Renyi Entropy for Local Quenches in 2D CFTs from Numerical Conformal Blocks

We study the time evolution of Renyi entanglement entropy for locally excited states in two dimensional large central charge CFTs. It generically shows a logarithmical growth and we compute the coefficient of $\log t$ term. Our analysis covers the entire parameter regions with respect to the replica number $n$ and the conformal dimension $h_O$ of the primary operator which creates the excitation. We numerically analyse relevant vacuum conformal blocks by using Zamolodchikov's recursion relation. We find that the behavior of the conformal blocks in two dimensional CFTs with a central charge $c$, drastically changes when the dimensions of external primary states reach the value $c/32$. In particular, when $h_O\geq c/32$ and $n\geq 2$, we find a new universal formula $\Delta S^{(n)}_A\simeq \frac{nc}{24(n-1)}\log t$. Our numerical results also confirm existing analytical results using the HHLL approximation.


Introduction and Summary
One useful way to characterize dynamical aspects of quantum field theories (QFTs) is to study properties of entanglement entropy [1,2,3,4,5,6,7]. Especially, the evolutions of entanglement entropy when we excite QFTs provide us with important pieces of information such as whether the quantum field theory is integrable or chaotic. Indeed, this has been manifest for locally excited states in conformally field theories (CFTs) as we briefly review below.
A famous example in this direction is the studies of global quantum quenches, which are translationally invariant excited states created by a sudden change of the Hamiltonian [8] (also refer to references in [9]). The time evolution of entanglement entropy under global quenches shows a causal and relativistic propagation of elementary excitations in generic CFTs. One way to study how the details of propagations of quantum entanglement are different beyond this universal behavior is to look at local excitations rather than global excitations. Motivated by this, the main purpose of this paper is to study the time evolution of entanglement when we locally excite a class of CFTs which are strongly interacting and have many degrees of freedom. Such a class of CFTs have holographic duals and is called holographic CFTs [10,11,12].
A locally excited state |Ψ (operator local quench) is defined by acting with a local operator O(x) on the CFT vacuum |0 in the manner 1 where N is the normalization factor. The infinitesimally small parameter ǫ > 0 provides a UV regularization as the truly localized operator has infinite energy. Consider the time evolution of the entanglement entropy S A = −Trρ A log ρ A and more generally Renyi entanglement entropy S (n) A = 1 1−n Tr log(ρ A ) n for the time evolved excited state |Ψ(t) = e −iHt |Ψ . We choose the subsystem A to be the half-space and ρ A is the corresponding reduced density matrix. The excitation is originally located in the subsystem B (i.e. complement of A), thus it creates additional entanglement between them. The main quantity of interest is the growth of entanglement entropy compared to the vacuum: Note that the n = 1 limit coincides with the von-Neumann entropy (or entanglement entropy) growth lim n→1 ∆S   Calculations of ∆S (n) A for massless scalar fields have been performed in [13,14,15,16] and it was found that the growth ∆S time. This is clearly interpreted as a system of entangled particles propagating at the speed of light [17]. The same behavior has also been found for rational CFTs in two dimensions, which is a typical example of integrable CFTs, [14,18,19,20,21,22].
On the other hand, for holographic CFTs, which are strongly interacting CFTs with large central charges [10,11,12], the evolution behavior changes drastically. In the calculation using the holographic formula [35,36], the local excitation corresponds to a massive particle falling in AdS 3 , whose mass m is related to the conformal dimension ∆ O of O(x) in (1) via the standard relation ∆ ≃ mR (R is the AdS radius). The holographic results for the (von-Neumann) entanglement entropy under time evolution at late time was obtained in [37] and this reads where c CF T is the central charge of the 2D holographic CFT. This time dependence has been precisely reproduced in [38] using a large central charge CFT analysis. Such a behavior is expected to stem from the chaotic nature of holographic CFTs, where the quasi-particle picture breaks down. Similar calculations in higher dimensional holographic CFTs have recently performed in [39] for a holographic computation in the AdS 4 /CFT 3 setup, where log t like behavior was observed.
In summary, the functional form of the growth of von-Neumann entanglement entropy ∆S A = ∆S (1) A depends on the nature of CFTs roughly in such a way that as interactions in the CFT get stronger, the rate of its growth increases. In [23], it was even conjectured that the logarithmic growth (3) found in the holographic 2d CFTs give an upper bound of the growth rate for any two dimensional CFTs. In this way, we already had enough knowledge about the von-Neumann entanglement entropy for the locally excited states.
However, to capture more information of quantum entanglement, we need to look at the Renyi entanglement entropies ∆S (n) A for various n. At present, we have only limited understanding of ∆S

(n)
A for the locally excited states as we will briefly review below. To obtain the full understanding of the time evolutions of the Renyi entanglement entropies is the main purpose of the present paper.
For the Renyi entanglement entropy in 2D holographic CFTs, the following behavior was derived in [40] when the (chiral) conformal dimension h O of local operator is small enough: In this way, interestingly, there are varieties of behaviors of (Renyi) entanglement entropy for the locally excited states defined by (1), depending on how much a given CFT is chaotic. Notice that the above operator local quench makes a significant contrast with the original class of local quenches introduced in [41] (see also [42,9]) defined by joining two semi-infinite CFTs, which always lead to logarithmic growth of entanglement entropy ∆S A (t) ≃ c CF T 3 log t ǫ for any two dimensional CFTs, both integrable and chaotic. In this paper, we would like to explore more on the analysis of ∆S

(n)
A for 2D holographic CFTs to obtain complete and systematic understandings without relying on special approximations. For example, one may notice that it is not immediately clear why the two results (3) for the von-Neumann entropy and (4) for the Renyi entropy with h O ≪ c CF T are related to each other. Indeed, the latter gets divergent in the von-Neumann limit n → 1. Moreover, there is no known result for the evolution of Renyi entropy with large To find a full control of the computability for any values of n and h O we will employ the powerful numerical program recently developed in [43] by Chen, Hussong, Kaplan and Yi, based on Zamolodchikov's recursion relation [44,45,46,47]. For example, this relation was applied to computations of entanglement entropy for multi-intervals in [48]. This powerful and non-perturbative method allows us to evaluate any conformal blocks with any values of conformal dimensions and central charges c. Note that for the replica computation of the Renyi entanglement entropy ∆S

(n)
A we set c = n · c CF T . Therefore in this paper we express the total (replicated) central charge as c when we talk about the conformal blocks.   A (t). The green and red regions can be well approximated by the HHLL conformal block approximation.
As we will explain later in the present paper, our analysis reveals that the behavior of conformal block changes drastically when the conformal dimensions are large. We find that the behavior of vacuum conformal block for the 4-pt function of the form In region (i) the coefficients c n of the power series of the function H(q) = 1 + ∞ n=1 c n q 2n , which is an important part of the contributions to the conformal block, grows polynomially. On the other hand in the region (ii) and (iii), c n grows exponentially.
In terms of our Renyi entanglement entropy calculations, these regions correspond to In particular, studies of the region (i) lead to the following new universal behavior of Renyi entropy growth On the other hand, in the region (ii) and (iii), we can apply the formula (3) when n ≃ 1, while in the region (ii) ′ and (iii) we can apply the formula (4) when h O ≪ c CF T . These behaviors are summarized in Fig.1. It will be an intriguing future problem to reproduce the new behavior (5) from holographic calculations. Also it is an important to understand better the nature of the non-trivial transition of the conformal blocks at h A,B = c/32. This paper is organized as follows: In section two, we review the computation of Renyi entanglement entropy in large central charge CFTs. In section three we analyse the vacuum conformal blocks by using the Zamolodchikov's recursion relation for various parameter regions. In section four, we apply the results of section three to the computations of Renyi entanglement entropy. In appendix A we summarize our conventions of four point functions and conformal blocks. In appendix B we briefly review the Zamolodchikov's recursion relation .

Renyi Entropy and Conformal Blocks
Here we review the general calculation of Renyi entanglement entropy for excited states in a 2D CFT in terms of conformal blocks. We express the 2D CFT we consider as M and its central charge is written as c CF T .

Growth of Renyi Entropy for Excited States
In the replica computation for Renyi entropy we introduce the replicated CFT with the central charge nc. The growth of Renyi entropy (2) for locally excite states (1) can be expressed by [49,38] ∆S (n) we take the subsystem A to be a semi-infinite interval and the twist operators are inserted at both end points of A.
Here O ⊗n is defined on the cyclic orbfold CFT (replicated CFT) M n /Z n (with the central charge c = n · c CF T ), using the operators in the seed CFT M (with the central charge c CF T ) as which is separated by the distance l from the boundary of A as shown in Fig.2. We take the subsystem A to be the half line [0, ∞]. In the complex plane (z,z), we insert the two twist operators each at z =z = 0 and z =z = ∞. The replicated operator O ⊗n , which produces the state |Ψ of the local excitation, is inserted at Fig.2. The rule of the analytical continuation from the Euclidean time to Lorentzian time follows from the prescription in [8]. Now, these insertions of four operators lead to the growth of the trace of the reduced density matrix Tr(ρ A ) n i.e. that of Renyi entropy as in (6). We define the chiral conformal dimension of O(x) as h O . Then the dimension of O n is written as nh O . The dimension of twist operator σ n is given by the standard formula By using the cross ratio z = z 12 z 34 z 13 z 24 , we can rewrite (6) as where G(z,z) is the four point function (refer to appendix A for more details of our conventions) and the cross ratio (z,z) is explicitly expressed as From these expressions, one find that the sign of the imaginary part of the cross ratio z changes at t = l. As a result, the cross ratio z picks up the factor e −2πi at t = l as Fig.3). This does not happen for the anti-chiral coordinatē z. We write the value of z after this monodromy transformation as z mo . For example, if f (z) := log(1 − z), then f (z mo ) = −2πi + f (z). In the late time region ǫ ≪ l ≪ t, which we are interested in, we find Therefore the time evolution of excited Renyi entropy at late time can be calculated as The 4-pt function G can be expressed as a summation over conformal blocks F hσ n ,h O ⊗n hp (z). In holographic CFTs we expect that due to its sparse spectrum, we can approximate G by restricting to the vacuum conformal block h P = 0. Thus we obtain This is the main target which we would like to evaluate in this paper.

Analysis of Conformal Blocks
In this section, after we review the numerical computation of conformal block based on Zamolodchikov's recursion relation, we explore properties of conformal block towards the calculation of Renyi entropy. We are especially interested in large central charge limit c ≫ 1.

Conformal Blocks and Numerical Approach
As in [44], the conformal block for the 4-pt function where and It is useful to note K(0) = π 2 and when z is small we have K(1 − z) ≃ 1 2 log 16 z . We can express z in terms of the theta functions as z = θ 2 (q) is found by solving the recursion relation in [44], whose numerical prescription was formulated in [43] (see appendix B of this paper). It is expanded as It is important to note that H h A ,h B h (q) has the symmetry: This follows from the relation (61) for the full 4-pt function is true for each conformal block. This relation is expected because it just flips A with B and does not change the structure of channel. One useful limit we take is q → 1. Accordingly z approaches to z = 1. We write this as z = 1 − ǫ with ǫ → 0. Then we have in this limit:

Simplest Example of H(q): Vacuum Primary
If we consider the trivial limit of vacuum primaries h A = h B = h = 0, obviously we have h=0 (z) = 1. This fixes the form of H(q) as follows When q is small (or equally z is small), this is expanded as follows: On the other hand, if we take the limit z = 1 − ǫ with ǫ → 0, we find where ∼ means the approximation up to a constant factor. By using the approximation formula of a summation (based on the saddle point approximation) (we assume A > 0): When From (24), we find the Cardy formula-like behavior of the coefficient c n defined in (18) when n ≫ c: c n ≃ β · n α · e A √ n , for a certain constant β which we are not interested in. Here A and α are given by

Behaviors of H(q)
Now we would like to examine the properties of H(q) for general h A and h B . We focus on the vacuum conformal block h = 0 as that is relevant for our later calculations of Renyi entropy. First we can analytically calculate the coefficient c n assuming the large c limit from the recursion relation: However note that here we ignored the lower powers of c and this approximation is only sensible for n ≪ c.
From this expression we find the following behavior of signs of c n (refer to Fig.4): We summarized these behavior in Fig.5. Note that c n is invariant under the exchange of h A and h B as follows from the symmetry (19).  Now we perform numerical computations of c n employing the computer program made in the paper [43] (for a short summary, see also appendix B of the present paper). First, our numerical calculations of c n for various values of (h A , h B ) precisely show the above behaviors of signs, even for the regions n ≫ c. By fitting the numerical result for c n (refer to the plots Fig.6) in the Cardy formula-like form and we evaluated the values of A and α for various (h A , h B ) and plotted in Fig.7.  Fig.7. This behavior is summarized as follows 2 Here the functions a 2 and a 3 are smooth monotonic functions bounded as |a 2 | ≤ 1 and |a 3 | ≤ 1. We have Note that a 3 (0, 0) = 1 in (34) following from (27). As we will show in the next subsections, we can evaluate A and α by using the heavy-heavy-light-light (HHLL) approximation of conformal blocks assuming . This leads to Indeed, we find a 3 (h L , c/32) ≃ 1 2 , which justifies (35) and shows that A is continuous at the border between (ii) and (iii). The value of α is found as which can be applicable to both (ii) and (iii) if h L is small enough. On the other hand, we would like to note that the result in the region (i), α is well fitted with the numerical data by the formula first considered in [43]: Note that the doubled coefficient of h A + h B compared with (39) can be understood from the factor two difference of the power of δ between the A > 0 formula (24) and A = 0 formula (25).

HHLL Approximation
When one of h A and h B is light and another is heavy, we can apply the heavy-heavylight-light (HHLL) approximation of conformal blocks [50,51]. We take In our case we have (we assume h = 0) Thus in the z → 1 limit: z = 1 − ǫ, we get In terms of the function H h H ,h L h=0 (q), this behavior is described as Now we focus on the region (iii) i.e. h H < c/32 so that c n is positive definite. By using the formula (24), we obtain the estimation of A and α given by (38) and (39).

q → i limit
When we consider the region (ii) i.e. h H > c/32, we need to worry about the alternating signs in (30), as c n behaves like c n ∼ (−1) n · n α · e A √ n . In such a case, it is not straightforward to obtain a formula like (24).
To cancel the signs, we consider another limit of q → i. This is obtained by z → 0 limit with a monodoromy transformation around z = 1. Explicitly we have This behaves like (we defineδ = π 2 log(16/ǫ) ): From the HHLL approximation (41), we obtain the following behavior for the modular By using the formula (24), we can read off from this behavior the advertised values of A and α given by (37) and (39). Actually, this limit q → i exactly corresponds to the one we need to calculate the Renyi entropy described in (12) i.e. q mo = q(z mo ). In this relation, we can identifyǫ = − 2iǫ t .

Evaluation of Renyi Entropy
Now we are in a position to study the Renyi entropy computed by the formula (14) based on our previous results for the vacuum conformal block. First note that to calculate the nth Renyi entropy for a large central charge CFT (holographic CFT) with a central charge c CF T by the replica method, we consider a CFT with the central charge c = n · c CF T , defined by taking n copies of the original CFT. Then we take The growth of Renyi entropy is symmetric under the exchange of h A and h B as Since we act the monodromy transformation only for z and not forz, in the limit (12) we have q → i as in (45) andq ≃z 16 → 0. Thus we can simply (14) as follows where we neglect subleading terms and keep such terms grow as ∼ | logǫ| ∼ log( t ǫ ) in the end. 3 In general, we find that ∆S

(n)
A grows logarithmically under time evolutions. Therefore, below, we are interested in the coefficient of the log t term, denoted by B(n, h O ): The behaviors of B(n, h O ) are summarized in Fig.1. Our numerical results of B(n, h O ) are plotted in Fig.8 as we will explain below.

HHLL Approximation
If we apply the HHLL approximation with h A ≪ c ∼ h B , we obtain from (46) If we choose the light operator O A to be the twist operator σ n in the limit n → 1, then we reproduce the formula (3). If we choose the light operator O A to be O, namely the operator for the excitation, then we reproduce another formula (4). Note that these approximated results are continuous at h A,B = c/32. The regions where we can apply this HHLL approximation is depicted in Fig.1 as the red and green regions.

Region (i)
In the region (i) defined by (29), we simply have namely the advertised formula (5). This is derived as follows. First note that H(q mo ) does not give any leading divergence which contributes to O(log t) entropy, remembering the behavior c n ∼ n α (i.e. no e A √ n factor). Then the formula (49) with nc CF T ≫ 1 leads to the formula (52). Indeed this behavior is confirmed in the left plot in Fig.8. At the special value n = 2 and h O = c CF T 32 , the conformal block coincides with a torus partition function [10,29,52]. By using this fact we can evaluate ∆S (2) A as computed in [53], which indeed agrees with (52). 4

Region (ii)
In the region (ii) defined by (30), we can apply the formula (24) in q → i limit as we have explained in the previous section. Thus we can calculate the coefficient B in (50) in terms of A: This function interpolates the HHLL approximated result and the formula (52) in the region (i). Examples of the plots of B(n, h O ) in this region can be found from the both two plots in Fig.8.

Region (iii)
The region (iii), defined by (31) also includes parameter spaces where we can apply the HHLL approximation. Also when n = 2 and h O = c CF T 32 , we should reproduce the behavior (52). Even though we do not have any analytical formula, our numerical results show that the function B(n, h O ) monotonically interpolates these boundary values, as depicted in the right plot of Fig.8.

Comment on OTOC
In this paper, we focus on entropy, but the vacuum block we derived can be also used to evaluate OTOC. As in [54], we can express the late time behavior of OTOC in holographic CFTs as The cross ratio z mo is given by where ǫ ij = i e 2πi β ǫ i − e 2πi β ǫ j and x is the separation between operators that keeps the ratioz/z fixed. This technique is very similar to that used to calculate entropy. Therefore we can apply our result to calculating OTOC straightforwardly and it leads to the late time behavior of OTOC for any two operators, in particular two heavy operators. The result is as follows.
and in the heavy-light limit, we can reproduce the results in [54,55]. Moreover we can conclude that the behaviors of OTOC show the late exponential decay for any operator at late time. This exponential decay cannot be seen in non-chaotic CFTs, where the OTOC approaches non-zero constant [56,57] or decays polynomially [58]. This may suggest that this late time behavior can also be used as a criterion of chaotic nature of a given quantum field theory, in addition to the existing arguments on the Lyapunov exponent [54,59,60]. It is also intriguing to note that the above exponential decay behavior of OTOC is directly related to the logarithmic growth of Renyi entanglement entropy (5).
takes the following form ∞) by taking the limit (w 1 , w 2 , w 3 , w 4 ) → (0, z, 1, ∞) and by absorbing the divergence as follows: In other words we find the relation On the other hand if we take the limit (w 1 , w 2 , w 3 , w 4 ) → (1, ∞, 0, z), we can define By comparing (58) and (60) This relation is very natural because it is just an exchange of two O A s with two O B s. The (normalized) 4-pt function can be written as the summation over all conformal blocks: where h p is the conformal dimension of the intermediate primary state.

B Recursion Relations
In this Appendix, we will review Zamolodchikov's recursion relation. In our case h 1 = h 2 = h A and h 3 = h 4 = h B , the Virasoro conformal block can be expressed as where h m,n is a zero of the Kac determinant and R m,n = 2 · p,q λ p,q 2 ′ k,l λ k,l · p,q (2λ A − λ p,q )(2λ B − λ p,q ).