Entanglement Entropy for Descendent Local Operators in 2D CFTs

We mainly study the R\'enyi entropy and entanglement entropy of the states locally excited by the descendent operators in two dimensional conformal field theories (CFTs). In rational CFTs, we prove that the increase of entanglement entropy and R\'enyi entropy for a class of descendent operators, which are generated by $\cal{L}^{(-)}\bar{\cal{L}}^{(-)}$ onto the primary operator, always coincide with the logarithmic of quantum dimension of the corresponding primary operator. That means the R\'enyi entropy and entanglement entropy for these descendent operators are the same as the ones of their corresponding primary operator. For 2D rational CFTs with a boundary, we confirm that the R\'enyi entropy always coincides with the logarithmic of quantum dimension of the primary operator during some periods of the evolution. Furthermore, we consider more general descendent operators generated by $\sum_{} d_{\{n_i\}\{n_j\}}(\prod_{i} L_{-n_i}\prod_{j}{\bar L}_{-n_j})$ on the primary operator. For these operators, the entanglement entropy and R\'enyi entropy get additional corrections, as the mixing of holomorphic and anti-holomorphic Virasoro generators enhance the entanglement. Finally, we employ perturbative CFT techniques to evaluate the R\'enyi entropy of the excited operators in deformed CFT. The R\'enyi and entanglement entropies are increased, and get contributions not only from local excited operators but also from global deformation of the theory.


Introduction
The entanglement entropy (EE) and the entanglement Rényi entropy (RE) are very helpful quantities to study global or non-local structures in quantum field theories. For example, computing topological entanglement entropy can characterize the topological order of the system [1]. In [2] a relationship between the topological entanglement entropy and boundary entropy has been pointed out. And in [3] the relationship between the boundary entropy and entanglement entropy has been explored. Both the entanglement and Rényi entropes of a subsystem A are defined with respect to the reduced density matrix ρ A , which is obtained by tracing out the degrees of freedom in the complement of A from the original density matrix ρ. Formally, the n-th Rényi entanglement entropy is defined by S A coincides with the von-Neumann entropy exactly, which defines the entanglement entropy. This leads to standard replica trick to in compute the entanglement entropy in the quantum field theory.
There could be topological contribution to the entanglement entropy even for gapless theories, e.g. conformal field theories (CFTs). In 2D rational CFTs, it was found [11] that for the locally primary excited states, the Rényi entropy difference is related to the quantum dimension [4] [5] of the primary operator, which is a kind of topological quantity. The computations of the entanglement entropies for locally excited states have been formulated in [6] [7] [8] in the field theory side. The entanglement entropy for local free scalar have been investigated in [8] [9] (see also [10] for the operator approach). In the large N CFTs , the entanglement entropies for locally excited states have been discussed in [12][13][14]. In a more recent paper [15], the Rényi entropy difference ∆S (n) A of the local excited states in a boundary conformal field theory (BCFT) has been studied. In [16] the left-right entanglement entropy for a boundary state has been discussed.
In all the study of quantum entanglement of local operator, only the primary operators have been discussed. 1 It is interesting to investigate the similar effect from a local descendent operators. Among the descendants, the stress tensor is of particular interest, as it is related to the energy density in the field theory, and is dual to a graviton in the holographic dual.
In this paper, we generalize the previous study [11] [15] on the Rényi entropy difference to the descendent operators with or without a boundary condition. First of all, we compute ∆S (2) explicitly for some simple descendent operators in minimal models, and we find that in these cases the entropy difference is exactly the logarithmic of the quantum dimension of corresponding primary operator. Moreover, we study the descendant operators in a general theory. The picture is that if the local operator is of the form L (−)L(−) O, the entropy difference 1 There has been some discussion on the entanglement entropy for descendant excited states in [7].
is the same as the one for the local primary operator. This could be explained by studying the divergent terms in the correlation function. However for a more generic operator like (1. 1) we find that the Rényi entropies and entanglement entropy have additional contribution. Such additional correction originates from the entanglement in the excited states. We give a systematic recipe to calculate the entanglement entropy for general descendant operators.
We also investigate the Rényi entropy for the states excited by a local descendent operator in the rational CFTs with a boundary. The boundaries introduced here do not break the conformal symmetry. Such kinds of theories are called the boundary conformal field theories (BCFT). We find that the Rényi entropies excited by the descendants in BCFTs are similar to the one discussed in [11] [15]. If the descendent states are generated by the operator (1.1), the boundary does change the time evolution of the Rényi entropy but it does not change the maximal value of the Rényi entropy. Finally, we investigate the Rényi entropies in a deformed CFT perturbatively. If the deformation can be seen as a perturbation, the Rényi entropies of locally excited states are just the summation of contributions from the local excitation and the one from global deformation.
The layout of this paper is as follows. In section 2, we would like to study the Rényi entropy of the local descendent state with or without a boundary. In section 3, we move to study the Rényi entropy of the local excited states in 2D CFT with deformation by chemical potentials. In these deformed theories, we obtain the Rényi entropy of a subsystem with time evolution. In section 4, we devote to the conclusions and discussions. In appendix, we list some technics which are very useful in our analysis.
While proceeding with this project, we noticed a recent papers [34] appearing in arXiv, which has overlap with our discussion in the sections 2.3 and 2.4.
2 Rényi entropy of descendent operator

Setup in 2D CFT
Consider an excited state which is defined by acting a primary or a descendent operator V a on the vacuum |0 in a two dimensional CFT. We make use of the Euclidean formulation and introduce the complex coordinate (w,w) = (x + iτ, x − iτ ) on R 2 such that τ and x are the Euclidean time and the space respectively. We insert the operator V a at x = −l < 0 and consider its real time-evolution from time 0 to t under the Hamiltonian H. The corresponding density matrix can be expressed as following: where N is fixed by requiring Trρ(t) = 1, and The infinitesimal positive parameter ǫ is an ultraviolet regularization factor. We treat ǫ ± it as purely imaginary numbers until the end of the calculations, as in [8,9,17].
To calculate ∆S (n) A , we can apply the replica method in the path-integral formalism by generalizing the formulation for the ground state [3] to our excited states [8]. It leads to a n-sheeted Riemann surface Σ n with 2n operators V a inserted. We choose the subsystem A to be an interval 0 ≤ x ≤ L at τ = 0. In section 2.2, 2.3, we would like to take L → ∞ as a warm up. For later parts, we will consider the situation with finite L.

Finally, the ∆S (n)
A can be computed as where (w 2k+1 , w 2k+2 ) for k = 1, 2, ..., n − 1 are n − 1 replicas of (w 1 , w 2 ) in the k-th sheet of Σ n . The term in the first line is given by a 2n-point correlation function on Σ n , while the one in the second line is given by n two-point functions on Σ 1 . Here ∆ a is the (chiral and anti-chiral) conformal dimension of the operator O a .

Convention
Let us study the second Rényi entanglement entropy in details to set up the convention. The calculations of ∆S (2) A is related to the four-point functions which we know pretty well for exactly solvable CFTs.
To compute ∆S (2) A we need to introduce the coordinate z i , which is related to w i by the conformal map w i = z 2 i . First the coordinates w i and z i behave like (2.5) Figure 1: This figure is to show our setup in two dimensional plane ω = x + it . The system will be triggered at x = −l and there are left-and right-moving quasi-particles at t = 0.
Thus we find The most important point is that we should treat ±iǫ + t as a pure imaginary number in all algebraic calculations and take t to be real only in the final expression of the entropy. This is a standard prescription to make analytical continuation of Euclidean theory into its Lorentzian version. Therefore we should identify (r cos θ 1 , r sin θ 1 ) = (−l, ǫ − it), (s cos θ 2 , s sin θ 2 ) = (−l, −ǫ − it).
This leads to To evaluate the four point-correlation functions, it is useful to focus on the ratio where z ij = z i − z j . Note that there is a useful relation (2.10) We are interested in the two limits (i) l >> t >> ǫ (early time) and (ii) t >> l >> ǫ (late time). From (2.9) we find that they correspond to Note that the second limit is non-trivial in that it does not respect the complex conjugate relation because of our analytical continuation of t.

Second Renyi entanglement entropy in terms of 4-pt functions
Let us first consider the Rényi entropy of the simplest descendent operator L −1 O, where O is a primary field. The four-point function on n-sheeted Riemann surface Σ n can be mapped to the one on R 2 by the conformal map w = z n . For the second Rényi entropy, we find where ∆ is the chiral conformal dimension of the primary operator O, and Since N i and M i do not contribute to the divergent terms in both the late time limit and the early time limit, we can take them as a factor to simplify our analysis. In (2.12), the last term in the summation diverge fastest in the small ǫ limit, so we only keep this term in the later calculation. Note also that rs behaves in the two limits as (i)Early time : rs ≃ l 2 , (ii)Late time : rs ≃ t 2 . (2.14) Due to the conformal symmetry, the four-point function on R 2 can be expressed as where (z,z) are given by (2.9).
The two-point function is 16) where N is the normalization of the primary operator O. Note that the four-point function is proportional to N 2 and the ∆S (2) A is of course independent of N . In the late time limit (ii), we finally find that the ratio Z 2 /(Z 1 ) 2 in (2.4) is expressed in terms of the four-point function on R 2 : (2.17)

Example I: (2, 1) operator in minimal model
Now we would like to study an explicit examples: the (2, 1) operator in the minimal models.
Let us consider a (p, p ′ ) minimal model with p > p ′ , where the primary fields φ (r 1 ,r 2 ) are specified by a pair of integers (r 1 , r 2 ) taking the values The central charge of (p, p ′ ) minimal model is 19) and the conformal dimension of the (r 1 , r 2 ) primary operator is given by Let us focus on the (2, 1) operator φ (2,1) in a (p, p ′ ) minimal model as it has a relatively simple four-point function. It has conformal dimension The function G(z,z) for the four-point function φ 2,1 (z 1 ,z 1 )...φ 2,1 (z 4 ,z 4 ) Σ 1 via the relation (2.15) is known to be [18] [19] where we have defined [20] [21] being the hypergeometric function. By taking the limit z 12 = z 34 → 0, we find that Thus we can identify the normalization factor N of the two-point function as follows Now we turn to the most interesting limit: late time limit t >> l >> ǫ, where we have Under this limit we can put (2.22) into the final formula (2.17). Notice that the I 2 term is more divergent in the small ǫ limit, and using the identities on the hypergeometric functions, we have In this way, we finally obtain 28) and the Renyi entanglement entropy difference For a general minimal model, the entropy difference is not vanishing for the descendent operator In a minimal model, the modular transformation maps the primary operator (s 1 , s 2 ) into (r 1 , r 2 ). Its S-matrix is defined to be S (r 1 ,r 2 ),(s 1 ,s 2 ) , given explicitly by The quantum dimension d (r 1 ,r 2 ) for the primary field (r 1 , r 2 ) is defined by Thus we can conclude that for the L −1 O(2, 1) operator the same as the one for the primary operator O(2, 1).

Example II: energy momentum tensor
Another simple case is the excitation of energy momentum tenor, which is the descendent state of the identity operator. Different from the case discussed above, it is L −2 rather than L −1 .
We only consider ∆S 2 here. The two-, three-and four-point correlation functions on the R 2 depend only on the central charge c [22], which are respectively (2.33) is the ratio. The transformation of the energy momentum tensor under the map z(ω) is given by with the Schwarzian derivative In the coordinate (2.5) the leading order of the two-point function is (2.38) After the transformation the four-point correlation function in the ω coordinate is where '...' denotes the less divergent terms, such as the product of three-point function and the Schwarzian derivative. During the time 0 < t < l or t > L + l, we can see also z ∼ O(ǫ 2 ). One could see from (2.34) the leading contribution of the four-point function is .
During the time l < t < L + l, (2.42) Now the leading order contribution of the four-point correlation function in the limit ǫ → 0 is (2.43) One could check (2.40) is the same as (2.43). Therefore, by using (2.4), we find ∆S (2) = 0 during the evolution. This is just as we expected, since T (z) is the descendent operator of identity I, the quantum dimension of which is 1.

2nd Rényi entropy for some descendant operators
In previous section, we computed the Rényi entropy of some specific descendent operators and found that these descendant operators had the same contribution as the primary operators.
In this subsection, we use the conformal block and operator product expansion (OPE) to show this is true for a large class of descendent operators. It is also a warm up for the study of more general descendant operators in the next subsection.
In this and next subsection, we will consider the case that the interval is finite, lying at where the correlation function Z n is defined on the n-sheeted surface, with V (w i ,w i ) on the i-th surface. We set V (w,w) can be any descendant operators. 2 We only need to set n = 2 for S (2) calculation. To evaluate the multi-point correlation function on n-sheeted surface, we need to take a conformal First we consider the descendent operator V (w,w) = ∂O a (w,w), where O a (w,w) is a primary operator, the two-point function equals to and the four-point function is Under the conformal transformation (2.48), for t < l For convenience the convention in this section and the next section is a little different from the one in other sections. The correspondence is w2j ) are close to each other. While for l < t < l + L, ) are close to each other. From the result in [11], for t < l, where we still only keep the leading divergent term for small ǫ. Consequently for l < t < l + L which leads to ∆S 2 = log d a for l < t < l + L.
Next we consider the descendent operators with the form Here the ellipsis denote the terms with lower conformal dimensions, which leads to less divergent terms in the limit ǫ → 0. 4 Under the conformal transformation (2.48), the four-point function transforms as Even though the operator V (w,w) is not a primary operator, the coefficient for the leading term is the same as the one for the primary operator. The terms with lower conformal dimensions are less divergent in the small ǫ limit, so do not contribute to the final result.
Because we assume that the operator V is a quasi-primary operator, it transforms homogeneously under a linear conformal transformation. Consider the conformal transformation then the two-point function is Now let us study the divergence in the correlation function of four descendants in the z coordinate. We set l < t < l + L then (z 1 , z ) are close to each other. The four-point correlation function of quasi-primary operators can be transformed into  [11]. In the fourth equation, we change the differential back into the Virasoro operators acting on the primaries in the correlation function. The last equation is based on the fact that the Ward identity is satisfied for the conformal blocks.
The most divergent term only comes from the one with m = n = 0 and c 0,0 = 1 da [11]. Actually even in the vacuum block, only the identity operator gives the most divergent term, (2.72) Changing back into the w-coordinate and keeping the most divergent term, we find (2.73) Therefore, for a quasi-primary operator we still have ∆S 2 = log d a .
In short, for two kinds of descendent operators: ∂O and L (−) O, the difference between the second Rényi entropy of the excited states and the ground state is universal, equals to the logarithmic of the quantum dimension. This is the same as the case of inserting a local primary operator.

n-th RE for generic descendent states
With the above study, we are ready to discuss the effect of most general descendent operators.
A descendent operator in a module generated from a primary operator O a may take the following generic form we see that z j is close to z ′ j+1 as in [11]. For the n-th Rényi entropy, we need to compute the 2n-point function. Similar to the previous calculation we need to take a conformal transformation to z coordinate and take proper channel to expand the 2n-point function into the holomorphic and the anti-holomorphic part, as graphically shown in Fig. 2 5 . In each channel only the identity operator contribute to the final result, so the 2n-point function breaks up into n two-point functions for the holomorphic part (and n for the anti-holomorphic part). We can take a conformal transformation back to the w coordinate, with the leading divergent term being transformed homogenously. Actually this recipe only needs us to know which two arguments are close to each other and the coefficients for the most divergent term. With this message we can directly do the calculation in the w coordinate and multiply the coefficients from the OPE and the channel changing (2.77) Each two-point function in the above relation can be computed directly. For example, for the 5 We have extracted this figure from [11].
holomorphic part, it is and defining the density matrix then the two-point and 2n-point function can be written as With the normalized density matrix The extra contribution to the entropy is remarkable. Formally it takes the form −trρ 0 log ρ 0 .
It is zero only when the matrix d m,j;r,k is of rank one. By matrix product, the matrix ρ 0 is also of rank one. under a conjugate transformation, ρ 0 can be diagnosed as (1, 0, 0 The crucial point here is that the holomorphic and anti-holomorphic parts have different limits. For example for the four-point function z 4 ) , the ǫ → 0 limit set (z 1 ,z 2 ) close to each other while (z 1 , z 3 ) close to each other. The OPE cannot be used directly. That's the reason why we need to transform into the z coordinate and use the relation between the conformal blocks in different channel.

Comments in BCFT
In this subsection, we would like to introduce a boundary (or defect) at x = 0 as shown in Fig. 1. This boundary preserves the conformal symmetry. There are two kinds of boundary preserving conformal symmetry, and they correspond to the Neumann boundary condition and Dirichlet boundary condition normally. The global property of the CFT with a boundary has been discussed in [17] [23] [24]. As we know, the correlation function in a BCFT are much different from the one in CFT without boundary. In terms of (2.4), the Rényi entropy may be sensitive to the boundary. In [15], the Rényi entropy of the primary states with a boundary has been studied and it was shown that the maximal value of Rényi entropy does not change.
The boundary effects just change the time evolution of Rényi entropy. It is also interesting to check what will happen to the Rényi entropy for the local descendent states in 2D CFT with a boundary. For simplicity, we only discuss the descendent operator whose entropy is the same as the one of corresponding primary operator.
As shown in [11] [15], the Rényi entropy highly depends on the conformal blocks of the theory for general rational CFTs in 2D. In terms of [23]  .., z n ; z * 1 , ..., z * n ), analytically continued to z * j =z j . In [11], the time evolution of Rényi entropy highly depends on the holomorphic part of conformal block. In 2D CFTs with a boundary, the boundary indeed changes the propagator of the primary field but does not change the fusion constants in the bulk. In this sense, one can expect that the time evolution of Rényi entropy of the descendent states in CFTs with a boundary is almost the same as the one in CFTs on the full complex plane. From the studies in the previous subsections, the Rényi entropy of the descendent states could be the same as the one of corresponding primary states. One can check that the boundary does not change the maximal value of the Rényi entropy of the descendent states. In terms of the quasi-particles picture given in [15], the boundary just changes the time evolution of Rényi entropy of the descendent states, just as the primary states.

Rényi Entropy in Deformed CFT
In the 2D rational CFT, we find that the Rényi entropy of local descendent operators could coincide with the logarithmic of the quantum dimension of the corresponding primary operator.
Now we would like to study these operators in the CFTs with additional deformations or interactions. In the deformed CFTs, there is no conformal invariance. Nevertheless the effect of deformations on the Rényi entropy of local excited states can then be studied within conformal perturbation theory [25], if the deformation is weak. Here we only consider the theory which is perturbed by the local interaction Φ(z). Namely, we consider the theory which is perturbed by an operator Φ(z) with the conformal dimension ∆ ≥ 2. The action is now where I 0 is the original CFT action, λ ≡ g/Λ 2−∆ with g being dimensionless and R 1 denotes the complex plane. Consequently the total Hamiltonian H of the theory is H = H 0 +δH, where H 0 is the Hamiltonian of original CFT. The system will be evolving under the Hamiltonian H in the real time approach. As usual, we consider an excited state by acting a primary or descendent operator O on the vacuum |0 , which is defined 6 by H |0 = 0.
We consider the parameter g ≪ 1 in (3.1). The two-point correlation function of the primary or the descendent operator is where Σ 1 and R 1 denote the complex plane in deformed CFT and original CFT respectively.
We expand the deformation with respect to the powers of λ The two-point correlation function (3.2) contains infinite towers of contribution from interaction between the operators O and Φ in CFT. Similarly, the 2n-point correlation function on Σ n can be expanded 6 We set the vacuum energy to be zero for convenience.
where Σ n is n copies of Σ 1 . To calculate the correlation function on R n , one can employ the following conformal transformation ξ = ω ω − l 1/n and ω = ω(ξ) = l ξ n ξ n − 1 .

(3.4)
Then R n is mapped to the complex plane R 1 . The points ω 1 , ..., ω 2n is mapped to z 1 , ..., z 2n respectively by (3.4). The 2n-point correlation function can be expressed as where C n is the Jacobian from O(ω 1 )...O(ω n ). We know that when L + l < t or t < L, . Then the leading order term in the 2n-point correlation function is where the ... stands for the higher order terms in g. In order to make the perturbation well defined, we need a regularization to deal with the divergent terms in the integration. Espe- For the second integration in (3.7), we should do a similar regularization to deal with the divergence. All the divergent terms can be absorbed into λ to make the perturbation well de- in the early time limit or the late time limit (2.11). Repeating the analysis, we can calculate the ∆S (2) A to higher order of λ formally. For example, In the relation (3.8), corresponds to the contribution from the local excitation during the time l < t < L + l.
If Φ is a primary field, Φ R 1 = 0. We consider the second order of λ. The two-point function is Similarly, the 2n-point function during the time L + l < t or 0 < t < l. Then where ∆S (n) CF T denotes the result in the CFT. The second part in (3.10) is just the correction to the Rényi entropy from the deformation. It is called global contribution in the paper [25] .
During the time l < t < L + l the only change is in ∆S

Conclusion and Discussion
In this paper, we have studied the Rényi entropy of local descendent operators in 2D CFT, extending the previous studies in [8] [11] [15]. In [8] [11] [15], it has been found that the Rényi entropy of a primary state is equal to the logarithmic of quantum dimension of the primary operator. It is a natural question to consider the quantum entanglement of the descendent states. Firstly, we showed that for the specific operator L −1 O with O being primary, its Rényi entropy is still the logarithmic of quantum dimension of the primary operator O. Secondly, we discuss the descendent state L (−) O|0 generated by a quasi-primary operator. For such quasi-primary states, we showed that their quantum entanglements are the same as their primaries. Despite the fact that the operators look quite complicated and their conformal transformations are involved, the leading divergent terms in the early time and late time limit are simple, behaving as the one for primary operators. As a result, the quantum entanglement of the quasi-primary operators are the same as the primaries. Moreover we discussed the most generic descendent operators of the form V = m,j,r,k d m,j;r,k (∂ m L (−,j) )(∂ rL(−,k) )O a (w,w). (4.1) Out of surprise, we found that the Rényi entropy of such operator is generally different from the one of the primary O a (w,w). Only when the rank of the matrix d m,j;r,k is one, the entropies are the same as the ones of the primary. A typical example of such operator is Otherwise there is extra contribution. A typical example with extra contribution is the operator of the form To clarify the entropy difference between two kinds of operators V (1) (w,w) and V (2) (w,w), it would be illuminating to consider the free scalar field theory φ in 2D. Consider the primary The Rényi entropy is log 2 when one of the sector spreads into the region of the subsystem. The first is still a maximally entangled state. Therefore, the Rényi entropy should be same as the original state. We could do the operation on the chiral sector by i L k i −i (i, k i = 1, 2, ...) any times. The result is just a new EPR state. The operation by L −i does not change the entanglement property, but just change the state in the chiral sector. This is the basic reason that the local excitation by the descendent operators works similarly as the primary operator in this class. The same argument could also be used for the operations  , the corresponding state is where we define With some normalization this is just the EPR state, the entanglement entropy of which is log 2 when one of the sector state spreads into the subsystem A. In our calculation this contribution comes from the second term in (2.86). For the operator O where we define We could see that after normalization the relation (4.8) is just the direct sum of two EPR states. The entanglement entropy is 2 log 2 when one of the sector state spreads into the subsystem A. This is also consistent with our calculation (2.86).
Indeed for the interacting theory it is not easy to find such a clear explanation. Actually we still need better understanding on the fact that the seemingly decoupled chiral and anti-chiral sector are entangled with each other, even for the primary excitation state, see [33]. 7 We ignore the constant 1 √ 2 for O2 below.
In this paper, we also introduced a boundary which do not break conformal symmetry as in [15]. For the descendent state of type (1.1), we estimated the maximal value of Rényi entropy of local descendent states, which is the same as the one in the theory without such kind of boundary, although the boundary change the time evolution behavior of Rényi entropy.
Finally, we deformed the original CFT by additional operator and discussed the Rényi entropy of generic local excited states. If the deformation is small, we can treat it as a perturbation of CFT. By this way, we can estimate that this deformation make a global contribution to the Rényi entropy up to g 2 , if the deformation is a primary field. Though there are divergences in the global contribution [25], such divergences can be regulated appropriately. First consider the conformal transformation of L −1 ψ(z) This is really the conformal transformation for the operator ∂ψ(z), which support our calculation.
For more generic descendent operator Actually, we can get a more general conformal transformation for the operators, which is an expansion for the previous result