thermalization.Another motivation to study Renyi reflected entropy is to compare the gravity side. Asmentioned in the main text, the Renyi reflected entropy has an obvious replica transitionas the replica number n is varied. (Similar transitions can be found in [56–61].) Thismight be related to the instability and we could find a transition accompanied by thisinstability in the bulk side. Further understanding of this transition is one of interestingfuture directions. Note that a sturdy of the Renyi reflected entropy is already startedin [45], however, the result is only perturbative, which does not enable us to observe thetransition. • joining quench, global quench, splitting quench, double quenchIn this paper, we only focus on the local operator quench introduced in [8]. Asidefrom this system, there are many variable ways to excite the vacuum state (e.g., join-ing quench [1], global quench [2, 3], splitting quench [4] and double quench [5–7]) Itwould be interesting to study dynamics of reflected entropy in these setups and identifysimilarities and di ff erences. • finite c An important future work is to understand how the dynamics of reflected entropy be-haves in other CFTs. This is motivated by the fact that the dynamics of entanglemententropy captures the chaotic nature of a given CFT. That is, its time-dependence in theholographic CFT [52, 53, 71], in RCFTs [86, 87], and in another irrational CFT [89] arevery di ff erent from each other. It is naturally expected for reflected entropy to be moreuseful to characterize CFTs, in particular, to identify the holographic CFT. In fact, our50ethod allows us to calculate the reflected entropy even in finite c CFTs and we haveshown a part of results in this paper. We hope to give complete results in a future paper.There is another motivation to study the reflected entropy in finite c systems. The reflectedentropy is very recently invented in [18], therefore, we have very limited knowledgeabout its properties (e.g., the monotonicity is satisfied or not). Against this backdrop, thischallenge gives a key to understanding the reflected entropy. • odd entanglement entropyOur natural expectation is that the odd entanglement entropy also contains informationabout correlations between two intervals and capture the chaotic nature in some sense.However, we have little knowledge about the odd entanglement entropy itself. An imme-diate future work is to investigate its properties in various setups and find out universality.It is particularly interesting for us to find a property which only holds in the holographicCFT. We expect that this quantity could be a good tool to identify the holographic CFT.Our result strongly suggests that the odd entanglement entropy in the holographic CFTperfectly captures the entanglement wedge cross section even in more general systems.We hope to prove this statement in a rigorous and general way in future. Acknowledgments
We thank Souvik Dutta, Jonah Kudler-Flam, Thomas Hartman, Masamichi Miyaji, MasahiroNozaki, Tokiro Numasawa, Tadashi Takayanagi and Koji Umemoto for fruitful discussions andcomments. YK is supported by the JSPS fellowship. KT is supported by JSPS Grant-in-Aidfor Scientific Research (A) No.16H02182 and Simons Foundation through the “It from Qubit”collaboration. We are grateful to the conference “Quantum Information and String Theory2019” in YITP and “Strings 2019”. 51
Semiclassical Fusion and Monodromy Matrix
In this appendix, we show the detailed derivation of the semiclassical monodromy matrix. Wehave the closed expression for the fusion and monodromy matrix, therefore, it is possible toevaluate their semiclassical limits by using them as in [69]. However, the simplest way tocalculate them is to make use of the closed form of the HHLL Virasoro block. We have toemphasize that what we need here is not the usual HHLL block introduced in [75], F LLHH ( h p | z ) = (1 − z ) h L ( δ − (cid:32) − (1 − z ) δ δ (cid:33) h p − h L F ( h p , h p , h p ; 1 − (1 − z ) δ ) , (A.1)but the semiclassical block derived by the monodromy method [95], F LLHH ( h p | z ) = (1 − z ) h L ( δ − (cid:32) − (1 − z ) δ δ (cid:33) h p − h L + (1 − z ) δ − h p , (A.2)where δ = (cid:113) − c h H . The former is derived in the large c limit with h H c , h L , h p fixed, on theother hand, the later is calculated in a di ff erent regime of parameter space, in the large c limitwith h H c , h L c , h p c fixed and set h H (cid:29) h L , h p (which is discussed in [79, 92]) . Therefore, these twoHHLL blocks are di ff erent from each other. For convenience, we call the former HHLL limit and the later semiclassical limit . We have to choose the later in our calculation because we takefirst the large c limit of the block with the twist operators, whose conformal dimensions areproportional to c . Note that the HHLL block and the semiclassical block can be related through F ( h p , h p , h p ; z ) −−−−→ h p →∞ (cid:32) + √ − z (cid:33) − h p , (A.3)which is shown by using the following identity, F ( h p , h p − , h p ; z ) = (cid:32) + √ − z (cid:33) − h p . (A.4)The fusion transformation leads to the relation, F LLHH ( h p | z ) −−−→ z → F α p ,α H (cid:34) α L α L α H α H (cid:35) (1 − z ) h L (1 − δ ) , (A.5)where we introduce the Liouville momentum as α L ( Q − α L ) = h L , α H ( Q − α H ) = h H , α p (cid:16) Q − α p (cid:17) = h p , (A.6)and F is defined in terms of the Virasoro fusion matrix F [51, 61] as F α p ,α H ≡ Res (cid:16) − π i F α p ,α ; α = α H (cid:17) . (A.7) The semiclassical conformal block with the z i -dependences, which are not fixed by the global conformaltransformation, is shown in [79, 96]. F α p ,α H (cid:34) α L α L α H α H (cid:35) −−−−−−−−→ semiclassicallimit δ h L (cid:32) δ (cid:33) h p . (A.8)In a similar manner, the Regge limit of the semiclassical block can be related to the monodromymatrix as F LLHH ( h p | z ) −−−−−−−−−−−→ z → − z ) → e π i (1 − z ) M ( + ) α p , α L (cid:34) α L α L α H α H (cid:35) . (A.9)Therefore, we obtain M ( + ) α p , α L (cid:34) α L α L α H α H (cid:35) −−−−−−−−→ semiclassicallimit (cid:32) i δ sin πδ (cid:33) − h L (cid:32) − i δ tan πδ (cid:33) h p . (A.10)Note that this is completely di ff erent from the monodromy matrix based on (A.1). The dimen-sion h p is order O (1), therefore, the large c limit does not change the hypergeometric functionpart of the HHLL block, unlike (A.3). As a result, we obtain M ( + ) α p , α L (cid:34) α L α L α H α H (cid:35) −−−−−−−−→ HHLL limitand h p → (cid:32) i δ sin πδ (cid:33) − h L (cid:32) − i δ sin πδ (cid:33) h p . (A.11)According to [74], the LHHL block with heavy intermediate state can be given by justprimary exchange. Therefore, the following type of the fusion matrix is trivial, F α H ,α L + α L (cid:34) α L α H α L α H (cid:35) −−−−−−−−→ semiclassicallimit . (A.12) B Semiclassical 5-point Block
B.1 Proof of (2.33)
In this Appendix, we show the detailed calculation of (2.33). From the expression (2.28), wefind that the Regge limit is given by˜ M ( − )0 , α m (cid:34) α m α m α O α O (cid:35) × (2 i (cid:15) ) h α m − nmh O (cid:3034) (cid:3250) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) (cid:3034) (cid:3251) (cid:3041)(cid:3040)(cid:2870)(cid:3080) (cid:3288) , (B.1)where h a = α ( Q − α ). If we take the limit m →
1, then the monodromy matrix simplybecomes one. At this stage, what we need to evaluate the reflected entropy is the followingasymptotics, Here we assume α min = α α m , which is naturally expected from the result in [51]. But this assumption is notnecessary because we obtain the same conclusion (B.8) without fixing α min . (cid:2870)(cid:3080) (cid:3080) (cid:3080) (cid:3080) (cid:3080)(cid:3043) −−−→ α → ? , (B.2)where we also take the large c limit with h p c fixed. It is important to note that the asymptotics ofthe 4-point semiclassical block is given by (see Appendix A), (cid:3043) −−−→ h → z h p (cid:32) + √ − z (cid:33) − h p = h p (cid:32) + √ − z − √ − z (cid:33) − h p , (B.3)which is used to reproduce the entanglement wedge cross section as in [18, 37]. In fact, we canderive this semiclassical block with the intermediate state of order c by the global block inthe following way (instead of relying on (A.2); z h p − h F ( h p , h p , h p ; z ) −−−−→ h p →∞ after h → z h p (cid:32) + √ − z (cid:33) − h p , (B.4)where the left-hand side is the well-known global block [99,100]. From this observation, we candeduce that the asymptotics of the 5-point block can be obtained by the 5-point global block,which has already calculated in [101] as (cid:2869) (cid:2869)(cid:3043) (cid:3117) (cid:2870) (cid:2870)(cid:2871) (cid:2871)(cid:2872) (cid:2872)(cid:2873) (cid:2873) (cid:3043) (cid:3118) = L h , ··· , h ( z , · · · z ) χ h p χ h p × F (cid:34) h p + h − h , h p + h p − h , h + h p − h h p , h p ; χ , χ (cid:35) , (B.5)where h i ( i = , · · ·
5) is the conformal dimension of the operator V i , the cross ratio is definedby χ i ≡ z i , i + z i + , i + z i , i + z i + , i + with z i , j = z i − z j , and the prefactor L is the leg factor as L h , ··· , h ( z , · · · z ) ≡ (cid:32) z z z (cid:33) h (cid:32) z z z (cid:33) h (cid:89) i = (cid:32) z i , i + z i , i + z i + , i + (cid:33) h i + . (B.6)The function F is the Appell function defined as F (cid:34) a , b , a c , c ; x , x (cid:35) = ∞ (cid:88) n , n = ( a ) n ( b ) n + n ( a ) n ( c ) n ( c ) n x n n ! x n n ! , (B.7) This approximated block is not the same as the regular part of the conformal block (i.e., the block with theheavy intermediate state) [97, 98]. The di ff erence between these two blocks is that the former has the intermediatedimension of order c , whilst the later is calculated in the limit h p (cid:29) c . a ) n = Γ ( a + n ) Γ ( a ) is the Pochhammer symbol and we define (0) n = δ n , . By using this result,we obtain (cid:3080) (cid:2870)(cid:3080) (cid:3080) (cid:3080) (cid:3080) (cid:3080)(cid:3043) −−−−→ h p →∞ after h α → χ h p + (cid:112) − χ − h p . (B.8)Here we leave only the linear term h p in the log of the block, like (A.2). This approximatedblock is what we want (2.33), where the explicit form of the cross ratio χ is given by χ = ( − v + t )( − u + v )( − u + t )( − v + v ) . (B.9) B.2 Proof of (2.43)
In this section, we show the asymptotics (2.43). The monodromy transformation in (2.43) canbe re-expressed as (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3034) (cid:3251) (cid:3040) (cid:3041) = (cid:3034) (cid:3250) ⊗(cid:3040)(cid:3041) (cid:3041)(cid:3034) (cid:3251) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993) (cid:3041) . (B.10)This is just the monodromy transformation of O ⊗ mn † around σ g − B g A . Let us recall that the orbifoldblock can be regarded as the square of the Virasoro block as explained in (2.15). Therefore, thismonodromy e ff ect comes from each Virasoro block (i.e., black and red in (2.15)) as M ( − )0 , α n (cid:34) α n α n α O α O (cid:35) M ( − )0 , α n (cid:34) α n α n α O α O (cid:35) × (2 i (cid:15) ) h α n − nh O × (cid:3041)(cid:3034) (cid:3251) (cid:3034) (cid:3250) (cid:3034) (cid:3251)(cid:3127)(cid:3117) (cid:3034) (cid:3250)(cid:3127)(cid:3117) (cid:3041)(cid:3080) (cid:3289) , (B.11)where we used the Regge limit of the block associated with Z n symmetry [51]. The explict formof this monodromy matrix is (A.10). To calculate the remaining 5-point conformal block, wecan again make use of the global block (B.5). In fact, we can easily show (cid:3080) (cid:2870)(cid:3081)(cid:3081)(cid:3080) (cid:3080) (cid:3080)(cid:3081) −−−−−−→ h α , h β → L h α , h α , h β , h α , h α ( z , · · · z ) χ h β χ h β , (B.12)and substituting this result into (B.11), we obtain (2.43).55 .3 Proof of (5.8) The conformal blocks in (5.7) is given by the square of the Virasoro block as (cid:3034) (cid:3250)(cid:3127)(cid:3117) ⊗(cid:3040)(cid:3041)(cid:2993)(cid:3034) (cid:3251) (cid:3041)⊗(cid:2870) = ⊗(cid:3041)⊗(cid:3041)(cid:2993)(cid:3041) , (B.13)where ∼ C Heavy-Heavy-Light OPE Coe ffi cient The Heavy-Heavy-Light OPE coe ffi cient can be calculated by the modular bootstrap equationfor a 1-point function on a torus [73], (cid:104) O H | O L | O H (cid:105) ∼ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ γ hL e − c − π (cid:18) − √ − c − h χ (cid:19) ˆ γ π (cid:32) − c − h χ (cid:33) − hL − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) χ | O L | χ (cid:105) , (C.1)where ˆ γ = (cid:113) c − h H − χ is the lightest one with (cid:104) χ | O L | χ (cid:105) (cid:44)
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