Holographic Quantum Circuits from Splitting/Joining Local Quenches
YYITP-18-122IPMU18-0197
Holographic Quantum Circuits from Splitting/Joining Local Quenches
Teppei Shimaji a , Tadashi Takayanagi a,b , and Zixia Wei aa Center for Gravitational Physics,Yukawa Institute for Theoretical Physics, Kyoto University,Kyoto 606-8502, Japan b Kavli Institute for the Physics and Mathematics of the Universe (WPI),University of Tokyo, Kashiwa, Chiba 277-8582, Japan
Abstract
We study three different types of local quenches (local operator, splitting andjoining) in both the free fermion and holographic CFTs in two dimensions. Weshow that the computation of a quantity called entanglement density, provides asystematic method to capture essential properties of local quenches. This allowsus to clearly understand the differences between the free and holographic CFTs aswell as the distinctions between three local quenches. We also analyze holographicgeometries of splitting/joining local quenches using the AdS/BCFT prescription.We show that they are essentially described by time evolutions of boundary sur-faces in the bulk AdS. We find that the logarithmic time evolution of entanglemententropy arises from the region behind the Poincar´e horizon as well as the evolutionsof boundary surfaces. In the CFT side, our analysis of entanglement density sug-gests such a logarithmic growth is due to initial non-local quantum entanglementjust after the quench. Finally, by combining our results, we propose a new class ofgravity duals, which are analogous to quantum circuits or tensor networks such asMERA, based on the AdS/BCFT construction. a r X i v : . [ h e p - t h ] M a r ontents / CFT . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Holographic Entanglement Entropy in AdS / CFT . . . . . . . . . . . . . 62.5 EE from AdS/BCFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 t Geodesic Length in Holographic Lorentzian Geometry 34 Introduction
Conformal field theories (CFTs) in two dimensions are very special among quantum fieldtheories (QFTs) in that infinite dimensional conformal symmetries constrain their prop-erties. Owing to this, we can analytically calculate much more physical quantities thanthose in ordinary QFTs. Nevertheless, we can observe a variety of qualitative differ-ence in the dynamics of two dimensional CFTs (2d CFTs). We can easily come up withtwo extreme examples of 2d CFTs. One is free CFTs such as massless free fermion orscalar theories. Another one is the strong coupling limits of 2d CFTs with large centralcharges c , namely the holographic CFTs, which have dual descriptions via the AdS/CFTcorrespondence [1, 2].One of the most interesting quantities to characterize the dynamical property of agiven quantum state is the entanglement entropy (EE) [3, 4, 5, 6, 7]. In this paper wewould like to explore how differences of 2d CFTs appear in the time evolution of certainexcited states. One useful class of excited states is called local quenches, where we exerta local excitation on a vacuum state. Since it initially modifies the state only locally, theyprovide clean examples where we can interpret the time evolution easier. We considerthree different types of local quenches: (i) local operator quenches, (ii) splitting localquenches, and (iii) joining local quenches. We sketched them in Fig.1.The first one (i) is simply defined by acting a local operator on a vacuum and weanalyze its time evolution. This was first introduced in [8, 9] and there have already beenmany related results both in field theory analysis and holographic analysis. The third one(iii) is defined by joining two semi-infinite lines, as first introduced in [10]. The secondone (ii) is triggered by splitting a connected line into two disconnected ones. This is anew setup which the present paper will discuss in detail.In this paper we will point out that it is very helpful and systematic to analyze notthe entanglement entropy itself but its second derivatives, called the entanglement density(ED) [11]. Indeed, this quantity extracts the essential behaviors of entanglement entropyunder local quenches in term of clear peaks in its graph. By studying the behaviorof entanglement density, we can clearly see both similarities and differences among theabove three types of local quenches in free CFTs and holographic CFTs, as we will explainlater. In particular, this observation resolves an apparent puzzle on the known logarithmicgrowing entanglement entropy under holographic operator local quenches.Another purpose of this paper is to explore the spacetime geometries of holographiclocal quenches. Since the gravity dual geometry of (i) local operator quenches was alreadygiven in [11], we will focus on (ii) and (iii). For (iii) joining local quenches, the constructionof gravity dual by employing the AdS/BCFT formulation [12] was given in [13] and wewill study more details of the spacetime geometry by using this description in this paper.In addition we will provide the gravity dual geometry for (ii) splitting local quenches.Refer to [14] for other classes of topology changing quantum operations in CFTs such asprojections and partial identifications. Our details geometric analysis clearly will explainthe two different sources of logarithmic time evolutions of entanglement entropy observedin the local quenches. 2ur analysis of holographic geometry of local quenches is also motivated by the con-jectured connection between the AdS/CFT and tensor networks [15] such as the MERA[16]. In tensor network descriptions, we normally consider discretized lattice theorieswhose continuum limits correspond to CFTs. Therefore, it is not directly related to thecontinuous AdS spacetimes, but to their discretized versions as in [15, 17, 18]. One wayto resolve this problem is to consider a continuous tensor networks as in the continuousMERA [19, 20] or path-integral approaches [21, 22, 23, 24, 25].However, it is also intriguing to try to realize a discretized version of AdS in a waythat it is naturally derived from the conventional AdS/CFT. The gravity duals of localquenches are useful for this purpose because we can model a class of quantum gates intensor networks by arranging the splitting and joining procedures in CFTs. As we willdiscuss in the final part of this paper, we presents a sketch of gravity duals of MERAtensor networks, by combining the holographic local quenches. This indeed qualitativelysupports the conjecture.This paper is organized as follows: In section 2, we review basic methods of calcu-lating entanglement entropy by conformal map and also by AdS/(B)CFT. In section 3,after we review the definition of entanglement density, we study its behaviors for globaland local operator quenches. In section 4, we analyze the splitting local quenches and itsholographic dual. In section 5, we analyze the joining local quenches and its holographicdual. In section 6, we explain how the logarithmic growth of entanglement entropy arisesfrom a geodesic length in each gravity dual. In section 7, we provide qualitative ten-sor network description of local quenches. In section 8, we combine the results in thispaper to provide a new gravity dual of a MERA-like (discretized) tensor network, inthe framework of AdS/BCFT. In section 9, we summarize our conclusions and discussfuture problems. In appendix A, we show the entanglement density can reproduce thecorrect entanglement entropy even when the subsystem consists of multiple disconnectedintervals in two dimensional massless Dirac fermion CFT. In appendix B, we will presentthe detailed computations of time evolutions of holographic entanglement entropy undersplitting/joining local quenches. Here we briefly review calculations of entanglement entropy (EE) in two dimensionalCFTs, based on conformal mappings and computations of holographic entanglement en-tropy (HEE) based on the AdS/CFT and AdS/BCFT. The entanglement entropy is de-fined by the von-Neumann entropy S A = − Tr[ ρ A log ρ A ], where ρ A is the reduced densitymatrix defined by tracing out the original quantum state over all parts of Hilbert spaceother than A . 3 <0t=0t>0 Operator Splitting Joining
Figure 1: The three local quenches are sketched: the local operator quench (left), thesplitting local quench (middle), and the joining local quench (right) in two dimensionalCFTs. The red points are locations where the energy density is very large.
Consider a two dimensional CFT (2d CFT) on a plane R , which is described by thecomplex coordinate ( w, ¯ w ). We define the time and space coordinate ( τ, x ) as w = x + iτ, ¯ w = x − iτ. (2.1)The time τ is Euclidean time and is analytically continued to the real time by τ = it. (2.2)A two point function of a primary operator O in a 2d CFT behaves as (cid:104) O ( w , ¯ w ) O ( w , ¯ w ) (cid:105) = 1 | w − w | h +¯ h ) , (2.3)where the primary operator O has the chiral/anti-chiral conformal dimension given by( h, ¯ h ). For the twist operator σ n , we have h = ¯ h = c ( n − /n ). When we compute theentanglement entropy (EE) for the subsystem A defined by the interval 0 ≤ x ≤ l at afixed time τ = 0, the relevant two point function is (cid:104) σ n ( l, l )¯ σ n (0 , (cid:105) = 1 l c ( n − /n ) . (2.4)Therefore the EE S A for the CFT vacuum reads S A = − ∂∂n log (cid:104) σ n ( l, l )¯ σ n (0 , (cid:105) (cid:12)(cid:12)(cid:12) n =1 = c l(cid:15) , (2.5)where (cid:15) is the UV cut off (lattice spacing). 4he R´enyi entanglement entropy, defined by S ( n ) A = 11 − n log Tr[( ρ A ) n ] , (2.6)can also be computed for the vacuum as S ( n ) A = c (cid:18) n (cid:19) log l(cid:15) . (2.7) To calculate the EE for a special class of excited states in 2d CFTs, we can employ theconformal transformation. We transform the original coordinate system ( w, ¯ w ) into a newone ( ξ, ¯ ξ ): ξ = f ( w ) . (2.8)Also we need to note that the original UV cut off (cid:15) , which is a lattice spacing in the w coordinate, is mapped to the one, called ˜ (cid:15) a,b , in the new coordinate ( ξ, ¯ ξ ) as (cid:15) = ˜ (cid:15) a | f (cid:48) ( w a ) | = ˜ (cid:15) b | f (cid:48) ( w b ) | . (2.9)Finally the EE for the excited state is found to be S A = c (cid:20) | f ( w a ) − f ( w b ) | (cid:15) | f (cid:48) ( w a ) || f (cid:48) ( w b ) | (cid:21) . (2.10)In our analysis of joining/splitting local quenches, the EE is computed from a twopoint function of twist operators in the presence of a conformal boundary, called boundaryconformal field theory (BCFT). Since two point functions in BCFT are similar to fourpoint functions in CFTs without boundaries, we do not have any universal expressionsuch as (2.3). However, we can obtain a definite analytical expression for special CFTssuch as the Dirac fermion CFT and the holographic CFTs, whose details will be discussedlater. / CFT The AdS/CFT correspondence argues that gravitational theories on AdS is equivalentto 2d holographic CFTs, which live on the AdS boundary [1]. The physical equivalenceis formulated such that the partition function in each side agrees with the other one [2],so called the bulk-boundary correspondence.Consider the Poincar´e metric of AdS , which is dual to the vacuum of 2d CFT (CFT ): ds = dη + dξd ¯ ξη , (2.11)5here we set the AdS radius to 1 for simplicity. The conformal transformation (2.8) isequivalent to the following coordinate transformation in AdS (see e.g.[26]): ξ = f ( w ) − z ( f (cid:48) ) ( ¯ f (cid:48)(cid:48) )4 | f (cid:48) | + z | f (cid:48)(cid:48) | , ¯ ξ = ¯ f ( ¯ w ) − z ( ¯ f (cid:48) ) ( f (cid:48)(cid:48) )4 | f (cid:48) | + z | f (cid:48)(cid:48) | ,η = 4 z ( f (cid:48) ¯ f (cid:48) ) / | f (cid:48) | + z | f (cid:48)(cid:48) | . (2.12)The metric in the coordinate ( w, ¯ w, z ) reads ds = dz z + T ( w )( dw ) + ¯ T ( ¯ w )( d ¯ w ) + (cid:18) z + z T ( w ) ¯ T ( ¯ w ) (cid:19) dwd ¯ w, (2.13)where T ( w ) = 3( f (cid:48)(cid:48) ) − f (cid:48) f (cid:48)(cid:48)(cid:48) f (cid:48) , ¯ T ( ¯ w ) = 3( ¯ f (cid:48)(cid:48) ) − f (cid:48) ¯ f (cid:48)(cid:48)(cid:48) f (cid:48) , (2.14)are the chiral and anti-chiral energy stress tensor. / CFT In Euclidean setups, the holographic entanglement entropy (HEE) [27, 28] is given by S A = L G N , (2.15)in terms of the length L of shortest geodesic which connects the two boundary points ofthe subsystem A . We also need to impose the homology constraint that this geodesic ishomologous to the subsystem A in the AdS geometry.If we consider a connected geodesic which connects w a and w b , then its length is givenby L ab = log | ξ a − ξ b | ˜ (cid:15) a ˜ (cid:15) b = log (cid:20) | f ( w a ) − f ( w b ) | (cid:15) | f (cid:48) ( w a ) || f (cid:48) ( w b ) | (cid:21) , (2.16)where note the relation (2.9), following from the above coordinate transformation around η = 0: η (cid:39) | f (cid:48) | z . Then the HEE L ab G N = c L ab reproduces the formula (2.10). Since our coming analysis of joining/splitting local quenches require us to consider aCFT, let us extend the AdS/CFT to a setup where a holographic CFT is defined on amanifold M with boundaries ∂M . In particular, when a linear combination of conformalsymmetry is preserved on ∂M , we call it a boundary conformal field theory (BCFT). The6olographic dual of a BCFT, called AdS/BCFT, can be constructed in the following way[12] (see also [29] for an earlier argument). For a typical AdS/BCFT setup, refer to theleft picture of Fig.2. Consider a surface Q which ends on ∂M and extends into the bulk.We impose the following condition on QK µν − Kh µν = − T BCF T · h µν , (2.17)where h µν is the induced metric on Q and K µν is the extrinsic curvature on Q ; K is thetrace h µν K µν . The constant T BCF T describes the tension of the ‘brane’ Q and can takeboth positive and negative values in general. This boundary condition (2.17) arises natu-rally in the AdS/BCFT setup as follows (for more detail refer to [12]). If we consider thestandard gravity action given by the Einstein-Hilbert action plus the Gibbons-Hawkingboundary term, it is well-known that there are two boundaries conditions: Dirichlet andNeumann. For AdS/BCFT, we choose the Neumann boundary condition as we want tokeep the boundary Q dynamical as is so in the BCFT boundary. The Neumann bound-ary condition is given by K µν − Kh µν = 0. To generalize this boundary condition, weadd the tension term of the surface Q to the bulk action, given by T BCF T (cid:82) Q √ h . Thismodified the Neumann boundary condition into the form (2.17). It is also useful to notethat in explicit examples, we can confirm that this boundary condition (2.17) preservesthe boundary conformal symmetry.The gravity dual of a CFT on M is given by the AdS gravity solution restricted on thespace N , defined by the bulk region surrounded by M and Q . To find such a solution, weneed to solve the Einstein equation with the boundary condition (2.17), where the presenceof Q gives back-reactions and modifies the bulk metric [12, 30, 31]. In our examples whichwe will discuss later, we can analytically find gravity duals of two dimensional BCFTs byusing the bulk extension of the conformal map (2.12).The region N gets larger as the tension T BCF T increases and this suggests that T BCF T estimates the degrees of freedom on the boundary ∂M . Indeed, as shown in [12] in theAdS case, the tension is monotonically related to the boundary entropy S bdy introducedin [32] as follows: S bdy = c T BCF T ) . (2.18)For notational simplicity, we introduce a positive parameter k ( >
0) by S bdy ≡ c
12 log k. (2.19)A larger k means a larger boundary entropy or tension T BCF T . In particular, we have k = 1 for T BCF T = S bdy = 0. Refer to e.g. [33, 34] for studies of HEE in higher dimensionalsetups.Later we will employ the AdS/BCFT to calculate the holographic entanglement en-tropy (HEE). Consider the holographic entanglement entropy S A for an interval A in theAdS /BCFT setup. As in the right picture, there are two possibility: connected one S conA and the disconnected one S disA . The latter arises because the geodesic which connects thetwo end points of A can end on the boundary surface Q in the middle as depicted in the7ight picture of Fig.2. The correct holographic entanglement entropy is given by the onewith a smaller length.The connected HEE S conA is not affected by the boundary surface Q . Therefore, thissimply agrees with the estimation (2.10) of the CFT without boundary. On the otherhand, the disconnected HEE S disA is highly affected by the boundary Q as the geodesicends on Q . In the Poincare AdS setup, S disA in Fig.2 is computed as (refer to [12] for thederivation): S disA = c (cid:18) s a ˜ (cid:15) a (cid:19) + c (cid:18) s b ˜ (cid:15) b (cid:19) + 2 S bdy , (2.20)where s a,b are the distance between the surface Q and the two end points of the interval A ; ˜ (cid:15) a,b are the UV cut off in the Poincare AdS at the two boundary points. Intuitively,as the surface Q gets further from the subsystem A , then geodesic length (i.e. HEE)gets larger. Thus as the tensor T BCF T gets larger, the HEE gets larger. This effect isuniversally described by the constant term proportional to S bdy in (2.20) as follows formthe result in [12]. Note that this is the sum of two disconnected geodesics and thereforehas the doubled contribution of S bdy .We can also understand this from the CFT viewpoints. In the holographic CFTs (orlarge central charge CFTs with sparse spectrum). As in the standard large c arguments[68], we can approximate a correlation function by a semiclassical saddle point. In oursetup with a boundary, there are two saddles. One of them is obtained by contracting twotwist operators, which give the connected EE S conA . The other is given by contracting eachof them with its mirror operator across the boundary, which leads to the disconnectedone S disA . Both of them agree with the results from AdS/BCFT. The S bdy dependence in(2.20) occurs because the branch cut which extends from a twist operator ends on theboundary as is already known in the standard calculation of EE in BCFT [6].More generally, including higher dimensional setups, we can calculate the holographicentanglement entropy in the following way. In the standard holographic entanglemententropy without any boundaries, S A is given by the area of minimal or extremal surfaceΓ A which ends on ∂A and which is homologous to A . In the presence of boundary surfaces Q , we impose the homology condition by regarding the surfaces Q as trivial spaces withthe zero size. Here we study a quantity called entanglement density (ED) introduced in [11]. First wewould like to note that this is not a new quantity as it follows from the values of the en-tanglement entropy (EE). However, the ED provides a helpful way to display the behaviorof EE in complicated systems, because we can capture the essence of time evolution fromthe behavior of ED as we explain later.The ED is defined from the data of EE in a 2d CFT when the subsystem A is aninterval. Since we have in mind generic excited states, we do not require the translational8 Q M (cid:1101)
M N
AdS Bdy x t A AdSBCFT Excluded
Q M z Figure 2: A sketch of AdS/BCFT analysis for AdS . A holographic CFT on M (withthe boundary ∂M ) is dual to gravity on N . The boundary of N consists of the surface Q and M . The right picture shows the calculation of holographic entanglement entropy.The blue curve gives the connected geodesic contribution and the green ones are thedisconnected geodesics which end on the boundary surface Q . The correct holographicentanglement entropy is given by the one with a smaller length.symmetry. When A is given by an interval a ≤ x ≤ b at a fixed time t , the EE is writtenas S A ( a, b, t ) (let us assume a < b ). The entanglement density n ( a, b, t ) is defined by n ( a, b, t ) = 12 ∂ S A ( a, b, t ) ∂a∂b = 12 (cid:18) ∂ ∂ξ − ∂ ∂l (cid:19) S A ( ξ, l, t ) , (3.1)where we introduced the center of the interval ξ and length l such that a = ξ − l , b = ξ + l . (3.2)For example, the ED for a CFT vacuum, denoted by n ( ξ, l, t ), takes the universal form: n ( ξ, l, t ) = c l . (3.3)By definition, the EE is represented as a double integral of ED: S A ( a, b, t ) = (cid:18)(cid:90) a −∞ dx (cid:90) ba dy + (cid:90) ∞ b dy (cid:90) ba dx (cid:19) n ( x, y, t ) . (3.4)Therefore, if we assume all entanglement comes from the bipartite quantum correlation,the ED measures the number of EPR pairs between x = a and x = b . However we shouldnote that this interpretation is too naive as in general we have multi-partite entanglement In this paper, ξ is also used to denote another quantity, the coordinate in AdS/BCFT given by (2.8)and (2.12). However, since the analysis of ED and discussions in AdS/BCFT do not appear at the sametime, its meaning can be easily figured out from the context.
9n QFTs. This issues becomes serious in the later time behavior of holographic localoperator quench [8], as pointed out in [35]. On the other hand, for the massless Diracfermion CFT, this interpretation works so well that we can reproduce correct results fromthe universal ED (3.3) even when A consists of multiple disconnected intervals as we willexplain in the appendix A. One of the purposes of the present paper is to emphasize thatthe entanglement density at least provides a useful and simple way to extract the essentialbehaviors of the entanglement entropy in a systematical way. We will see this in manyexamples.We are especially interested in the difference∆ n ( ξ, l, t ) = n ( ξ, l, t ) − n ( ξ, l, t ) , (3.5)where n ( ξ, l, t ) is given by (3.3). In other words, ∆ n ( ξ, l, t ) is the entanglement densityfor the entanglement entropy growth∆ S A = S A − S A , (3.6)where S A is the entanglement entropy for the ground state. As already noted in [11], the growth of ED ∆ n ( ξ, l, t ) enjoys several interesting properties.First of all, owing to the first law of entanglement entropy [36, 37, 38], in the small sizelimit l →
0, we have ∆ n ( ξ, , t ) = − π T tt ( ξ, t ) , (3.7)where T tt is the energy density.Moreover, the growth of ED satisfies a sum rule: (cid:82) da (cid:82) db ∆ n ( a, b, t ) = 0 if the totalstate is pure. In [11], this was proved when we impose the periodic boundary condition,which identifies x = a with x = b . More generally, we can prove this even when the totalspace is an interval 0 ≤ x ≤ L as we will explain below. The precise statement of thesum rule is (cid:90) L db (cid:90) b da ∆ n ( a, b, t ) = 0 , (3.8)at any time t . To prove this, we can rewrite it as follows (cid:90) L db (cid:90) b da ∂ a ∂ b ∆ S A ( a, b, t ) = (cid:90) L db (cid:104) ∂ b ∆ S A ( a, b, t ) | a = b − ∂ b ∆ S A ( a, b, t ) | a =0 (cid:105) . (3.9)First of all, the quantity ∂ b ∆ S A | a = b is vanishing because the first law (3.7) tells us thebehavior ∆ S A ∝ ( b − a ) when | b − a | is very small. The second term in the right handside also follows because (cid:90) L db ∂ b ∆ S A ( a, b, t ) | a =0 = ∆ S A ( a, b, t ) | b = L, a =0 − ∆ S A ( a, b, t ) | b =0 , a =0 = 0 , (3.10)10here we employed the first law behavior and the pure state property S A = S A C ( A C isthe complement of A ).It is also intriguing to note that we can also define the entanglement density for R´enyientanglement entropy (2.6). Since this quantity also has the property ∆ S ( n ) A ∝ ( b − a ) when | b − a | is very small, the R´enyi entanglement density ∆ n ( n ) ( a, b, t ) also satisfies thesum rule (3.8). One of the simplest but non-trivial examples of homogeneous excited states in CFTs is theglobal quenches. This is triggered by a sudden change of the Hamiltonian from a gappedone to a critical one H at a specific time t = 0. As argued in [39], we can model thisprocess by approximating the state just after the quench by a regularized boundary state | Ψ( t = 0) (cid:105) = e − αH | B (cid:105) , where α is an infinitesimally small parameter of the regularizationand | B (cid:105) is a boundary state [40, 41]. In general, The EE of a subsystem with length l shows the linear growth and the saturation [39]∆ S A = πc α t (0 < t ≤ l/ , ∆ S A = πc α l ( t > l/ . (3.11)This linear growth can be explained by the entangled pair creations and their relativisticpropagations [39].In this case the ED looks like∆ n ( ξ, l, t ) = πc α δ ( l − t ) + ∆ n ( ξ, l, t ) UV . (3.12)The UV contribution ∆ n ( ξ, l, t ) UV ∼ O ( T tt ) is localized at short distances l ≤ α such thatit obeys the sum rule (3.8).If we consider the massless free Dirac fermion CFT in two dimension ( c = 1) as asolvable example, we explicitly find the following expression of the EE S A = 16 log (cid:34) απ(cid:15) · cosh (cid:0) πt α (cid:1) sinh (cid:0) πl α (cid:1) cosh (cid:0) π α ( t + l/ (cid:1) cosh (cid:0) π α ( t − l/ (cid:1) (cid:35) . (3.13)Thus, the ED reads∆ n ( l, t ) = − l + π α (cid:32) (cid:0) πl α (cid:1) + 1cosh (cid:0) π α ( t − l/ (cid:1) + 1cosh (cid:0) π α ( t + l/ (cid:1) (cid:33) . (3.14) This is found by employing the conformal map ξ = exp (cid:16) πw (cid:15) (cid:17) and the expression (4.3) and (4.4).The same expression was obtained in [39]. For the boundary state we can choose either Dirichlet orNeumann boundary condition for the scalar field obtained from the bosonization, both of which lead tothe same entanglement entropy by an appropriate choice of twist operators as shown in [42]. l - D n Figure 3: The behavior of entanglement density (ED) under the global quenches. Theleft graph describes the ED ∆ n ( l, t ) as a function of l at t = 3 (we chose α = 0 . n ( l, t ) as a function of the time t (horizontalaxis) and the subsystem size l (depth axis), where we set α = 0 . n ( l, t ) in Fig.3. We can also confirm that the sum rule (cid:82) ∞ dl ∆ n ( l, t ) = 0,explicitly. Similar arguments can also be applied to holographic calculations of globalquenches [43, 44, 45, 46]. Next we consider the time evolution of an excited state produced by a local insertion ofa primary operator O at x = 0 at the time t = 0 [8, 9]: | Ψ( t ) (cid:105) = N O e − iHt · e − αH O (0) | (cid:105) , (3.15)where α is again an infinitesimally small regularization parameter; N O is the normalizationfactor so that the state has the unit norm. We call its time evolution a local operatorquench (refer to the left picture of Fig.1). We will discuss other types of local quenches(splitting and joining ones) in later sections, which are the main setups we consider inthis paper. However, here we briefly discuss the operator quenches because they areinstructive for our later arguments and the behaviors of their ED have not been studiedwell before.Let us focus on two dimensional free CFTs or more generally rational CFTs (RCFTs)such as free scalar fields or minimal models, for simplicity. We again choose the subsystem A to be the interval a ≤ x ≤ b . As found in [8, 9, 47], the evolution of (R´enyi) EE looks Refer also to the papers [47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66]for further related calculations. α →
0) ∆ S ( n ) A = 0 (0 < t < | a | , t > b ) , ∆ S ( n ) A = log d O ( | a | < t < b ) , (3.16)for any n , including the EE n = 1. Here we assumed b > | a | . The quantity d O ( ≥ x = 0 and thenit splits into the left and right moving quanta, which are entangled with each other asargued in [8, 9, 47].The entanglement density (ED) is found to be ∆ n ( a, b, t ) = − log d O · sgn( ab ) · δ ( t − | a | ) δ ( t − | b | ) (3.18)It is also straightforward to reproduce the behavior (3.16) by integrating this entangle-ment density over a suitable region of ( ξ, l ) as we explain in the left and middle picturesof Fig.4.Let us confirm this in an explicit example of a massless free scalar field φ . To obtainan analytical expression, we calculate the growth of 2nd R´enyi entanglement entropy∆ S (2) A . We choose the operator O to be O = e iφ + e − iφ . In this case we have d O = 2 and∆ S A = log 2 for a < t < b . This is simply because this operator creates the genuine Bellpair [8, 9, 47]. The explicit form of ∆ S (2) A was obtained in [47] and is given by∆ S (2) A = log (cid:18)
21 + | z | + | − z | (cid:19) , (3.19)where z = ( z − z )( z − z )( z − z )( z − z ) is the cross ratio of the four locations: z = − z = (cid:114) a − t − iαb − t − iα , z = − z = (cid:114) a − t + iαb − t + iα . (3.20)Indeed, in the limit α →
0, we can confirm ( z, ¯ z ) → (0 ,
0) for the regions 0 < t < | a | and t > b , while we have ( z, ¯ z ) → (1 ,
0) when | a | < t < b . This leads to the result (3.16).We plotted the ED obtained from this R´enyi entropy in Fig.5 for α = 0 .
1. Indeed, wecan numerically confirm the behavior (3.18) as well as the sum rule. It is also useful to notethat the ED is vanishing completely at t = 0 as the positive and negative delta-functionalpeaks coincide and cancel each other. Holographic Case To see this, note that for any a , b , we can write∆ S A = log d O · ( θ ( | b | − t ) θ ( t − | a | ) + θ ( t − | b | ) θ ( | a | − t )) , (3.17)by using the Heaviside step function θ ( x ). l (cid:647) l a b t -t 2b 2a 2t 2t Figure 4: The integration range (cid:82) dldξ of the ED (red regions) and causality range (brownshaded region). The left picture shows the region of integration which computes S A forthe subsystems A = [ a, b ]. The middle one shows the same one when A is the half line: a = 0 and b = ∞ . The brown shaded region in the right picture describe the range of( ξ, l ) where the local excitation at x = t = 0 can make any physical influence assumingcausality. In all these three pictures, the purple dot at ( ξ, l ) = (0 , t ) represents the peakof ED due to the local quench. In the right picture, the blue curve near this dot (givenby l = 2 (cid:112) ξ + t ) describes the delta functional peaks which are peculiar to holographiclocal quenches. Note that they are outside of the causality region. The calculations of S A for a (cid:28) t (cid:28) b correspond to integrating the red region in the right picture. We note theleft part of the blue curve gradually gets into the red region, which gives the logarithmicgrowth of EE.Figure 5: The profile of entanglement density for the 2nd Renyi entropy under the localoperator quench in c = 1 free scalar CFT at the time t = 0 (left) and t = 1 (right) with α = 0 .
1, trigger by the operator O = e iφ + e − iφ . The horizontal and depth coordinatecorrespond to ξ and l , respectively. 14ne may wonder if a similar result is true for local operator quenches in holographicCFTs. A holographic calculation for the operator local quench was given in [11] (see also[67]) and its entanglement entropy was reproduced from CFT analysis in [68]. Especially,the resulting entanglement entropy shows a logarithmic growth S A (cid:39) c tα + c l(cid:15) , (3.21)when | a | (cid:28) t (cid:28) b . We will explain the geometric explanation in gravity dual in section 6.This logarithmic growth is clearly different from the previous RCFT case (3.16). Thislooks puzzling because the relativistic propagation picture leads to the behavior (3.16)and there seems to be no way to explain the slow logarithmic growth (3.21) in relativistictheories.Interestingly, we can resolve this puzzle by looking at the ED instead of EE. Ourargument here will be very brief because we will study this more closely in the joininglocal quench model, which has the similarity on this aspect. The plot of ED shows thatthe peak at ( ξ, l ) = (2 t,
0) is not localized as opposed to the Dirac fermion case, but it iscontinuously distributed on a curve, which is depicted as the blue curve in the right pictureof Fig.4. We observe that the left part of peak curve (blue) enters into the integrationregion (red) as time evolves. Thus, we find that S A increases gradually under the timeevolution. This qualitatively explains the behavior (3.21).Note that the continuous peaks are outside of the region which we expect from thecausality propagation of the local excitation inserted at x = t = 0. The holographicanalysis in [11] shows they are on the curve l = 2 (cid:112) ξ + t . In particular, at t = 0, thepeaks are distributed on l = 2 | ξ | . This is clear different from the RCFT examples, wherethe ED vanishes everywhere at t = 0. From this, we learn that in holographic CFTs, theoperator local quench (3.15) initially generates highly non-local entanglement, as opposedto what we expect from the naive causality argument. The relativistic propagation ofthis non-local entanglement is the reason why we find the logarithmic growth (3.21).This property is also expected from the fact that the insertion of local operator is not alocal unitary transformation [56]. Note that this initial non-local entanglement can benegligible in the operator local quench in RCFTs, whose ED only has the localized peakat ( ξ, l ) = (2 t, Now we move on to the main analysis of this paper. Consider the splitting process ina 2d CFT. Namely, we start with a 2d CFT on a connected line −∞ < x < ∞ and at t = 0 we cut it into two halves at x = 0, as sketched in the middle picture of Fig.1. Thepreparation of the quantum state at t = 0 just after the splitting process can be done byconsidering the Euclidean path-integral as in Fig.6. The parameter α corresponds to theregularization of local quench. This setup is described by the conformal transformation: ξ = i (cid:114) w + iαw − iα ≡ f ( w ) , (4.1)15 =Im w x =Re w x t -i (cid:626) i (cid:626) -i (cid:626) Lorentzian Euclidean
Figure 6: The geometries which realize the splitting local quenches. The left figuredescribes the space on which we perform the Euclidean path-integral. This is mappedinto an upper half plane by the map (4.1). The right picture describes the path-integralrealization of the time evolution after the split process happened at t = 0, where theEuclidean path-integral for τ < − α < Im[ w ] < α and Re[ w ] = 0 (theleft picture of Fig.6) into an upper half plane Im[ ξ ] ≥ One example where we can calculate the time evolution of EE under the splitting quenchis the massless Dirac fermion CFT in two dimensions ( c = 1). We choose the subsystem A to be an interval a ≤ x ≤ b , or equally l = b − a and ξ = ( a + b ) /
2. Due to the paritysymmetry x → − x , we can assume | a | < b, b > , (4.2)without losing generality. In the rest of this paper, we always focus on the entanglemententropy S A for this interval.In the Dirac free fermion CFT, the twist operator σ n is described by a bosonization σ n = e in X , where X is the free massless scalar field dual to the Dirac fermion ψ via thestandard relation ψ = e iX . Since we know the analytical form of correlation functions offree scalar on an upper half plane, we can calculate the two point function of the twistoperators even in the presence of the boundary as follows (this is identical to eq.(2.31) of[14] see also [7, 69, 42]) : S A = −
16 log(
F (cid:15) ) , (4.3)where F is given by F = (cid:12)(cid:12)(cid:12)(cid:12) dξ a dw a (cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12) dξ b dw b (cid:12)(cid:12)(cid:12)(cid:12) · ( ξ a − ¯ ξ b )( ¯ ξ a − ξ b ) | ξ a − ¯ ξ a || ξ b − ¯ ξ b | ( ξ a − ξ b )( ¯ ξ a − ¯ ξ b ) . (4.4)16 t D S
10 20 30 40 50 t - - - D S - - Ξ- - - - - - - D S Figure 7: The plots of the entanglement entropy growth S A − S (0) A for α = 0 .
1. The leftgraph describes the time evolution when we take ( a, b ) = (15 , a, b ) = (0 . , ξ when we fix l = 2 and t = 0.The analytical expressions of S A when α is infinitesimally small are given as follows.In the early time period, 0 < t < | a | , we have S A = 13 log( b − a ) /(cid:15). (4.5)When | a | < t < b , we have S A = 16 log 4 | a | ( b − a )( t − | a | )( b − t )( a + b )( t + | a | ) α(cid:15) . (4.6)At late time, t > b , we have S A = 16 log 4 | a | b ( b − | a | ) ( b + a ) (cid:15) . (4.7)First, from (4.5),(4.6) and (4.7), we find that when the subsystem A is symmetricaround the origin i.e. ξ = 0, the EE coincides with that for the ground state S (0) A : S A (0 , l, t ) = S (0) A = 13 log l(cid:15) . (4.8)We also plotted the numerical values of EE in Fig.7. When ξ (cid:54) = 0, we find S A − S (0) A gets non-trivially positive only for the period | a | < t < b . This agrees with the relativisticparticle propagation picture of free massless Dirac fermion CFT. For 0 < t < | a | , the EEis the same as that for the vacuum. At the late time region t > b , it approaches to theEE for the ground state of the separated system. This is manifest in the left picture inFig.7. The middle picture in Fig.7 shows that when we take | a | (cid:28) b , the EE is graduallydecreasing just after the sudden initial rise. Indeed, from (4.6), we find the analyticalprofile for | a | (cid:28) t < b : S A (cid:39)
16 log 4 | a | ( b − t ) α(cid:15) . (4.9)The right picture in Fig.7 shows that the EE is reduced when either end point of A gets closer to x = 0, i.e. at the splitting point. This is because in this case, the EE onlycomes from that for a half-line. 17 - - Δ n < Δ n > ξ l - - Figure 8: The plots of the entanglement density ∆ n ( ξ, l, t ) for α = 0 . t = 0 (left twographs) and t = 1 (right two graphs). The upper and lower graphs are the contour and3D plots. The horizontal and depth coordinate corresponds to ξ and l , respectively.The entanglement density (ED) can also be computed from the second differentiationof the EE (3.1). The results are plotted in Fig.8. We can clearly see the positive peakat ( l, ξ ) = (2 t,
0) as well as the negative peaks at ( l, ξ ) = (0 , ± t ), similarly to the localoperator quench in RCFTs (3.18). The former agrees with the picture of propagatingrelativistic particles. The latter coincides the peaks of the energy density predicted fromthe first law of EE. In addition, there are also peaks near the origin ξ = l = 0 and this isdue to the cut along x = 0 where the space is divided into the left and right part, whichis special to the splitting local quenches.It is also interesting to note that a negative value region ∆ n < .2 Splitting Local Quench in Holographic CFT Another example which allows us to calculate the EE analytically is 2d holographic CFTs.For the holographic CFT, we can apply the coordinate transformation (2.12) to calculatethe EE. The dual Euclidean geometry is given by Im[ ξ ] ≥ S conA of thesubsystem A (= [ w a , w b ]) is computed by using (2.16) as follows: S conA = c (cid:20) | f ( w a ) − f ( w b ) | (cid:15) | f (cid:48) ( w a ) || f (cid:48) ( w b ) | (cid:21) , (4.10)where f ( w ) is given by (4.1).On the other hand, the HEE for the disconnected geodesics is found by applying theAdS/BCFT as we explained in eq.(2.20). The final result reads S disA = c (cid:18) f ( w a ))(Im f ( w b )) (cid:15) | f (cid:48) ( w a ) || f (cid:48) ( w b ) | (cid:19) + 2 S bdy . (4.11)Note that the boundary entropy S bdy contribution arises because the location of boundarysurface gets tilted according to the value of the tension T BCF T following (2.18), so thatit solves the boundary condition (2.17). The boundary entropy S bdy is parameterized by k as in (2.19). Since S dis is the sum of two disconnected geodesic, we have the doubledcontribution of S bdy in (4.11). Refer to the right picture of Fig.2.The final HEE is given by the smaller of the two: S A = min { S conA , S disA } . (4.12)The behaviors of EE are plotted in Fig.9 by choose the vanishing tension T BCF T = 0or equally vanishing boundary entropy S bdy = 0.When α is infinitesimally small, we obtain the following analytical expressions (wetake the boundary entropy arbitrary) as explained in the appendix B:In the early time period, 0 < t < | a | , we have S conA = c b − a ) /(cid:15),S disA = c a − t )( b − t ) α (cid:15) + 2 S bdy . (4.13)When | a | < t < b , we have S conA = c b − a )( t − a )( b − t ) α(cid:15) , ( a > c b − a )( t + a )( b + t ) α(cid:15) , ( a < S disA = c | a | ( b − t ) α(cid:15) + 2 S bdy . (4.14)19t late time, t > b , we have S conA = c b − a ) /(cid:15), ( a > c t − a )( t − b ) α (cid:15) , ( a < S disA = c | a | b(cid:15) + 2 S bdy . (4.15)It is clear that at the late time limit t → ∞ , the EE approaches to a constant value S A ( t → ∞ ) = min (cid:20) c b − a(cid:15) , c | a | b(cid:15) + 2 S bdy (cid:21) , (4.16)which agrees with the expected result for the separated two half lines.The behavior of entanglement entropy is also numerically plotted in Fig.9 by choosingthe vanishing tension T BCF T = 0 or equally vanishing boundary entropy S bdy = 0. Theleft graph looks very similar to the one in the Dirac fermion case, which is interpretedby the propagation of relativistic particles. However, it is intriguing to note that if theboundary entropy S bdy is positive and very large, then S conA can dominate in some regionand gives a qualitative discrepancy from the Dirac fermion result.It is interesting to ask the time evolution of EE when we choose the subsystem A to bealmost a half of the total system i.e. a (cid:28) t (cid:28) b . If we choose the boundary entropy S bdy is very large such that k (cid:29) k (2.19)), then S conA is favoredfor the period a (cid:28) t (cid:28) (2 k + 1) a , which results in the logarithmic growth S conA (cid:39) c tα + c l(cid:15) . (4.17)This is peculiar to the holographic CFTs. However, if k is order one, this logarithmicgrowth is missing and the EE monotonically decreases as in the previous Dirac fermionexample.The right picture (assuming S bdy = 0) in Fig.9 is qualitatively similar to what weobtained for the free Dirac fermion CFT. For finite α >
0, one may notice that theentanglement entropy for the connected curve S conA is not continuous as in the third graphin Fig.9. However, this occurs when one of the endpoints of the subsystem A coincideswith x = 0 and thus this discontinuity should happen because the left and right at x = 0are disconnected in the splitting quench. This can be easily understood if we go to thePoincar´e coordinate (2.11).The entanglement density is plotted in Fig.10. In the upper graphs, we smearedthe derivatives in (3.1) by replacing it with the finite difference: e.g. ∂ a S ( a, b, t ) → ( S ( a + δ/ , b, t ) − S ( a − δ/ , b, t )) /δ . We did so because in this holographic case, we expectgenuine delta-functional behaviors due to the phase transitions of HEE. In addition tothe expected positive peak at ( ξ, l ) = (0 , t ) and the two negative ones at ( ξ, l ) = ( ± t, t D S
10 20 30 40 50 t - - - D S - - Ξ- D S Figure 9: The plots of the holographic entanglement entropy growth S A − S (0) A at α = 0 . Q is vanishing T BCF T = 0. The blue/redgraph describes the connected/disconnected contribution. The left graph describes thetime evolution when we take ( a, b ) = (15 , a, b ) = (0 . , ξ when we fix l = 2 and t = 0.we observe continuous peaks along the following two curvesCurve 1 : l = − t + | ξ | k + (cid:115)(cid:18) t − | ξ | k (cid:19) + 4 kk + 1 ( t + | ξ | ) | ξ | , (4.18)Curve 2 : l = t + | ξ | k + (cid:115)(cid:18) t + | ξ | k (cid:19) − kk + 1 ( t − | ξ | ) | ξ | , (4.19)where we used the positive parameter k defined in (2.19). These two curves are locatedat phase transition points between S con and S dis as we can see from (4.14), where (4.18)and (4.19) are situated in a > a <
0, respectively. Their profile is plotted in thelower left picture of Fig.10. Both of them coincides with l = 2 | ξ | at t = 0. In the limits t → ∞ and | ξ | → ∞ , the curves are approximated byCurve 1 : l (cid:39) kk + 1 | ξ | , ( t → ∞ ) , l (cid:39) | ξ | − t k + 1 , ( | ξ | → ∞ )Curve 2 : l (cid:39) t − k − k + 1 | ξ | , ( t → ∞ ) , l (cid:39) | ξ | + 2 t k + 1 . ( | ξ | → ∞ )The presence of these peak curves is peculiar to holographic CFTs and is missing inthe free Dirac fermion CFTs or more generally RCFTs. They are outside of the causalityzone and therefore the initial state already has highly non-local entanglement. In thelower right picture of Fig.10, we sketched the calculation of EE, which is given by theintegration of ED over the red region. For this, we notice that the blue curve (4.19)gets into the red region, while the green curve (4.18) goes out. Thus, we can explainthe absence of the logarithmic growth of EE for the splitting local quenches due to thecancellations between these two opposite effects. Also we will provide the geometricexplanation of the log t behaviors in section 6. However, if we consider the case k (cid:29)
1, then (4.18) approaches l = 2 ξ and thus this curve is fullyincluded in the red region until t gets very large. This explains the logarithmic growth (4.17), peculiarto the large k case. =0.5 k=1 k=2k=0.5 k=1k=2 Curve 2Curve 1 - - Ξ l Figure 10: The behaviors of a smeared entanglement density (ED) for the holographicsplitting local quench. The upper left and right graph are the plots of ∆ n ( ξ, l, t ) at t = 0and t = 2, respectively with α = 0 . T BCF T = 0). The horizontal anddepth coordinate corresponds to ξ and l , respectively. Here we smeared the originallydelta functional behavior of the ED by replacing the second derivatives w.r.t l and ξ withfinite-differences δ = 0 .
2. We observe delta functional behaviors on the two curves (4.18)and (4.19) at k = 1. The lower left picture shows these two curves at t = 2 for k = 0 . , , S A for a (cid:28) t (cid:28) b by integrating theED over the red region. The green and blue curves represent curve 1: (4.18) and curve 2:(4.19). We find that the blue curve gets into the red region, while the green curve goesout. The cancellations between them leads to the absence of logarithmic growth.22 .3 Holographic Geometry of Splitting Local Quench Now let us study the holographic geometry dual to the splitting local quench. The metricof the gravity dual is given by (2.13) with T ( w ) = 3 α w + α ) , ¯ T ( ¯ w ) = 3 α
4( ¯ w + α ) , (4.20)where w = x + iτ and ¯ w = x − iτ . For its Lorentzian extension, we can set w = x − t and ¯ w = x + t . The important ingredient of AdS/BCFT as reviewed in section 2.5 isthe boundary surface Q which extends from the AdS boundary z = 0 to the bulk AdS.At z = 0, Q coincides with the cut in Fig.6, which describes the splitting process inthe 2d CFT. Therefore, we have to be careful in its global geometry i.e. which part ofthe geometry we should pick up and where the boundary surface Q in the AdS/BCFTprescription is located.Let us consider the Euclidean geometry dual to the left CFT picture of Fig.6. It isimportant to note that the gravity dual is precisely defined by mapping an upper halfIm[ ξ ] ≥ (2.11) using the transformation (2.12) with (4.1), assumingthat the tension is vanishing T BCF T = 0. The boundary Q in the AdS/BCFT, given byIm[ ξ ] = 0 in the latter setup, is mapped to a region on x = 0.First we focus on the time slice τ = 0 (i.e. Im[ w ] = 0) of the gravity dual. The metricon this time slice is given by ds = dz z + (cid:18) z + 3 α z α + x ) (cid:19) dx . (4.21)We can confirm that this slice is mapped into a quarter of the sphere given by | ξ | + η =1 with η ≥ ξ ] ≥
0. However, to realize one to one map, we need to remove theregion z > x + α ) α , (4.22)and identify x with − x for any x along the curve z = x + α ) α . A sketch of the map (2.12)at t = 0 is depicted in Fig.11. The boundary surface Q extends from the AdS boundary z = 0 toward the IR region but it ends at z = 2 α due to the identification. This showsthat the two half lines x > x < z = 0, is connected in thebulk through the horizon given by the identified curve z = x + α ) α . Therefore the CFTson these two half lines are entangled and the entanglement entropy is estimated as thelength of this curve: S A = c (cid:90) x ∞ dx √ x + α (cid:39) c z ∞ α , (4.23)where z ∞ = 2 x ∞ /α is the IR cut off. This agrees with our expectation because the statebefore the splitting of the system was the ground state of the CFT on a line. Note that23 w,w,z) ( (cid:647)(cid:853)(cid:647)(cid:853)(cid:635) ) (cid:655) x z (cid:635) i 1 -1 1 i (cid:626) -i (cid:626) (cid:626) Identify
Remove
0 0
IdentifyIdentify
RemoveRemoveRemove Figure 11: The map (2.12) between the gravity dual of splitting local quench (left) andthe upper Poincar´e AdS (right). The red curve in the left is given by z = x + α ) α andthe region inside this curve should be removed with the identification x with − x on thecurve. The length of red curve gives the HEE between two separated lines. The greenand dark brown curve describe the boundary surface Q .in the Poincar´e coordinate, this is equal to a quarter of the largest circle in the sphere | ξ | + η = 1, depicted as the red curve in the left picture of Fig.11.Next we would like to examine how the boundary Q in the AdS/BCFT looks like inour gravity dual. To see this we analyze which region is mapped into Im[ ξ ] = 0. It isstraightforward to see that the boundary Q should be on the slice x = 0 (i.e. Re[ w ] = 0).Notice that we need to distinguish the two segments Q + and Q − of the boundary surface Q at x = + δ and x = − δ ( δ > ξ ] < ξ ] > − πα < τ < πα . In Fig.12,we sketched how the boundary surface Q + looks like in the ( w, ¯ w, z ) coordinate and howit is mapped into the boundary Im[ ξ ] = 0 and Re[ ξ ] < x = + δ .The tip z = 2 α of the identification region (4.22) at the time slice τ = 0 is extendedalong z = 2( α − τ ) √ α − τ . (4.24)for − α < τ < α . This is the red curve in Fig.12. Note that this tip is the end point ofthe boundary surface Q in the case of T BCF T = 0. Therefore the two sheets ( x = + δ and x = − δ ) are identified along this curve (4.24) with each other.There is one more non-trivial issue. To secure one to one mapping into the half ofPoincar´e AdS, we need to remove the white region in the left picture in Fig.12 and join thetwo blue curves with each other such that it agrees with the right picture i.e. Im[ ξ ] = 024 (cid:655) (cid:626) - (cid:626) (cid:2009) (cid:3398) (cid:2009) (cid:1878) = 2( (cid:2009) (cid:2870) (cid:3398) (cid:2028) (cid:2870) ) (cid:2009) (cid:2870) (cid:3398) (cid:2028) (cid:2870) (cid:1006)(cid:626) (cid:1878) = 2( (cid:2009) (cid:2870) (cid:3398) (cid:2028) (cid:2870) )3 (cid:2009) (cid:626) (cid:1006)(cid:626) (cid:2009) (cid:90)(cid:286)(cid:647) (cid:635) - (cid:2869)(cid:2871) (cid:3398) - Figure 12: The boundary surface Q + for T BCF T = 0 in the gravity dual of splitting localquench. The left picture describes in the coordinate ( w, ¯ w, z ) setting x = + δ →
0. Theright picture does in the Poincar´e coordinate ( ξ, ¯ ξ, η ) on the boundary Im[ ξ ] = 0. Thered curve in the left picture is mapped into the Re[ ξ ] = 0 in the right one. The two bluecurves in the left are pasted with each other in a way they are mapped into the blue curvein the right picture. The eight colored regions in the left are mapped into those in theright. Note that there are one more boundary surface corresponding to x = − δ → ξ ] > − α < τ < α should also be removed to secure the map is one to one.in the Poincar´e coordinate. Finally we can confirm that the colored regions in the leftpicture are mapped into those in the right one in Fig.12. The full boundary surface Q isgiven by joining two copies of such a space.We will not get into details of the geometry in other parts as we can understand howthe Lorentzian space looks like from the above observations. Nevertheless, it is helpful topoint out that the identification surface (4.24) at non-zero x is found to be z = 2 (cid:112) x + ( α + τ ) (cid:112) x + ( α − τ ) (cid:114) α − (cid:16)(cid:112) x + ( α + τ ) − (cid:112) x + ( α − τ ) (cid:17) ≡ G E ( x, τ ) . (4.25)In the Lorentzian signature, this surface looks like z = 2 (cid:112) x + ( α + it ) (cid:112) x + ( α − it ) (cid:114) α − (cid:16)(cid:112) x + ( α + it ) − (cid:112) x + ( α − it ) (cid:17) ≡ G L ( x, t ) . (4.26)Thus the Lorentzian geometry for t > w, ¯ w, z ) is given by removing the part z > G L ( x, t ) and by identifying each two points on (4.26) as ( τ, x, z ) ∼ ( τ, − x, z ). Theboundary surface Q consist of identical two surfaces Q + and Q − which extend in z < α + t ) √ α +4 t and are localized at x = δ and x = − δ , respectively. The final spacetime of thegravity dual is depicted in Fig.13. 25t late time we can approximate this as z (cid:39) t. (4.27)This behavior z ∼ t at x = 0 clearly shows that the two half lines at the boundary arenot causally connected in a marginal sense. Therefore the region x > x <
0, whichare separated at the AdS boundary z = 0 are connected in a space-like way (i.e. it is anon-traversable wormhole) through this horizon in the bulk. This is consistent with thefact that in the CFT side there is no direct interaction between these two CFTs for t > T BCF T = 0 for simplicity, we wouldlike to briefly mention how the gravity dual is modified when T BCF T (cid:54) = 0. When T BCF T is non-zero, the boundary surface Q in ( ξ, ¯ ξ, η ) will be tilted asIm[ ξ ] = − T BCF T (cid:112) − T BCF T η. (4.28)When T BCF T <
0, the boundary surfaces Q + and Q − tilted in the bulk such that they stillcoincide at z = 0 and the angle between them takes a fixed positive value. Therefore, thebulk region gets squeezed and the entanglement entropy S A (4.23) is decreased. When T BCF T >
0, the boundary surfaces are modified in an opposite way and the bulk regionexpands. The entanglement entropy S A gets increased.In this subsection we studied the Lorentzian motion of the boundary surface Q . For theholographic entanglement entropy, we actually need the profile of geodesic in Lorentziangeometry, though in the previous subsection we employed the formal Wick rotation argu-ments to avoid this analysis. We will explicitly study how the Lorentzian geodesic lookslike in section 6. Before we go on, we would like to present numerical results for a splitting local quenchin a simple spin model for a reference. We start with a spin-1 / L sites and free boundary condition. Cut the interaction between site j and j + 1 toseparate it into two independent chains at t = 0. In this case Hamiltonian before quench H before and that after quench H after turns out to be H before = − J L − (cid:88) i =1 σ zi σ zi +1 + h L (cid:88) i =1 σ xi (4.29) H after = − J (cid:18) L − (cid:88) i =1 σ zi σ zi +1 − σ zj σ zj +1 (cid:19) + h L (cid:88) i =1 σ xi (4.30)Then let us start with the ground state of H before and investigate its dynamics under H after at t ≥
0. We presented our numerical results in Fig.14. We can see the behavior26 t - (cid:626) Lorentzian (t>0)
Euclidean (cid:894)(cid:655)(cid:1092)(cid:1004)(cid:895) Q+Q-
Identified (cid:563)(cid:1085)(cid:563) - Figure 13: The spacetime geometry of gravity dual of the splitting local quench. Thegreen surface ( Q + ) and brown one ( Q − ) are the boundary surface Q in the AdS/BCFT.The red region defined by z > G L ( x, t ) should be removed with the dotted red curveand the doubled red curve identified. Thus this identified red region corresponds to thehorizon. The region x > x <
0, which are separated at the AdS boundary z = 0are connected (in a space-like way) through this horizon in the bulk.of its EE at an early time is very similar to the result in free fermion (4.6) and thedisconnected contribution in holographic CFTs (4.14), qualitatively. At late time weobserve oscillatory behavior because our spin system has a finite size. t S A (a) 0 ≤ t ≤
50 100 150 200 t S A (b) 0 ≤ t ≤ π Figure 14: Splitting local quench in spin-1/2 transverse Ising model: time evolution ofEE S A for J = h = 1, L = 6, and j = 3, where the subsystem A is chosen to be A = { i | i = 4 , } . We can see oscillation in long time range, while the short-time behaviorof S A is very similar to those in CFT cases. Consider a 2d CFT on two semi-infinite lines x > x <
0. We join each endpoint x = 0 at t = 0 as in the right picture of Fig.1. This setup is described by the path-integral27 =Im w x =Re w xt -i (cid:626) i (cid:626) -i (cid:626) LorentzianEuclidean
Figure 15: The geometries which realize the joining local quenches. The left figure de-scribes the space on which we perform the Euclidean path-integral. This is mapped intoan upper half plane by the map (5.1). The right picture describes the path-integral re-alization of the time evolution after the joining process happened at t = 0, where theEuclidean path-integral for τ < ξ = i (cid:114) iα − wiα + w ≡ f ( w ) . (5.1)The parameter α again corresponds to the regularization of local quench. We choose thesubsystem A to be a ≤ x ≤ b at time t as before. This corresponds to w a = a − t, ¯ w a = a + t, w b = b − t, ¯ w b = b + t, (5.2)where we performed the analytical continuation of the Euclidean time τ = it . Consider a massless Dirac fermion CFT in this joining quench. The time evolution of EEcan be computed from the formula (4.3) and (4.4) as in the previous analysis of splittingquench.The analytical expressions of S A at the time t when α is infinitesimally small, aregiven as follows (we choose A to be the interval a ≤ x ≤ b and we can again assume (4.2)without losing generality). In the early time period, 0 < t < | a | , we have S A = 16 log 4 | a | b ( b − a ) ( b + | a | ) (cid:15) , (5.3)When | a | < t < b , we have S A = 16 log 4( b − a ) b ( b − t )( t − a ) α ( a + b )( b + t ) (cid:15) , (5.4)28 t (cid:1) S
10 20 30 40 50 t - - - (cid:1) S - - (cid:1) - - - - - - (cid:2) S Figure 16: The plots of the entanglement entropy growth S A − S (0) A for α = 0 . a, b ) = (15 , a, b ) = (0 . , ξ when we fix l = 2 and t = 0.At late time, t > b , we have S A = 13 log( b − a ) /(cid:15). (5.5)It is useful to note that when b = − a > S A takes a constant value for any t : S A =(1 /
3) log (2 b/(cid:15) ) = S (0) A , as we found also for the splitting quench.The behavior of entanglement entropy is also numerically plotted in Fig.16. The leftgraph shows that the EE gets larger during only the time period where one of the entangledpair is included in the subsystem A, while the EE is the same as its vacuum value for theother time period. This fact is also true in the above analytical results as well as in all ofour numerical plots. This property is consistent with the relativistic particle propagationpicture for the massless Dirac fermion CFT.The middle graph shows a logarithmic growth at an early time. Indeed, from (5.4),we find that when | a | (cid:28) t (cid:28) b , the EE shows the logarithmic behavior: S A (cid:39)
13 log 2 t(cid:15) + 16 log lα , (5.6)which agrees with the known local quench behavior [10].The entanglement density (ED) can also be computed by taking differentiations ofthis entropy. The results are plotted in Fig.17. We can clearly see the positive peakat ( l, ξ ) = (2 t,
0) as well as the negative peaks at ( l, ξ ) = (0 , ± t ). The former agreeswith the relativistic particle picture. The latter coincides the peaks of the energy density.Even though in the previous splitting case of the Dirac fermion CFT, the negative region∆ n < l, ξ ) = ( t,
0) was expanding, in the present joining case, the expandingregion at the same place is positive ∆ n >
0. This indeed violates the naive causalityconstraint (refer to the right picture in Fig.4) and we can again interpret this as thepresence of non-local entanglement in the initial state just after the quench. This leadsto the logarithmic growth at late time in the joining case (5.6) as opposed to the splittingcase. 29igure 17: The plots of the entanglement density ∆ n ( ξ, l, t ) for α = 0 . t = 0 (left) and t = 1 (right). The horizontal and depth coordinate corresponds to ξ and l , respectively.30 .2 Joining Local Quench in Holographic CFT For the holographic CFT, we can substitute the coordinate transformation (5.1) to (2.12)in order to calculate the EE. The dual geometry is given by Im[ ξ ] ≥ S conA and S disA at the time t when α is infinitesimallysmall, are given as follows (we choose A to be the interval a ≤ x ≤ b with (4.2)). In theearly time period, 0 < t < | a | , we have S conA = c b − a ) /(cid:15), ( a > ,c a − t )( b − t ) α (cid:15) , ( a < S disA = c (cid:0) b | a | /(cid:15) (cid:1) + 2 S bdy . (5.7)When | a | < t < b , we have S conA = c b − a )( t − a )( b − t ) α(cid:15) , S disA = c b ( t − a ) α(cid:15) + 2 S bdy . (5.8)At late time, t > b , we have S conA = c b − a ) /(cid:15), S disA = c t − b )( t − a ) α (cid:15) + 2 S bdy . (5.9)In the late time limit t (cid:29) b , we always find S conA = S (0) A dominates as we expect.The behavior of entanglement entropy is also numerically plotted in Fig.18. By com-paring the holographic results (Fig.18) for vanishing boundary entropy S bdy = 0 with thosefor the Dirac fermion’s one (Fig.16), we find they agree with each other qualitatively, ifwe ignore the phase transition behavior in the former. If we take | a | (cid:28) t (cid:28) b , then wefind the logarithmic growth: S A = S disA (cid:39) c t(cid:15) + c lα , (5.10)which indeed agrees with the Dirac fermion result (5.6).However if the boundary entropy is large enough for S conA to be favored, then we finda slower logarithmic growth for | a | (cid:28) t (cid:28) b : S A = S conA (cid:39) c tα + c l(cid:15) . (5.11)The entanglement density is plotted in Fig.19. In the upper two graphs, we againsmeared the derivatives in (3.1) by replacing it with the finite difference. We observe thatin addition to the expected positive peak at ( ξ, l ) = (0 , t ) and the two negative ones at31 (cid:2)(cid:1) (cid:3)(cid:1) (cid:4)(cid:1) (cid:5)(cid:1) (cid:6)(cid:1) (cid:1) (cid:1)(cid:7)(cid:6)(cid:2)(cid:7)(cid:1)(cid:2)(cid:7)(cid:6)(cid:3)(cid:7)(cid:1) (cid:1) (cid:8) (cid:1)(cid:2) (cid:3)(cid:2) (cid:4)(cid:2) (cid:5)(cid:2) (cid:6)(cid:2) (cid:1) - (cid:2)(cid:7)(cid:6)(cid:2)(cid:7)(cid:6)(cid:1)(cid:7)(cid:2)(cid:1)(cid:7)(cid:6)(cid:3)(cid:7)(cid:2)(cid:3)(cid:7)(cid:6) (cid:1) (cid:8) - (cid:1) - (cid:2) (cid:2) (cid:1) (cid:1) - (cid:3)(cid:4)(cid:2)(cid:3)(cid:4)(cid:2)(cid:3)(cid:4)(cid:1)(cid:3)(cid:4)(cid:5)(cid:3)(cid:4)(cid:6) (cid:2) (cid:7) Figure 18: The plots of the holographic entanglement entropy growth S A − S (0) A for α = 0 . T BCF T = 0. Theblue and red curve describes S conA and S disA , respectively. The left graph describes thetime evolution when we take ( a, b ) = (15 , a, b ) = (0 . , ξ when we fix l = 2 and t = 0.( ξ, l ) = ( ± t, k as in (2.19)): l = t − | ξ | k + (cid:115)(cid:18) t − | ξ | k (cid:19) + 4 kk + 1 ( t + | ξ | ) | ξ | . (5.12)The profile of this curve is plotted in the lower left picture of Fig.19. This continuouspeak on (4.19) arises since there is a phase transition from S dis to S con as we can see from(5.8). At t = 0, this curve is reduced to the line l = kk +1 | ξ | . In the limit t → ∞ and | ξ | → ∞ we have l (cid:39) t + 2( k − k + 1 | ξ | , ( t → ∞ ) , l (cid:39) kk + 1 | ξ | + 4 k k + 1 t, ( | ξ | → ∞ ) . (5.13)The presence of the codimension one peak is again peculiar to holographic CFTs and ismissing in the free Dirac fermion CFTs or more generally RCFTs. Its presence shows thatinitial entanglement gets modified in highly non-local way. Moreover, this time, the curve(5.12) extends to the region outside of the region l > | ξ | (i.e. the red region). Thereforethe blue curve enters into the red region from both the left and right side as in the lowerright picture of Fig.19. This explains the logarithmic growth of S A (5.10) with the doubledcoefficient compared with that for the local operator quench (3.21). Moreover, if k getsvery large, the curve (5.12) approaches to l (cid:39) | ξ | in the limit | ξ | → ∞ . Therefore thesituation gets very similar to the local operator quench and therefore this explains thebehavior (5.11). Also we will provide the geometric explanation of the log t behaviors insection 6. Now let us study the geometry of the gravity dual for the joining local quench. Sincethis has many similarities with that for the splitting local quench, our explanation will be32 = = = (cid:1) l = (cid:1) l =- (cid:1) - - Figure 19: The behaviors of a smeared entanglement density for the holographic joininglocal quench. The upper left and right graph are the plots of ∆ n ( ξ, l, t ) at t = 0 and t = 2, respectively, with α = 0 . T BCF T = 0). The horizontal and depthcoordinate corresponds to ξ and l , respectively. Here we smeared the originally deltafunctional behavior of the entanglement density by replacing the second derivatives w.r.t l and ξ with finite-differences δ = 0 .
1. We observe a delta functional behavior on thecurve (5.12) at k = 1. The lower left picture describes the curve (5.12) for k = 0 . k = 1 (orange) and k = 2 (green) at t = 2 as well as the line l = 2 | ξ | (red). The lowerright picture shows the computation of S A for a (cid:28) t (cid:28) b by integrating the ED overthe red region. The blue curve represents the peaks on (4.19). As the time evolves, theblue curve enters into the red region from both the left and right side. This explains thelogarithmic growth of EE with the doubled coefficient.33rief. Indeed since the energy stress tensor (4.20) remains the same, the metric of gravitydual for the joining quench is the same as that for the splitting one. The differencecomes from the location of the boundary surface Q . The time slice t = 0 in the ( w, ¯ w, z )coordinate is again mapped into the quarter sphere | ξ | + η = 1 in the half Poincar´eAdS: Im[ ξ ] >
0. However the details of the mapping is different from the previous one assketched in Fig.20. In the present case, the region (4.22) corresponds to Im[ ξ ] < z = 2( x + α ) /α nowrepresents the boundary surface Q . The region − iα < τ < iα and x = 0 is now mappedinto a region Re[ ξ ] = 0 in the half Poincar´e AdS . Since this map is identical to that forthe splitting quench i.e. Fig.12, we will omit its detail.Finally we obtain the spacetime geometry sketched in Fig.21. The boundary surface Q , expressed by the red surface, is given by the same expression (4.25) and (4.26) in theEuclidean and Lorentzian case, respectively. In the Euclidean geometry, there is the greensurface Q (cid:48) which extends from the CFT boundary and this is actually identified with thered surface Q , whose details we will omit as they are involved with complicated numerics.In some sense, in this geometry, the location of the boundary surface Q is opposite tothat for the splitting local quenches.In summary, the time evolution after the joining local quench is described by thespacetime with the boundary surface Q which moves toward the horizon at the speed oflight. This intuitively agrees with our expectation that after the joining, the CFT stategradually approaches to the vacuum state on a connected line. It is also intriguing tonote that in the Lorentzian description, the boundary surface Q does not end on the AdSboundary but is localized in the bulk.So far we implicitly assumed T BCF T = 0. For more general cases T BCF T (cid:54) = 0, theAdS/BCFT setup in ( ξ, ¯ ξ, η ) coordinate is given by the boundary surface (4.28). Thenwe can map Q into the original coordinate ( w, ¯ w, z ) as in Fig.20. When T BCF T > T BCF T < Q is moved in the larger (smaller) z direction. Inother words, the physical region of the gravity dual, surrounded by the boundary surface Q and the AdS boundary z = 0, expands as T BCF T gets larger as expected.In this subsection we studied the Lorentzian motion of the boundary surface Q . For theholographic entanglement entropy, we actually need the profile of geodesic in Lorentziangeometry, though in the previous subsection we employed the formal Wick rotation argu-ments to avoid this analysis. We will explicitly study how the Lorentzian geodesic lookslike in section 6. t Geodesic Length in Holographic LorentzianGeometry
In our previous calculations of holographic local quenches we mainly worked in Euclideansetups and we took the analytic continuation to Lorentzian time evolutions only for the This motion of boundary surface is analogous to a string world-sheet description of holographic localquenches proposed in [70], though the choice of geodesics responsible for the HEE is different from ours. w,w,z) ( (cid:647)(cid:853)(cid:647)(cid:853)(cid:635) ) (cid:655) x z (cid:635) i 1 -1 1 i (cid:626) -i (cid:626) (cid:626) Remove
0 0
RemoveRemoveRemove Boundary
Figure 20: The map (2.12) between the gravity dual of joining local quench (left) andthe upper half of the Poincar´e AdS (right). The red curve in the left is given by z =2( x + α ) /α and this corresponds to the boundary surface Q . Thus the region inside thiscurve should be removed. xt - (cid:626) Lorentzian (t>0)
Euclidean (cid:894)(cid:655)(cid:1092)(cid:1004)(cid:895) Q x Q’ (cid:563) Figure 21: The spacetime geometry of gravity dual of the joining local quench. The redregion defined by z > G L ( x, t ) should be removed. The boundary of this region, whichwe call the red surface, corresponds to the boundary surface Q . Though the green surface Q (cid:48) is also boundary surface, it should be identified with the red one.35nal results of entanglement entropy. This procedure went straightforwardly or mechan-ically due to the simplicity of Euclidean gravity duals, while this raises the questionwhether we really have sensible Lorentzian gravity duals whose geodesic lengths give thecorrect holographic entanglement entropy. We will confirm this by explicitly finding outthe Lorentzian geodesics in this section. Note that in the true Lorentzian gravity dual,coordinates ( t, x, z ) should take real values, while the coordinates in Poincare coordinate( ξ, ¯ ξ, η ) take complex values in general due to the coordinate transformation (2.12) getscomplex valued under the Wick rotation.The results of the time evolutions of HEE under (i) local operator quench S ( O ) A (3.21),(ii) splitting local quench S ( S ) A (4.13)-(4.15), and (iii) joining local quench S ( J ) A (5.7)-(5.9)are characterized by two logarithmic behavior log t/(cid:15) and log t/α . Moreover, we can findthe same expression of S A for these different types of local quenches. Our analysis ofLorentzian geodesics in this section, will also explain such logarithmic behaviors in asystematical way. First note that in the late time limit t (cid:29) b > a , we find S con ( S ) A ( t ) (cid:39) S dis ( J ) A ( t ) (cid:39) c t (cid:15) + c t α . (6.1)First we would like to point out that both are identified with the twice of the geodesiclength between the boundary point ( t, x, z ) = ( t, ,
0) and the tip of the surface Q , givenby z = 2( α + t ) √ α + 4 t ≡ g ( t ) . (6.2)When t (cid:29) α , we have g ( t ) (cid:39) t + 7 α t + · · · . (6.3)These geodesics are sketch as Γ ± in Fig.13 and Γ in Fig.21. In the former (splittingquench) case, we consider a connected geodesic and therefore should avoid touching onthe boundary surface Q . On the other hand, in the latter (joining) case, the disconnectedgeodesic should end on the boundary surface Q . Therefore these two should approximatelycoincide in the late time limit, by setting a, b → t, x, z ) coordinate, it is easier to start with the transformedcoordinate (Re[ ξ ] , Im[ ξ ] , η ). However, we have to note that since we are considering ananalytical continuation into the Lorentzian geometry, the coordinates (Re[ ξ ] , Im[ ξ ] , η ) takecomplex values in general, though ( t, x, z ) should take real values. By remembering theconformal map (5.1) for the joining quench and the holographic transformation (2.12), it36s straightforward to find that the correct geodesic is given by ξ ¯ ξ + η = e − iβ , (6.4)where the real parameter β ( >
0) is related to the boundary time t at z = 0 via t = α · tan β . (6.5)Therefore for t (cid:29) α we have β (cid:39) π − αt . (6.6)In terms of the original coordinate, (6.4) is equivalent to z = 2( t + 1) (((1 + cos β ) t − α sin β ) + ( t sin β − α (1 − cos β )) ) / (((1 + cos β )(3 t + 4 t ) + α sin β ) + (sin β (3 t + 4 t ) + α (1 − cos β )) ) / , (6.7)which is plotted in Fig.22.At late time, we have z ( t ) (cid:39) t − t + O (1 /t ) and thus the geodesic extends to infinityalmost in a light-like way. In the limit t → t (cid:29) α we have z ( t ) (cid:39) (cid:112) t ( t − t ) + ( t − t ) / √ t + · · · . (6.8)Indeed we can confirm that this is a solution to the geodesic equation in the holographicspacetime, obtained from (4.20) and (2.13), given by (on x = 0): ds = 1 z (cid:2) dz − (1 + f ( t, z ) z ) dt (cid:3) , (6.9) f ( t, z ) = 9 α z − α ( t + α ) t + α ) . (6.10)We can estimate the geodesic length of (6.7) as (cid:90) ∞ t dt (cid:112) ( z (cid:48) ) − − f z z ( t ) (cid:39) c t(cid:15) . (6.11)The twice of this length can explain only one of the logarithmic term in (6.1).The missing contribution actually comes from another part of the geodesic which isnot covered by the coordinate (6.9). Indeed, if we parameterize the geodesic (6.4) as ξ = e − iβ/ cos θ, η = e − iβ/ sin θ, (0 ≤ θ < π . (6.12) We can find this form by noting that the solution to the geodesic equation takes the form ξ ¯ ξ + η =const. and by plugging the coordinate values at the AdS boundary. θ = 0 corresponds to the AdS boundary η = 0 and θ = π/ Q on Im[ ξ ] = 0. Actually from the map (2.12), we find that thetrajectory (6.7) for t ≤ t ≤ ∞ only covers a part of the full geodesic (6.12) given by0 ≤ θ ≤ β/
2. The geodesic length for this part is computed as (cid:90) cos( β/ e − iβ/ ˆ (cid:15) e − iβ/ dηη (cid:112) e − iβ − η = log (cid:18) e − iβ/ ˆ (cid:15) (cid:19) + log (cid:18) cos( β/ β/ (cid:19) , (6.13)where η = ˆ (cid:15) is the cut off corresponding to the original one z = (cid:15) . The first term in theright hand side is the full contribution (cid:90) e − iβ/ ˆ (cid:15) e − iβ/ dηη (cid:112) e − iβ − η = log (cid:18) e − iβ/ ˆ (cid:15) (cid:19) (cid:39) log t (cid:15) + log t α , (6.14)which agrees with a half of (6.1). The second term is estimated when t (cid:29) α aslog (cid:18) cos( β/ β/ (cid:19) (cid:39) − log t α . (6.15)This shows that the part of geodesic length in the Poincar´e-like coordinate patch (6.9)gives (identical to (6.11)) L poincar ´ e = log t (cid:15) , (6.16)and the one outside of the Poincar´e patch does L outside = log t α . (6.17)To understand this outside contribution well , let us ignore the back reaction andfocus on the Poincar´e AdS (restricted on x = 0) by setting f ( t, z ) = 0 in (6.9). This isembedded in the global AdS (refer to Fig.22) ds = − ( r + 1) dT + dr r + 1 + r dφ , (6.18)at θ = 0, via the map (refer to e.g.[8]) √ r + 1 cos T = α + ( z − t ) /α z , √ r + 1 sin T = tz . (6.19)Here the parameter α is introduced such that the static trajectory at r = 0 is mappedinto that for an accelerated particle z = √ t + α in the Poincar´e AdS, which imitatesthe behavior (6.3). Such a contribution is also familiar in holographic global quenches [43, 44, 45] and local operatorquenches [63]. T r r=0r=0r=0r=0Q
Poincare Outside (cid:660) (cid:651) - (cid:651) P P Figure 22: Profiles of the geodesic. In the left graph, we show the geodesic (blue) andthe boundary surface Q (orange) in the Poincar´e coordinate ( z, t ) with t = 10 and α = 1. The right picture shows the outside part of the disconnected geodesic L outside (green dotted line) as well as the one in the Poincar´e patch L poincar ´ e (blue line) in theglobal AdS coordinate. The point P is situated at 0 < x (cid:28) t , while the point P is at0 < t (cid:28) x . The blue region corresponds to the Poincar´e patch. The red vertical line is thetip (6.2) of the boundary surface Q . The brown curve describes the connected geodesicthat corresponds to (6.21).At the AdS boundary, the late time limit t (cid:29) α in the Poincar´e AdS corresponds tothe time T (cid:39) π − α/t and the cut off r ∞ = t / ( α(cid:15) ) in the global AdS. We are consideringa geodesic which starts from this boundary point and ends on the tip of the surface Q ,namely the straight line r = 0 in the global AdS . It is clear that this geodesic is givenby T =constant and the radial coordinate r runs from r = r ∞ to r = 0. Along thisgeodesic, the part outside of Poincar´e patch is 0 < r < r ∗ , where r ∗ is the solution to √ r + 1 cos T + r = 0, from which we find r ∗ (cid:39) t /α (cid:29) L poincar ´ e = (cid:90) r ∞ r ∗ dr √ r + 1 (cid:39) log t (cid:15) ,L outside = (cid:90) r ∗ dr √ r + 1 (cid:39) log t α . (6.20)These reproduce the previous results from the exact geodesic (6.16) and (6.17). Now we move on to the middle time period and we focus on the region a (cid:28) t (cid:28) b tosee the logarithmic behavior clearly. Then we find the following coincidences between the39hree different types of local quenches : S con ( O ) A (cid:39) S con ( S ) A (cid:39) S con ( J ) A (cid:39) c t α + c b(cid:15) . (6.21)On the other hand, the disconnected geodesic length in joining quench behaves differently: S dis ( J ) A (cid:39) c t (cid:15) + c t α + c b(cid:15) . (6.22)The coincidence (6.21) can be easily understood because they are described by ageodesic which connects the two end points, where the existence of slits is not a crucialobstruction, and because the background metric is the same (refer to the brown curve inFig.22). We can regard the first term ( c/
6) log( t/α ) as the geodesic length outside of thePoincar´e patch as in (6.17). The second term ( c/
3) log( b/(cid:15) ) is simply interpreted as thefamiliar geodesic length in the Poincar´e AdS .On the other hand, the geodesic length (6.22) is divided into two contributions: theone which connects x = a and Q and the other one which connects x = b and Q . Theformer one is clearly equal to ( c/ L poincar ´ e + L outside ) in (6.16) and (6.17). The latterone is simply given by the standard result of AdS/BCFT: ( c/
6) log( b/(cid:15) ). These explainthe behavior (6.22).
Here we would like to present a sketch of tensor network description of time evolution ofour local quench states and their entanglement entropies so that they matches with ourgravity duals. Our argument in this section will be heuristic and qualitative.We employ the MERA (Multiscale Entanglement Renormalization Ansatz) [16] for ourqualitative interpretation, though our arguments can equally hold for other holographictensor networks. MERA is a class of tensor networks which are constructed by two kindsof tensors: isometry and disentangler (Fig.23). This is manifestly scale invariant andis known to describe ground states of critical quantum systems with high accuracy. Intensor networks, by contracting all interior tensors, a desired quantum state is realized atthe boundary. We regard the physical system as the boundary, and the tensor networkstructure as the bulk in AdS/CFT [15].Consider a curve γ in the bulk, whose length can be defined by the number of tensorsit crosses . For a subsystem A on the boundary, its EE S A is upper bounded by S A ≤ C × (length of γ A ) (7.1) Note that for the holographic local operator quench, we need only connected geodesics as there is noboundary surface Q . It would be more precise to define the length of a curve in a tensor network by the number of tensorlegs it crosses. In our case, however, since we only care about the shortest curve in MERA, this definitionis also allow for convenience. A is a subsystem of the boundary of MERA, and γ A is a curvein the bulk who share the boundary with A . ( z, x ) forms a Poincar´e coordinate, while w = log ( z/(cid:15) ) corresponds to the number of layer.where C is a constant and γ A is an arbitrary curve who shares the boundary with A (i.e. ∂γ A = ∂A ).Particularly, in typical examples of MERA, such as that of a critical transverse Isingmodel, S A is roughly equal to S A (cid:39) C × (cid:0) length of γ minA (cid:1) (7.2)where γ minA is the minimal curve, whose length is proportional to log L A , with L A the size(i.e. the number of spins) of subsystem A .A qualitative correspondence between MERA and AdS is as follows [15]. If we set thecoordinate x parallel to the layers, and the coordinate w = log ( z/(cid:15) ) vertical to them (theright picture in Fig.23), then the geometry of MERA is interpreted as a discrete versionof the time slice of AdS (i.e. hyperbolic space), whose metric is ds = (cid:18) dz + dx z (cid:19) = R (cid:15) − dw (log e ) + 2 − w dx , (7.3)where (cid:15) is introduced as an UV cutoff. Refer to [25] for a possible interpretations ofvarious slices in AdS spacetime as tensor networks.Now we would like to consider an tensor network interpretation of our three types of lo-cal quenches (local operator/splitting/joining). We will take a heuristic strategy: first weassume specific structures of tensor networks under the time evolutions and then confirmif they reproduce the correct time evolutions of entanglement entropy. In our construction41igure 24: A tensor network description of local operator quench at t = 4 (cid:15) . The blue dotis the large modification of tensor dual to the falling particle. The blue curves describethe image of shock wave deformation of tensors. The parameters of tensors are expectedto be modified, and tensors close to the blue curve are expected to accept the strongestmodification.of tensor networks corresponding to local quenches, we assume that an appropriate timeslice t = const. of a gravity dual spacetime after a local quench, is approximately a tensornetwork at the time t . We note that the information of tensor network at each time canbe separated in to two ingredients: one is the geometric structure of tensor network itself,and another one is the parameters of each tensor. One essential observation we willmake below is that we can divide the growth of EE into a “shock wave contribution”and “splitting/joining contribution”, which can also be suggested from the holographicanalysis in the previous section. In the discussion below, we will regard the “shock wavecontribution” as modifications of the parameters and “splitting/joining contribution” asmodifications of the network structure. We will see that under this assumption, by choos-ing appropriate tensor network structure and neglecting modifications on the parameters,we can realize qualitative features of the entanglement structure in the AdS/CFT setup. In a MERA corresponding to the ground state of a critical system, for example, network structuremeans the way tensors linked with each other. Besides, all the isometries (disentanglers) carry the sameparameters due to the scale invariance of the state. Network structure and parameters together gives acorresponding quantum state. .1 Local Operator Quench A local operator quench in a holographic CFT can be regarded as a particle falling towardsthe AdS black hole horizon on the gravity side [11]. This particle trajectory is analogousto the tip of the boundary surface Q in the splitting/joining local quench. In this picture,its back-reaction on spacetime spreads as a shock wave [71], and the EE of a subregionon the boundary will change when the shock wave crosses its minimal surface.Accordingly in the tensor network, we expect the local operator quench causes amodification of tensor at x = 0 and w = 0 due to the initial operator insertion, thoughthe structure of tensor network does not change. As the time evolves, this modificationpropagates into the interior as w ∼ log t . At the same time, tensors are modified alongthe shock wave. These modifications increases quantum entanglement locally. These aredepicted in Fig.24.The behavior of the EE is consistent with the causal behavior that the EE gets non-trivial only for the time interval a < t < b , as we found both for the RCFT and holographicCFT. In the holographic case (3.21), however, we observed a logarithmic growth S A (cid:39) c tα (cid:124) (cid:123)(cid:122) (cid:125) shock wave + ..., (7.4)if a (cid:28) t (cid:28) b . Our entanglement density analysis showed such a logarithmic contribution isdue to the presence of non-local entanglement at the initial state. Our geometric analysisin the gravity dual shows this contribution comes from outside of the Poincar´e patch(6.17). In tensor network, this hidden contribution is expected to come from modificationsof tensor parameters, though we cannot figure it out precisely. Since this is owing to thehuge back-reactions, we would like to still call this a “shock wave contribution”. Tofind out the details of this contribution, we need to make the conjectured correspondencebetween the AdS/CFT and tensor networks more precisely beyond qualitative arguments,which is not available at present. Now we moved on to the splitting quench. We split the disentangler in the MERA at x = 0 , z = (cid:15) ( w = 0), initially (refer to [72] for introducing boundaries in MERA). Thisis motivated by the gravity dual where the boundary surface extends to the bulk as inFig.13. We identify the boundary in the gravity dual as the termination of tensor network(i.e. cutting the disentanglers or unitaries). This elimination of disentanglers propagatesunder the time evolutions as in the gravity dual. Therefore at the time t = 2 n (cid:15) , thedisentangler at x = 0 , w = n is eliminated. At the same time, from the gravity dual, weexpect there are shock wave propagations as in the local operator quench case. Thus wealso expect that in our tensor network description a shock wave spreads from w = n atthe same time t = 2 n (cid:15) . Both of them will contribute to the change of EE. We call them“shock wave contribution” and “splitting contribution” respectively.43rom the CFT viewpoint, we can heuristically understand this prescription as follows.When we triggered the splitting quench, we eliminate the nearest neighbor interactionsbetween two adjacent lattice sites. This is equivalent to cutting the disentangler in thevery UV. After the time evolution this cutting of entanglement propagates towards theIR. We show in Fig.25 how the network changes after splitting quench. If we focus on thesubsystem A = { x | x ∈ (0 , l(cid:15) ) } and neglect modifications of tensor parameters (equallyneglect “shock wave contribution”), we will find S A is given by the length of minimaldisconnected curve, which decreases logarithmically in time at first, and then stays asa constant. In other words, splitting contribution gives a negative logarithmic growth ∝ − log t/(cid:15) to S disA . On the other hand, as we have mentioned in the case of local operatorquench (7.4), the shock wave contribution should give a positive logarithmic growth ∝ log t/α to EE. In this way, we reach the following estimation: S disA (cid:39) c tα (cid:124) (cid:123)(cid:122) (cid:125) shock wave − c t(cid:15) (cid:124) (cid:123)(cid:122) (cid:125) splitting + ..., (7.5)where we determined the coefficient of the second logarithmic term so that we do nottotally have any logarithmic growth as in our previous calculation of HEE (4.14). t = (cid:15) t = 2 (cid:15) Figure 25: Time evolution of the tensor network after splitting quench. The red curveshows the minimum curve in the bulk. The blue dot and green curve are dual to thefalling particle and boundary surface Q . Here, A = { x | x ∈ (0 , (cid:15) ) } . For the joining local quench, we start with two copies of semi infinite MERA, whichare disconnected initially. The quench is triggered by adding a new disentangler on44 = 0 , z = (cid:15) ( w = 0), at t = (cid:15) . Then, besides a shock wave spreading at w = n in thebulk, a new disentangler on x = 0 comes out to connect the n th layer at t = 2 n (cid:15) . Both ofthem will contributes to the EE evolution. We call them “shock wave contribution” and“joining contribution” respectively.We show in Fig.26 how the network will change after joining quench. If we focus onblock A = { x | x ∈ (0 , l(cid:15) ) } and neglect modifications of tensor parameters (equally neglect“shock wave contribution”), we will find S A is firstly given by the length of minimaldisconnected curve, which increases logarithmically in time. In other words, the joiningcontribution gives a positive logarithmic growth to S disA , which is expected to be equalto the absolute value of splitting contribution, since they should cancel with each otherif they occur at the same time. On the other hand, since the shock wave contributionshould give a positive logarithmic growth that cancels with split contribution, it is properto think that the two contributions together give a double logarithmic growth, which isexactly consistent with the results in Section 5.2. That is, S disA (cid:39) c tα (cid:124) (cid:123)(cid:122) (cid:125) shock wave + c t(cid:15) (cid:124) (cid:123)(cid:122) (cid:125) joining + ... (7.6)In this way, we can explain the logarithmic growth we saw in (5.8)Figure 26: Time evolution of the tensor network after joining quench. The left, middleand right picture correspond to t = (cid:15) , 2 (cid:15) and 4 (cid:15) , respectively. The red curve shows theminimum curve in the bulk. The blue dot and green curve are dual to the falling particleand boundary surface Q . Here, A = { x | x ∈ (0 , (cid:15) ) } . We can simply summarize gravity dual spacetimes of the splitting and joining local quenchas in the left and middle picture of Fig.27 in the limit α →
0. If we combine them suchthat we join two CFTs at t = t and split them again at t = t , then the expectedholographic dual is given by the right picture in Fig.27. The important lesson from thisexample is that we can create a slit ( S in that picture) which looks “floating” in the bulk45 x Q t x Q t=0 t=0 Q x Q =0 t x Q t=t t=t Q t= S Figure 27: A simplified sketch of the gravity dual spacetimes for the splitting (left), thejoining (middle) local quench and their combination (right). The brown surfaces describethe boundary surfaces Q . In the right picture, the boundary surface Q intersects with theblue horizontal plane which describes the time slice t = 0. Their intersection S makes aslit on the time slice.on a time slice of gravity dual. Of course, the location of such a floating slit moves underthe time evolution following the condition (2.17).As an ambitious attempt which finalizes this paper, we would like to propose a holo-graphic counterpart of tensor network states in the AdS/BCFT setup by repeating suchprocedures. Consider the MERA tensor network [16] and try to realize an analogous ge-ometry on a time slice of a gravity dual. As we explained in section 7, the MERA networkconsists of the disentangling and coarse-graining operations, which cut the entanglementin the ground state of a critical spin chain. We depicted a simple setup to achieve thisin Fig.28. In this analogy, each “spin” in MERA corresponds to the (blue) region be-tween two (brown) slits. A coarse-graining and disentangler correspond to terminatingand creating a hole in a (brown) slit, respectively. The standard estimation of EE S A in the MERA nicely agrees with the HEE calculation in the AdS/BCFT as we show inFig.28, where we simply ignored the back-reactions. The entanglement entropy reducedby the coarse-graining/disentangler operation correspond to the green/red dotted line inthe pictures.Notice that in this section we consider a different interpretation of the AdS spacetimein terms of tensor networks, as compared with the arguments in section 7. Here weapproximately regard the narrow strip in AdS space as a link in a tensor network and fillthe gap between the discretized lattice network (such as MERA) between the continuousAdS spacetime by cutting the latter spacetime into lattices by inserting various boundarysurfaces of AdS/BCFT. This idea opens up a new approach to the conjectured connectionbetween the AdS/CFT and tensor networks.Next we discuss how to realize a gravity dual whose time slice is given by the geometryin Fig.28. One way is to perform an Euclidean path-integration on the manifold givenby the left picture in Fig.28 with z interpreted as the Euclidean time − τ and considerits gravity dual. Note that the initial state z = − τ = ∞ is expected to be a completely For another connection between BCFTs and tensor networks refer to [73]. x
4p 2p p 4q 2q z x
4p 2p p 4q 2q (cid:563) A A A
Figure 28: A sketch of the time slice of our proposed AdS / BCFT setup dual to a MERA-like tensor network (left two pictures), and the estimation of EE S A in the AdS/BCFTand tensor network (right two pictures). The blue region in each picture is the bulk timeslice and we added its tensor network description in its next right. The brown thick linesare the slits created by the boundary surface Q in the AdS/BCFT. The red lines and greentriangles in the tensor network describe the disentanglers and the coarse-graining tensors.The red and green dotted lines are the geodesics whose length compute the holographicentanglement entropy, corresponding to the entanglement removed by the disentanglersor coarse-graining tensors. We ignore the back-reactions just for simplicity and choosethe locations of the slits such that the values of the EE for each red/green dotted linesare the same, which leads to the scale invariance. By comparing the right two pictures,we find that the calculation of HEE in our AdS/BCFT setup is equivalent to that for thetensor network.disentangled state. Therefore we choose the state at τ = −∞ to be the (regularized)boundary state | B (cid:105) [74, 75]. Indeed, we expect that the time slice τ = 0 of such aEuclidean gravity dual is given by what we want, according to what we learn from theholographic local quenches summarized in Fig.27.Another idea is based on a Lorentzian path-integration in the dual CFT. Now weregard z as the real time − t in Fig.28 and again choose the initial state at t = −∞ to bethe boundary state | B (cid:105) . Since each end points of the slits moves toward the horizon atthe speed of light, we again expect that the gravity dual of this time-dependent setup isapproximately given by the one in Fig.28. In this paper, we studied the time evolutions of entanglement entropy (EE) under threetypes of local quenches (local operator, splitting and joining) in two dimensional CFTs.Our main examples of CFTs are the free massless Dirac fermion CFT and holographicCFTs in two dimensions. The typical behavior of EE for each of the three types oflocal quenches in holographic CFTs is the logarithmic growth for a large subsystem. Onthe other hand, for operator local quenches in free or RCFTs, we observe simple step47unctional behaviors. However, for splitting and joining local quenches, we still havelogarithmic time evolutions even for free or RCFTs. We also noted that there are twotypes of logarithmic growth: ( c/
6) log( t/α ) and ( c/
6) log( t/(cid:15) ), where α is a regularizationparameter of the local quench and (cid:15) is the lattice spacing of the CFT. In this paper, wegot systematic understandings of these differences from various points of view, includingentanglement density (ED), holographic geometry, and tensor network descriptions. Wealso calculated evolution of EE in spin systems for splitting quenches.In the holographic CFT case, the EE is computed as the length of geodesics in thedual AdS spacetime based on the holographic entanglement entropy (HEE). Furthermore,in the splitting/joining quench case, these geodesics can end on the boundary surfaces Q , following the AdS/BCFT prescription. Therefore there are two kinds of geodesics:connected and disconnected. Thus the HEE is given by min { S conA , S disA } . One of them, S disA , depends on the tension parameter T BCF T in AdS/BCFT.In the splitting and joining quench, both gravity duals share the same metric but havedifferent boundary surfaces Q . In order to see the logarithmic growth of HEE, we haveto consider a region which is not covered by the Poincar´e patch because the geodesicpenetrates the Poincar´e horizon. We found that the growth ( c/
6) log( t/(cid:15) ) comes from thegeodesic length in the Poincar´e patch, while the other one ( c/
6) log( t/α ) comes from thatnear the surface Q , which is hidden inside the Poincar´e horizon.In the splitting quench, the boundary surface Q expands from the AdS boundarytoward the bulk at the speed of light. We can regard the two semi-infinite lines at eachtime are connected through an expanding horizon or equally (non-traversal) wormhole.Note that since Q is time-like, there is no causal influence between them. In the joiningquench, the boundary surface Q can be regarded as a “falling string” towards the horizon,which is analogous to the falling particle in the holographic local operator quench. In bothexamples, the precise location of Q depends on the tension T BCF T in a way that when T BCF T > T BCF T < c/
6) log( t/α ) and ( c/
6) log( t/(cid:15) ) cannot be easily distinguished fromthis perspective.The gravity duals of local quenches also have a qualitative interpretation in termsof MERA-like tensor networks, where ∼ log( t/(cid:15) ) behavior can be understood as split-ting/joining disentanglers. However, we could not find a clear realization of the ∼ log( t/α )behavior, called “shock wave” contribution. One possibility is to explain this by insert-ing extra tensors into the networks and we would like to leave more details for a futureproblem.In the final part of this paper, we presented an analogue of quantum circuits using bothof splitting and joining quenches and gave their gravity duals based on the AdS/BCFT.48his offers a method of discretizing gravity dual spacetimes. It is expected that this canfurther lead to a deeper understanding of the conjectured AdS/tensor network correspon-dence. Acknowledgements
We are grateful to Pawel Caputa, Masahiro Nozaki, Tokiro Numasawa and Tomonori Uga-jin for useful comments. TT is supported by the Simons Foundation through the “It fromQubit” collaboration and by World Premier International Research Center Initiative (WPIInitiative) from the Japan Ministry of Education, Culture, Sports, Science and Technology(MEXT). TT is supported by JSPS Grant-in-Aid for Scientific Research (A) No.16H02182and by JSPS Grant-in-Aid for Challenging Research (Exploratory) 18K18766.
A Vacuum Entanglement Density in Massless DiracFermion CFT
Here we would like to show that the simple profile of entanglement density (we set c = 1in (3.3)) n ( x, y ) = 16( x − y ) , (A.1)reproduces arbitrary entanglement entropy in the massless Dirac fermion CFT, includingcases where the subsystem A consists of multiple disconnected intervals: A = I ∪ I ∪ · · · ∪ I n . (A.2)We choose the n intervals are parameterized by I i = [ a i , b i ] for i = 1 , , · · · , n . Forconvenience, we introduce b = −∞ and a n +1 = ∞ . Then, by summing all bipartiteentanglement, using the entanglement density (A.1), S A is estimated as follows: S A = n (cid:88) k =0 n (cid:88) l =1 (cid:90) a k +1 b k dx (cid:90) b l a l dy x − y ) , = 16 n (cid:88) k =0 n (cid:88) l =1 (cid:18) log (cid:18) a k +1 − b l a k +1 − a l (cid:19) + log (cid:18) b k − a l b k − b l (cid:19)(cid:19) , = 13 n (cid:88) i,j =1 log | b i − a j | − n (cid:88) i,j =1 log | a i − a j | − n (cid:88) i,j =1 log | b i − b j | . (A.3)We interpret the diagonal terms in the final expression by introducing the UV cut off (cid:15) as a i − a i = b i − b i = (cid:15) . Thus finally we obtain S A = 13 n (cid:88) i,j =1 log | b i − a j | (cid:15) − n (cid:88) i>j =1 log | a i − a j | (cid:15) − n (cid:88) i>j =1 log | b i − b j | (cid:15) . (A.4)49his indeed agrees with the known expression of entanglement entropy in the masslessDirac fermion CFT [7]. A closely related property is that the multi partite mutual infor-mation is vanishing [76]. B Detailed Computations of Evolutions of HolographicEntanglement Entropy
Here we present the details of calculations of HEE under splitting/joining local quenches.We define the Euclidean coordinate ( τ, x ) and complex coordinate ( w, ¯ w ) as w = x + iτ, ¯ w = x − iτ. (B.1)Both splitting/joining quench setup are described by almost the same conformal maps:(splitting) f + ( w ) = i (cid:112) ζ ( w ) arg( ζ ) ∈ ( − π, π ] (B.2)(joining) f − ( w ) = i (cid:112) − ζ ( w ) arg( − ζ ) ∈ ( − π, π ] . (B.3)Here we define ζ ( w ) by ζ ( w ) = w + iαw − iα . (B.4)The HEE for the connected/disconnected geodesics S conA, ± , S disA, ± for the subsystem A (=[ w a , w b ]) is computed by (when the tension T BCF T is vanishing): S conA, ± = c
12 log (cid:18) | f ± ( w a ) − f ± ( w b ) | (cid:15) | f (cid:48)± ( w a ) | | f (cid:48)± ( w b ) | (cid:19) , (B.5) S disA, ± = c
12 log (cid:18) f ± ( w a )) (Im f ± ( w b )) (cid:15) | f (cid:48)± ( w a ) | | f (cid:48)± ( w b ) | (cid:19) . (B.6)The plus and minus subscript in S A, ± corresponds to splitting and joining quench, respec-tively.When computing these quantities, it will be helpful to use the results below. ζ = w + iαw − iα = x + i ( τ + α ) x + i ( τ − α ) = x + τ − α + i (2 xα ) x + ( τ − α ) (B.7) | ζ | = (cid:112) ( x + τ − α ) + (2 xα ) x + ( τ − α ) = (cid:115) x + ( τ + α ) x + ( τ − α ) (B.8) d √± ζdw = − iαw + α (cid:112) ± ζ (B.9) | w + α | = (cid:112) ( x + ( τ + α ) )( x + ( τ − α ) ) (B.10)= (cid:112) ( x + τ − α ) + (2 xα ) (B.11) (cid:12)(cid:12)(cid:12)(cid:12) d √± ζdw (cid:12)(cid:12)(cid:12)(cid:12) = α ( x + ( τ − α ) ) (cid:112) ( x + τ − α ) + (2 xα ) (B.12)50or simplicity we define R, A ± as R = (cid:112) ( x + τ − α ) + (2 xα ) (B.13) A ± = (cid:112) ( x + τ − α ) + (2 xα ) ± ( x + τ − α ) . (B.14)And we get more results, (cid:112) ζ = √ A + + i √ A − (cid:112) x + ( τ − α ) ) ( x > √ A + − i √ A − (cid:112) x + ( τ − α ) ) ( x < . (B.15) (cid:112) − ζ = − i √ ζ ( x > i √ ζ ( x < . (B.16)Im( i (cid:112) ± ζ ) = (cid:115) A ± x + ( τ − α ) ) (B.17)(Im( i (cid:112) ± ζ )) = (cid:112) ( x + τ − α ) + (2 xα ) ± ( x + τ − α )2( x + ( τ − α ) ) (B.18)Also we can compute | f ( w a ) − f ( w b ) | in the connected EE. If a, b > | (cid:112) ± ζ a − (cid:112) ± ζ b | = (cid:12)(cid:12)(cid:12) ( √ A + a + i √ A − a ) (cid:112) b + ( τ − α ) − ( √ A + b + i √ A − b ) (cid:112) a + ( τ − α ) (cid:12)(cid:12)(cid:12) a + ( τ − α ) ) ( b + ( τ − α ) ) (B.19)If a < < b , | (cid:112) ± ζ a − (cid:112) ± ζ b | = (cid:12)(cid:12)(cid:12) ( √ A + a − i √ A − a ) (cid:112) b + ( τ − α ) ∓ ( √ A + b + i √ A − b ) (cid:112) a + ( τ − α ) (cid:12)(cid:12)(cid:12) a + ( τ − α ) ) ( b + ( τ − α ) ) . (B.20)Here we write ζ ( x, τ ) = ζ x , A ± ( x, τ ) = A ± x .Now we can derive Euclidean HEE for the connected/disconnected geodesics, with R ( x, τ ) = R x , S conA, ± = c
12 log R a R b ( Q − ( √ A + a A + b + √ A − a A − b )) α (cid:15) ( a, b > R a R b ( Q ∓ ( √ A + a A + b − √ A − a A − b )) α (cid:15) ( a < < b ) (B.21) Q = (cid:112) ( b + ( τ − α ) )( a + ( τ + α ) ) + ( a ↔ b ) (B.22)= √ (cid:112) ( a + τ + α )( b + τ + α ) − (2 τ α ) + R a R b (B.23) S disA, ± = c
12 log 4 R a R b A ± a A ± b α (cid:15) (B.24)51e do analytic continuation to the real time τ → it , and then we approximate x, t, | x − t | (cid:29) α . Under this approximation we get R x ∼ | x − t | + x + t | x − t | α (B.25) A ± x ∼ (cid:0) | x − t | ± ( x − t ) (cid:1) + (cid:18) x + t | x − t | ∓ (cid:19) α (B.26)Consequently we derive the Lorentzian HEE formula. Here we write the results onlyin splitting quench case (i.e. S A, + case), S disA, + ∼ c a − t )( b − t ) α (cid:15) ( t < | a | < | b | )4 | a | ( b − t ) α(cid:15) ( | a | < t < | b | )4 | a || b | (cid:15) ( | a | < | b | < t ) (B.27)And for connected HEE we have to consider the sign of a . 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