R\'enyi entropy of locally excited states with thermal and boundary effect in 2D CFTs

We study R\'enyi entropy of locally excited states with considering the thermal and boundary effects respectively in two dimensional conformal field theories (CFTs). Firstly we consider locally excited states obtained by acting primary operators on a thermal state in low temperature limit. The R\'enyi entropy is summation of contribution from thermal effect and local excitation. Secondly, we mainly study the R\'enyi entropy of locally excited states in 2D CFT with a boundary. We show that the evolution of R\'enyi entropy does not depend on the choice of boundary conditions and boundary will change the time evolution of R\'enyi entropy. Moreover, in 2D rational CFTs with a boundary, we show that the R\'enyi entropy always coincides with the log of quantum dimension of the primary operator during some periods of the evolution. We make use of a quasi-particle picture to understand this phenomenon. In terms of quasi-particle interpretation, the boundary behaves as an infinite potential barrier which reflects any energy moving towards the boundary.


Introduction
Many kinds of observables can be defined in Quantum field theories (QFTs). When we study global or non-local structures, entanglement entropy (EE) or the entanglement Rényi entropy (RE) are very helpful quantities. For a subsystem A, both of them are defined as a function of the reduced density matrix ρ A . The reduced density matrix ρ A can be defined from the original density matrix ρ by tracing out the subsystem B which is the complementary of A.
One may be curious about weather there is a kind of topological contribution in entanglement entropy even for gapless theories, particularly for conformal field theories (CFTs). For example, topological properties can be qualified by computing topological contributions in entanglement entropy called topological entanglement entropy [1]. In this paper, we focus on extracting such kind of topological quantity from both Rényi entropy and von-Neumann entropy of locally excited states in two dimensional rational CFTs with thermal effect and boundary effect. An earlier work [2] about a relationship between the topological entanglement entropy and boundary entropy. The relationship between the boundary entropy and entanglement entropy have also been explored [3].
The n-th Rényi entanglement entropy is defined by S (n) A = log Tr[ρ n A ]/(1 − n) formally. The limit n → 1 coincides with the von-Neumann entropy. This is standard replica trick method to calculate the entanglement entropy. The difference of S (n) A between the locally excited states and the ground states with introducing thermal and boundary effects are main interest in this paper. The difference is denoted by ∆S (n) A . Replica approach calculations of ∆S (n) A for states excited by local operators have been given in [4,5,7]. We will closely follow the construction in [5] [6] [7]with introducing thermal effect and boundary effect.
We would like to review the thermal effect and boundary effect respectively. There are many studies about thermal effect in 2D CFTs. For thermal states, EE is not a good measurable quantity, which is contaminated by the thermal entropy of the subregion. In high temperature limit, the EE will be dominated by thermal entropy. To reveal the quantum entanglement of system with thermal effect, one should identify the thermal contribution and other contributions of EE. [8]conjectured universal form of correction of EE in any quantum system with mass gap. [9] provided the form of the coefficient of such correction in 2D CFT. To generalize studies in [9] to higher dimensions, [10] [11] considered thermal corrections to the entanglement entropy on spheres. On the other hand, the dynamics in 2D CFTs with a boundary have many new features comparing with 2D CFTs in full complex plane. The original works have been done by Cardy, who discussed surface critical behavior of correlation functions [12]. [13] studied the constraints on the operator content by imposing by boundary conditions and also the classification of boundary states in terms of the modular transformation. In [14] [15], the concept of boundary operators have been introduced. In [16], an additional boundary condition was discovered in the non-diagonal minimal model. [17] showed that the resulting set of boundary conditions to be complete. There are also nice correspondence called as AdS/BCFT proposed by [18] [19]. The boundary effect can be also studied holographically, which is beyond the issue considered in this paper.
In 2D rational CFTs, the authors in [6] get an amazing result for the locally excited states, which relates the Rényi entropy to the quantum dimension of the primary operator which is kind of topological quantity. In this paper, we generalize the previous study [6] on Rényi entropy with thermal and boundary effects. Firstly, there is a simple sum rule between the thermal correction and the local excitation in low temperature limit. That is to say the total Rényi entropy are summing over Rényi entropy of local excitation and the one of thermal excitation in low temperature limit. Such kind of relation is similar to the sum rule related to the Rényi entropy in [7]. It is easy to generalize the result to local excitation in pure state in 2D CFT. We make use of a different approach [8] to obtain the thermal correction to Rényi entropy which can be reduced to [20]. Secondly, we investigate the the Rényi entropy for states excited by local primary operators in the rational CFTs with a boundary. These boundaries introduced here do not break the conformal symmetry. Such theories are called as BCFTs. We show the time evolution of the Rényi entropy in 2D free field theory and Ising model. Then we generalize to rational CFTs with a boundary. The boundary changes the time evolution of the Rényi entropy, but does not change the maximal value of the Rényi entropy. All these cases studied in this paper show that the Rényi entropy does not depend on the choice of boundary conditions. In 2D rational CFTs with a boundary, we also show that the maximal value of the Rényi entropy always coincides with the log of quantum dimension of the primary operator during some periods of the evolution. We give the physical understanding of boundary effect, which support the quasi-particles explanation of the local excitation. The boundary behaves as an infinite potential barrier which reflects the quasi-particle moving towards it.
The layout of this paper is as follows. In section 2, we study the thermal effect on the Rényi entropy of the local excited state in low temperature limit. In section 3, we set up the local excitation in 2D CFT with a boundary and obtain the Rényi entropy of a subsystem with time evolution. We study the 2D free scalar, and Ising model as examples, then generalize the result to 2D rational CFT. In section 4, we devote to the conclusion and physical interpretations of such kinds of effects shown in this paper.

Local excitation in non-vacuum states
In this section, we would like to study the local excitation of thermal state. We consider a system with temperature T = 1/β and assume the excitation is local at x = −L by primary operator O shown in fig. [1]. In this section, we just only consider the low temperature case with large β [10]. The subsystem A is −l < x < 0. The density matrix ρ(t) is where we have considered the real time evolution, ǫ is the ultraviolet regularization, N (t) is fixed by normalization condition trρ(t) = 1, E n are the energy of the excited states. The complex coordinates in ω plane are listed as follows.
To simplify our analysis, we only consider the system with low temperature. For later Figure 1. This figure is to show our setup in two dimensional complex plane ω = x + it. The system will be triggered at x = −L and there are left-and right-moving quasi-particle at t = 0. This setup is same as the one in 2D CFTs without boundary [6].
convenience, we define which can be taken as the reduced density matrix related to the vacuum and first excited state respectively, where we normalize the vacuum energy to be zero, B is the complementary part of subsystem A. In the low temperature expansion with βE 1 ≪ 1 In terms of the definition of the Rényi entropy One could check that when there is no local excitation, i.e., the operator O = I, the result is the same as [9]. The second and third terms of the last line in (2.5) involve in the coupling between the local excitation and thermal environment. In terms of the state operator correspondence, one can denote |1 = lim t→−∞ ψ(x, t) |0 [9]for the excited state with energy E 1 . Using the path integral language, (2.5) is where C n is the n-sheet cylinder with circumference Λ and O(ω 1 ,ω 1 )...O(ω 2n ,ω 2n ) are the operators that are inserted in the suitable place in the n-sheet cylinder. Here we consider the whole system has an infrared cut-off Λ, and l/Λ ≪ 1. The first term (2.6) is given by [6] for the local excitation in vacuum. We will study the second and third term of (2.6) in detail.
The following transformation can map the n-sheet cylinder to a cylinder with circumference Λ, (2.7) The points ω = −∞ and ω = ∞ are mapping to z −∞ = e −2iπl/Λ and z +∞ = 1 respectively. For simplifying our analysis, we only consider the range |ω| ≪ Λ. Otherwise, the following calculation (2.9) will be much more complicated. The final statement does not change without this approximation. Thus (2.7) reduces to which is same as the one that is used in [6]. The points z 1 , ..., z 2n are given by (2.9) In t > l + L or 0 < t < L, In L < t < L + l, In t > l + L or 0 < t < L the second term of (2.6) is In the above formula, we label ψ ′ (z,z) as the map of the operator ψ(ω,ω) 1 . In the second step due to (2.10) we have used (2.13) 1 The operator may not be a primary operator , such as the energy-momentum tensor T . A Schwarzian derivative term related to energy momentum tensor operator [9][21] will present due to conformal transformation.
In L < t < L + l, the correlation function can not be factorized as (2.13) directly due to (2.11). Following logic in [6], one can make use of n − 1 times fusion transformation (z 1 , z 2 )(z 3 , z 4 )...(z 2n−1 , z 2n ) → (z 2 , z 3 )(z 4 , z 5 )...(z 2n , z 1 ). The second terms of (2.6) is still given by in ǫ → 0 (2.14) Here we assume that there are no nontrival correlation between O and ψ for simplifying analysis. The third term of (2.6) in the limit ǫ → 0 is The sum of the second and third term is the same as the thermal correlation [9] for the short interval limit. (2.6) is the summation over the thermal correction of the vacuum state and local excitation in the vacuum state in low temperature limit. The Rényi entropy is summation of thermal effect and local excitation. This relationship can be considered as a sum rule for Rényi entropy of low temperature thermal state with local excitation, which is different with the sum rule proposed in [5]. For the pure state one could also get a similar sum rule. The local excitation in thermal states have been also studied in [20] for free boson and Ising model. With a different method [9], we can reproduce the Rényi entropy [20] for short interval in the low temperature with taking limit ǫ → 0 for general 2D rational CFT.

Local excitation in 2D CFTs with a boundary
In this section, we would like to study the Rényi entropy of locally excited state in 2D CFT with a boundary which preserves conformal symmetry. The global property of the CFT with a boundary has been discussed in literature [22][23] [24]. As we know, the Rényi entropy is sensitive to correlation function. The boundary will change the correlation functions. And the boundary conditions also effect the correlation functions. It is interesting to check what will happen to the Rényi entropy for the local excitation in 2D CFT with a boundary.

Set-up of local exciation
We begin with a CFT with a boundary at x = 0 and the CFT is living in the range x ≤ 0. We divide the this region into two parts, one part is −l < x < 0 denoted by A and the other is complement to the region A denoted by B. The time t vary from −∞ to +∞, and the Hamiltonian H is well defined as a operator to generate the time evolution. We assume that the local excitation of vacuum is at x = −L and consider the Rényi entropy of the subsystem A. The time dependent density matrix can be written as where the coordinates are, This figure is to show our setup in two dimensional left half plane ω = x + it with a boundary x = 0. The system will be triggered at x = −L and there are left-and right-moving quasi-particle at t = 0.
We still make use of replica trick to study the variation of the Rényi entropy of the subsystem A in this section. By definition, the variance of n-th Rényi entropy can be calculated as where B n is the n-sheet Riemann surface that consists of n copies of original plane x ≤ 0 with gluing together along −l x 0, t = 0. With the following conformal transformation, the n-sheet Riemann surface can be mapped to a disc |z| 1 which is smooth surface. The boundary x=0 corresponds to |z| = 1. Furthermore, we can map disc to the upper half plane (UHP) t 0 with the other conformal map After two conformal maps, one can make use of well known results of 2D CFT on UHP. Finally, the variation of the Rényi entropy (3.3) is This is the key formula in the remain context of this paper. As a simple example, we consider the 2nd Rényi entropy firstly. These coordinates on UHP can be expressed by the original spacetime coordinate as follows.
One can analyze the above formula carefully in two different time regions. When 0 < t < L − l or t > l + L with ǫ → 0, and the Jacobi factor of conformal transformation, When L − l < t < L + l with ǫ → 0, 12) and the Jacobi factor of conformal transformation,

2nd Rényi entropy for free boson
We will concern the following local operators in the free scalar field firstly, The time evolution of Rényi entropy for such operators have already studied in [6] in 2D CFT on the complex plane. There are two kinds of boundary conditions for 2D free scalar field theory. One is ∂φ ∂n | B = 0 called Neumann boundary condition and the other is φ| B = 0 called Dirichlet boundary condition. Since the boundary condition is homogenous, it is invariant under the conformal transformation. The image method [25] 2 is an efficient way to obtain the correlation function on UHP from correlation function on the full complex plane. The two kinds of boundary conditions correspond to different parity transformation in image method. Due to the present of boundary, there are constraints on local conformal transformation, the anti-holomorphic and the holomorphic sectors in correlation function are no longer independent. More precisely, the correlation function on the upper half plane can be expressed by holomorphic part of conformal block on the full complex plane with including the 'images' of the holomorphic coordinates. That is to say, where φ andφ refer to the holomorphic and anti-holomorphic part of the field φ 3 . After a parity transformation the anti-holomorphic part become a holomorphic field with conformal dimension of the original anti-holomorphic part. In terms of image method, we should introduce parity transformation. For the free boson the parity transformation is η = 1, −1 corresponds to the Neumann boundary condition and Dirichlet boundary condition respectively. In terms of (3.6), to obtain Rényi entropy, it is necessary to know the two-point and four-point correlation function on the UHP.

Local excitation O 1
Let's consider the operator O 1 firstly. Using the image method, we could get the two-point function is the field with parity transformation. The subindex 'holo' means that we only keep the holomorphic part of the correlation and set the anti-holomorphic part to be constant 4 which is determined by boundary condition in general. For two-point function one could normalize the field and take constant to be 1. The 4-point correlation function could be obtained by similar procedure.
In the region 0 < t < L − l or t > l + L, as (3.8) shows, the correlation function (3.20) can be factorized as in ǫ → 0 limit. The image method leaves us with a constant C. To fix the constant C, we take the limit In terms of image method, we can also obtain where C is constant. Comparing (3.22) with (3.23), we can fix C = 1 which is consistent with the normalization of the two point correlation function. For n = 2, the variation of the Rényi entropy (3.6) is where we have use (3.19), (3.20), (3.23) and Jacobi factor (3.9)(3.10).
In the other region L − l < t < L + l, we can not factorize the correlation function as (3.21) directly. For 2D free scalar theory, the situation becomes much simpler. The correlation function could be expressed as [25] with the neutral condition α 1 + α 2 + ...α n = 0 and z ij = z i − z j ,z ij =z i −z j . Thus where we have used fact C = 1. To get ∆S A we only need to change 2 ↔ 4 in (3.24). Using (3.12)(3.19)(3.26) and Jacobi factor (3.13)(3.14), we get ∆S (2) (3.27) in L − l < t < L + l. To close this subsection, one more thing should be noted that ∆S does not depend on the choice of parity transformation. Actually in our above calculation we do not use the value of η, which is related to the boundary condition. The leading order of the correlation function is same. ∆S A is always zero for operator O 1 . To see the effect of the boundary, we should consider more complicated example.

Local excitation O 2
O 2 is a linear combination between O 1 and O † 1 . We can calculate ∆S (2) A with following the logic in previous section. The two-point correlation function of O 2 , We could use the result of four-point function of O 1 which has been studied in 3.2.1. From (3.18)and (3.17) one could see that the four-point function is dependent on the boundary conditions or parity transformations. We will calculate ∆S (2) A with Neumann and Dirichlet boundary condition respectively.
In (3.29), terms containing equal number of O 1 and O † 1 will survive due to the neutrality condition. Thus there are 6 terms making contribution to the 4-point correlation function. In the region 0 < t < L − l or t > L + l, (3.29) can be expressed by factorized form on the R 2 . For example, the first term in (3.29) as a leading term is , (3.30) in the limit ǫ → 0. The second terms as sub-leading term is In L − l < t < L + l with ǫ → 0, (3.12) shows ξ 14 ∼ ξ 23 ∼ ǫ. Terms making non-vanishing contribution to 4-point correlation function will change in the limit ǫ → 0. For example the third term in (3.29) as a sub-leading term is, where '...' stands for the terms that are same order in the limit ǫ → 0. One could count the leading contribution term by term in (3.29), there are only two terms that are of O(ǫ −1 ). Thus the variation of the Rényi entropy ∆S 2. For the Dirichlet boundary condition, i.e., η = −1. All the terms in (3.29) are non-vanishing in term of the neutrality condition. However, there are still four different terms in (3.29) contributing to the leading order in the region L + l < t or t < L − l in ǫ → 0 limit. Thus the ratio (3.6) is 1 and ∆S (2) A = 0. In the region L − l < t < L + l, there are two terms in (3.29) with ǫ → 0 limit, which is the same as the situation of Neumann boundary condition. Thus the ratio (3.6) is 1 2 and ∆S (2) A = log 2. In this example we could see that ∆S (2) A does not depend on the choice of boundary conditions.

n-th Rényi Entropy for free boson
In this subsection, we would like to generalize our studies to n-th Rényi entropy which involves in the the 2n-point correlation function on B n . The conformal transformation (3.4) (3.5) can map B n to UHP. Finally one can calculate the 2n-point correlation function on UHP by the 'method of images' in terms of 4n-point correlation function on R 2 . The points ξ 1 , ξ 2 ...ξ 2n on B n are where 0 ≤ k ≤ n − 1. In the region t > L + l or t < L − l, one can obtain and the Jacobi factor of conformal transformation, In L − l < t < L + l, one could find . (3.39) and the Jacobi factor of conformal transformation, (3.41)

Local excitation O 1
We consider the operator O 1 firstly. In the region L − l < t or t > L + l with ǫ → 0, the 2n-point correlation function of O 1 is whereÕ 1 is defined as e −iηφ/2 with parity transformation and ξ 2n+1 =ξ 1 ,...,ξ 4n =ξ 2n . Using (3.6), the variation of the n-th Rényi entropy can be obtained as follows.
In the region L − l < t < L + l with ǫ → 0, the 2n-point correlation function on UHP is Where we have made use of (3.44) and Jacobi factor (3.13)(3.14).

Local excitation O 2
For operator O 2 , there are 2 2n terms making contribution to the correlation function like the ones in (3.29). Firstly let us consider the case with Neumman boundary condition. Obviously, these terms with equal number of O 1 and O † 1 can survive in the limit ǫ → 0. The 2n-point correlation function of O 1 on B n can be expressed by 4n-point correlation function on R 2 , (3.46) To be convenient, we use the symbol +1 referring to O † 1 and −1 referring to O 1 in the correlation function to simplify our analysis. Then the 2n-point correlation function on UHP can be formally written as where we have made use of the image method in the second line and i j = ±1 stands for the operator O 1 or O † 1 with the coordinate (ξ j ,ξ j ). The constraints i 2n+j = i j with 1 ≤ j ≤ 2n corresponds to the Neumann boundary condition in terms of image method. For the correlation function in the Neumann boundary condition case, the non-zero terms in (3.47) should satisfy neural condition In the region t < L − l or t > L + l with the limit ǫ → 0, due to (3.36), the leading contribution of the 2n-point correlation function (3.47) are there are 2 n terms that are leading divergence after considering the constraints, and these terms are all equal to each other. Equivalently, (3.47) can be written by the notation O 1 and O † 1 as  In L − l < t < L + l with the limit ǫ → 0, due to (3.39) the leading contribution of (3.47) should satisfy following constraints i 2n+1 + i 2n+2 = 0, i 2n+3 + i 2n+4 = 0, ..., i 4n−1 + i 4n = 0. In terms of (3.39), the leading terms of (3.47) should also satisfy following constraints With these constraints (3.53)(3.54)(3.55), the correlation function is where the factor 2 is given by constraints (3.53)(3.54)(3.55). Putting (3.56) into (3.6) with considering Jacobi factor (3.13)(3.14), the variation of the n-th Rényi entropy is In CFT with the Dirichlet boundary, the only difference with the Neumann boundary condition is the constraints i 2n+j = i j → i 2n+j = −i j . One could check that the leading order correlation function is same as the Neumann boundary condition. Thus the n-th Rényi entropy is not dependent on the choice of boundary condition. We do not repeat here.

Rényi Entropy in Ising model
It is natural to ask how about the Ising model which is simplest unitary minimal model. There are three kinds of primary operators, i.e., the identity I, the spin operator σ and the energy operator ǫ. There are also two kinds of parity transformation which involve in two kinds of boundary conditions, which correspond to two different parity transformations.
One of the parity transformation [25] is where µ is the disorder operator. The other parity transformation [25] is We would like to study the local excitation by the spin operator σ with conformal dimension (h = 1 16 ,h = 1 16 ) in the same setup given in section 3.1. The variation of the Rényi entropy for the subsystem −l < x < 0 is given by (3.6).

67)
F bc is the fusion matrix. Making the fusion is equal to ξ 2 ↔ ξ 4 . In summary, the leading divergence of the correlation function is related to c = 0 as identity as follows In terms of (3.12), one can find leading contribution of express (3.66) as following form in L − l < t < L + l with ǫ → 0 where 0 stands for the identity operator. Then the variation of the Rényi entropy for n = 2

∆S
(2) where we have used the fact F 00 = 1 √ 2 [32] in two-dimensional Ising model. Note that the Rényi entropy does not depend on the choice of boundary condition in the Ising model, since the leading behavior of F + is the same as one of F − with ǫ → 0.
One alternative way to understand the phenomenon is to make use of diagram representation of conformal block as fig. [3]. Since the behavior of coordinatesξ j does not change when L − l < t < L + l, we only need one time fusion transformation, which is different with [6]. In the fig.[3], In L − l < t < L + l with ǫ → 0, the leading contribution of (3.66) originates from the above conformal block involving in identity operator. Because the fusion factor F 00 can not be canceled in the ratio (3.6), ∆S

Rényi Entropy in General Rational CFTs
In this subsection, we would like to generalize the analysis to the rational CFTs with a boundary in our previous set-up shown in subsection 3.1. In terms of (3.6), we should know the 2-point correlation on B 1 and 2n-point correlation fuction on UHP as usual. In generic rational CFTs with a boundary, the 2-point correlation can be expressed as (3.73) In the limit z → 0 Where "..." stands for higher order terms and z = z 12 z 34 z 13 z 24 and h b is conformal dimension of primary class b.
whereξ 2n+i = ξ i , 0 ≤ i ≤ 2n. With using n − 1 times fusion transformation as the one shown in fig.[3], we get We could calculate ∆S n A by using (3.6). The variation of the n-Rényi entropy is A . As we know, the 2npoint correlation function on UHP can be expressed by linear combination of chiral part of conformal block associated with 4n-point correlation on full plane. In this subsection, we do not make use of parity transformation containing the boundary information. The boundary data has been encoded in the coefficient of conformal block in this subsection, i.e., A p in (3.73). We could see that that ∆S n A also does not depend on the choice of boundary.

Conclusion and Discussion
In this paper, we have investigated the two kinds of effects on the locally excited states with time evolution. In the first case, we study the locally excited states with thermal effect in low temperature system. We have figured out the thermal correlation which is the same as [9] for the short interval limit. The Rényi entropy is equal to summation over the logarithmic of quantum dimension and thermal entropy in low temperature. In this paper, we just only confirm that such kind of sum rule is only true for the short interval l in the low temperature limit. We make use of different approach [8] to obtain the thermal correction to Rényi entropy which can be reduced to [20] in low temperature. One can also calculate the Rényi entropy in the large interval [9] limit, the higher temperature limit [33] as well as beyond the leading order of the perturbation (2.4). We expected the sum rule relation is not hold in those cases. But one should note that we actually do not consider the back-reaction of the locally excited states to the thermal environment. When the energy of the local excitation is much lower than the thermal environment, it is safe to ignore the back-reaction. But in some special situation we expect the sum rule will break down. It is an interesting topic to consider in the future.
In the second case, we have studied the Rényi entropy of local excited states in 2 dimensional CFTs with a boundary. For 2D CFTs with a boundary, to obtain Rényi entropy can be converted to obtain the correlation function on UHP by using of conformal transformation technique. As a warm up, the Rényi entropy has been calculated with help of image method in the 2D free field theory with a boundary. The Rényi entropy is vanishing for operator O 2 (3.15) in t > L + l or t < L − l and log 2 in L − l < t < l + L, which is the same as previous study [6] in full complex plane without boundary. To confirm this fact, the Rényi entropy have been calculated in Ising model and more generic rational CFTs. Although the correlation function and conformal blocks in 2D CFTs with boundary are totally different from the ones in 2D CFT without boundary, we get a same maximal value of the Rényi entropy for the rational CFTs without a boundary [6]. In generic 2D rational CFTs with a boundary, we confirm that the maximal value of Rényi entropy is the same as the one in 2D rational CFTs without boundary. [34] also try to understand the fact which is not contract with that the left-and right-moving chiral sectors are decoupled. [34] generalize the result in [6] to irrational CFT, for example, Liouvile CFT. They found that the left-right entanglement entropy saturates the Cardy entropy. In terms of standard view, the Cardy entropy counts the microscopic entropy of actual CFT spectrum. The Cardy entropy seems to suggest two chiral sectors are decoupled. The authors in [34] proposed a pragmatic point of view to reconcile [6] with the fact that there should also be a comparably large EE between the two chiral sectors of CFT. For example, Non-chiral local operators will be left-right entangled. In BCFTs, the two chiral sectors are no longer independent. We have shown some additional examples to confirm the pragmatic perspective.
For general rational CFTs in 2D, the Rényi entropy highly relys on the conformal blocks of the theory. The n-point correlation functions in 2D CFTs with boundary are related to the holomorphic part of conformal blocks of the 2n-point correlation functions on the 2D full complex plane. This relationship had been studied by the image method [23][24] very well. More precise relation is that an n-point function in the UHP, which is a function of the coordinates (z 1 , , z n ;z 1 , ...,z n ) behaves under conformal transformations in the same way as the holomorphic factor of a 2n-point function in the full plane which depends on (z 1 , ..., z n ; z * 1 , ..., z * n ), analytically continued to z * j =z j . In [6], the time evolution of Rényi entropy highly depends on the holomorphic part of conformal block. In 2D CFTs with a boundary, the boundary changes the evolution of the Rényi entropy but does not change the value of the Rényi entropy, which is closely related to fusion constants in the bulk. Because the behavior of the 'image' coordinates (anti-holomorphic coordinates) does not change as the holomorphic coordinates when L − l < t < L + l, we get the same Rényi entropy as [6].
The boundary introduced here works as the infinite potential barrier for the time evolution of the entangled quasi-particles pairs [5][6] triggered by local excitation as shown in fig.[4]. The the Rényi entropy measures the entanglement between the quasi-particles generated by local excitation. After entangled pairs are created at −L, the two quasiparticles will propagate in two opposite directions, i.e., left-moving and right moving. When the right-moving particle enter the interval −l < x < 0 denoted by A, the Rényi entropy takes maximal value due to entanglement between two entangled particles. In fig.[4], the blue wave lines mimic the interaction of two entangled quasi-particles. When the right-moving quasi-particle reach the boundary, the quasi-particle will be reflected by the boundary without losing energy. As the calculation in [35] shows the locally excited states carry the energy of O(ǫ −1 ). The conformal transformation, z → z + ǫ(z) andz →z +ǭ(z), should keep the boundary conformal invariant, which lead to the constraint T =T on the boundary. In the Cartesian coordinates, the constraint becomes T xt = 0, which means that no energy can flow across the boundary. This is main reason the quasi-particle must be reflected no matter what is the conformal invariant boundary condition. In this sense, the boundary change the time evolution of Rényi entropy.In our paper we take the scale ǫ as the minimal scale, and keep the leading order of ǫ in the calculation. So we must miss some information when t ∼ L, i.e., the quasi-particle is close to the boundary. In this case, we should make use of bulk boundary correlation functions and boundary structure constants in BCFT to figure out the time evolution of entangled quasi-particles. The next leading order calculation of ǫ may give us more insight on this point.