Higher Spin Entanglement Entropy

In this paper, we develop a perturbation formulation to calculate the single interval higher spin R$\acute{e}$nyi and entanglement entropy for two dimensional conformal field theory with $\mathcal{W}_{\infty}(\lambda)$ symmetry. The system is at finite temperature and is deformed by higher spin chemical potential. We manage to compute higher spin R$\acute{e}$nyi entropy with various spin deformations up to order $\mathcal{O}(\mu^2)$. For spin 3 deformation, we calculate exact higher spin R$\acute{e}$nyi entropy up to $\mathcal{O}(\mu^4)$. When $\lambda=3$, in the large $c$ limit, we find perfect match with tree level holographic higher spin entanglement entropy up to order $\mu^4$ obtained by the Wilson line prescription. We also find quantum corrections to higher spin entanglement entropy which is beyond tree level holographic results. The quantum correction is universal at order $\mu^4$ in the sense that it is independent of $\lambda$. Our computation relies on a multi-valued conformal map from $n$-sheeted Riemann surface $\mathcal{R}_n$ to complex plane and correlation functions of primary fields on complex plane. The method can be applied to general conformal field theories with $\mathcal{W}$ symmetry.


Introduction
Entanglement entropy and its generalization Rényi entropy are important quantities to study quantum systems. Entanglement entropy is a good parameter to characterize the effective degree of freedom of a region which is entangled with the rest of the system. It has interesting applications in condensed matter systems [3,4,5]. In general, they are hard to compute. Surprisingly, inspired by AdS/CF T correspondence, Ryu and Takayanagi [38] proposed that entanglement entropy in a conformal field theory could be calculated from a minimal surface in the dual bulk. Their beautiful work provides an effective way to compute entanglement entropy. At the same time, it opens a new window to study AdS/CF T , especially the emergence of spacetime in gravity.
On the other side, another important development in AdS/CF T in recent years is the higher spin holography [6,7,8], which suggests explicit duality between Vasiliev higher spin theory [9,10] and vectorial conformal field theory in the large N limit. Among various proposals,HS/CF T 2 [11], which relates higher spin theory in AdS 3 [12] to W N minimal model in large N limit, is of particular interest. This duality triggered various studies in AdS 3 gravity and CFT with W symmetry, please find nice reviews [13,14] on this topic.
Higher spin entanglement entropy arises from the combination of the previous two separate branches. In the field theory side, it is interesting to consider the deformation of entanglement entropy when there are other conserved currents. In the bulk, as the holographic entanglement entropy relates to a geometric object in usual Einstein-Hilbert theory, a generalization to holographic higher spin entanglement entropy may provide insights on mysterious higher spin geometry. Also, they are expected to provide nontrivial check of HS/CF T correspondence. Several works on this issue have been done. In [15,16], for the theory with W symmetry, two interval Rényi entropy without classical higher spin deformation has been calculated in the short interval limit, the quantum oneloop results in the gravity and CFT side match exactly up to O(x 8 ), where x is the cross ratio constructed from the two intervals. When there are higher spin chemical potential deformations, it is believed that the bulk configuration becomes a higher spin black hole [23], generalizing the conventional BT Z black hole. In these cases, a holographic higher spin entanglement entropy has been proposed by Wilson line priscription [28,39]. It has been checked up to O(µ 2 ) for the CFT with W 3 symmetry [40] while a general method to evaluate higher spin entanglement entropy in the CFT side is still lack so far.
In this paper, we develop a general prescription to calculate single interval higher spin Rényi entropy(HSRE) and entanglement entropy(HSEE) from CFT side perturbatively. Instead of inserting twist operators, we use a multi-valued conformal map from a n-sheeted Riemann surface to a complex plane. For spin 3 deformation, we find the higher spin Rényi entropy and entanglement entropy of W ∞ (λ) theory up to O(µ 4 ). We show that the O(µ 2 ) correction of higher spin Rényi entropy and entanglement entropy are indeed universal, in the sense that they are independent of the value of λ up to a normalization constant. At the O(µ 4 ), we find the classical and all loop quantum corrections to HSRE and HSEE. For λ = 3, the classical part matches with the gravity result exactly. Our method is sufficiently general to extend to all kinds of higher spin deformations.
The structure of this paper is as follows. In section 2, we review the general method to calculate N interval Rényi entropy and entanglement entropy in two dimensional conformal field theory, emphasizing the importance of a conformal map from n-sheeted Riemann surface to complex plane and its multi-valued property. In section 3, we review pure higher spin gravity in AdS 3 , including its asymptotic symmetry, higher spin black hole and its partition function. Some important results on W symmetry and black hole partition function are shown at the same time. In section 4, we briefly review the holographic higher spin entanglement entropy and expand the results of sl(3) theory up to O(µ 4 ). In section 5, we calculate perturbative partition function of higher spin black hole, which is a good exercise for HSRE and HSEE. In section 6, we calculate HSRE and HSEE up to O(µ 4 ) and match HSEE to holographic results in section 4. Discussion and conclusion are included in the last section. Some technical details are collected in three Appendixes.

Rényi and Entanglement Entropy
To define entanglement entropy, we suppose the system has a density matrix ρ and then divide the system into a subsystem A and its complement B. The total Hilbert space is factorized into H A ⊗ H B . We trace out the information of B and obtain the reduced density matrix of A, then the entanglement entropy of A is defined as the standard von Neumann entropy, A useful notion associated to the entanglement entropy is the n-th Rényi entropy S (n) , By analytical continuation of n → 1, we find the entanglement entropy, The procedure by introducing Rényi entropy to find the entanglement entropy is called the replica trick [1]. In two dimensions, the replica trick is applied as follows. Suppose the original CFT is defined on a Riemann surface 1 R 1 . We uplift the theory to n disconnected sheets and there is a branch cut along A in each sheet. We glue the branch cut successively and find an n-sheeted Riemann surface R n . The partition function in the n-sheeted Riemann surface is denoted as Z n .
In general, the subsystem A consists of N intervals, The partition function 2 Z n,N could be determined by introducing primary twist(and anti-twist) operators [2] with conformal dimension h n =h n = c 24 (n − 1 n ) at each point u i (v i ), such that Z n,N is a 2n point function of an orbifold conformal field theory on R 1 , The expectation value of arbitrary operator O on R n,N is For the single interval, suppose the original CFT is defined on a complex plane C, there is a direct conformal transformation which maps the n-sheeted Riemann surface R n,1 to the complex plane, where z and ω are the coordinates defined on R n,1 and C respectively. Hence we find an identity due to the conformal transformation 3 , ω ′ is the derivative to z, h O is the conformal dimension of O, · · · is the inhomogeneous term in the conformal transformation for a general quasi-primary operator. Let N = 1 in (7) and compare it with (9), we can solve the three point correlator < Oσ nσn >. For example, for the stress tensor, the Schwarz derivative will contribute to < T σ nσn > and the result is consistent with those from Wald identity of T with two primary operator with conformal dimension h = c 24 (n − 1 n ). This fact actually determines the partition function Z n,1 and hence the one-interval Rényi entropy and entanglement entropy are When the system is at a finite temperature 1 β , there is another conformal transformation which maps the complex plane to the cylinder, where s is a cylinder coordinate with 4 s = σ + iτ , σ ∈ (−∞, ∞), τ ∈ [0, β). Then the transformation of the twist operator under (11) leads to the one-interval Rényi entropy and entanglement entropy at finite temperature, For one-interval, the essential part is the existence of a conformal map (8). This conformal map is multi-valued, whose j-th solution is where n . Before we close this section, we emphasize that the conclusion (10) and (12) rely on the partition function is where q = 2πiτ . Once there is another continuous global symmetry, for instance, a higher spin symmetry, we should insert another zero mode operator weighted by its corresponding chemical potential, then the total partition function (14) should be modified. One can expand the modified partition function according to the order of the chemical potential, and solve the problem perturbatively. We will discuss this problem more concretely in the later sections.

AdS Higher Spin Gravity
The AdS 3 nonlinear higher spin gravity [12] describes the interaction between scalars(and fermions when promoting supersymmetry) and an infinite tower of spins with s ≥ 1. The matter multiplet can be truncated and one finds pure higher spin AdS 3 theory. This pure higher spin theory is described by two flat connection equations where 5 F = dA + A 2 and the connection one form A is valued in some higher spin algebra, which includes spin 2 algebra sl(2) as a subalgebra. A well-known higher spin algebra is hs[λ] which includes each spin s ≥ 2 once. Another view is to write 3d gravity as a difference of two Chern-Simons theory [17,18], with the Chern-Simons action . k cs is related to Newton constant by comparing (16) to Einstein-Hilbert action. The pure higher spin gravity is constructed by embedding sl(2) into a larger algebra [19].
where θ(z 1 , · · · , z 4 ) = x + 1 x − 2 = Actually, the four point correlator of spin J is This can be understood as follows. The exchange symmetry of 1 and 2 restrict the function f (x) as a function of θ = (x+ 1 x )−2, while the maximal power of θ is determined by examining the most singular behavior as z 1 → z 3 . Constants a[J, j] are found by matching (29) with the spin J four point function from the Wald identity.

Higher Spin Black Hole
A higher spin black hole [23] is a solution of (15) which have higher spin charges and higher spin chemical potentials. Choosing the three coordinates to be x ± , ρ, a spin 3 black hole in sl(3) theory is with Here we split the sl(3) generators into the spin 2 part(L i , i = 0, ±1) and the spin 3 part(W m , m = 0, ±1, ±2). Two parameters L, W are related to the spin 2 and spin 3 charges 6 . The parameter µ is the spin 3 chemical potential. There is another implicit parameter in this solution, which is the inverse temperature τ . It appears in the periodic To ensure the smoothness of the solution, one should impose trivial honolomy condition around the thermal circle. This condition relates the chemical potentials to the higher 6 We should mention that in different formalism, the identification of the charges is different. But in any case they can be determined by these two parameters. 7 Here we have transfered the solution to Euclidean version and replaced x + (x − ) with z(z).
spin charges, hence there are two freely parameters in the smooth solutions. A spin 3 black hole in hs[λ] theory can be constructed analogously.
Though the usual notion of geometry is lack in this context, one can still define consistent thermodynamics for higher spin black holes. So far, several methods have been developed to study the first law thermodynamics of higher spin black holes. These includes, 1. Dimensional analysis [24]. In this method, one requires the consistency of first law of thermodynamics and relies on the dimensional counting of the quantities in the theory. The striking formula follows from Euler's theorem on homogeneous functions. One can regard this method as a higher spin generalization of Smarr's formula [25] for usual black holes.
2. Action variational principle [26]. In this method, the action is thought as a saddle point approximation of the partition function. One adds suitable boundary terms to the action to ensure the variation of the action to be consistent with the first law of thermodynamics. It is a natural generalization of Gibbons-Hawking's analysis of black hole thermodynamics [27].
3. Conical singularity method [31]. The gravitational entropy can be calculated by the conical singularity [32,33] method, the authors in [31] found an extension to the higher spin gravity.
4. Wilson line approach [28,39]. This method can be generalized to calculate classical holographic entanglement entropy straightforwardly, we will discuss it in more detail in the following sections.
5. Noether charge method [29]. After suitability reformulating the Noether charge method [30] in Chern-Simons language, the authors in [29] can define the entropy of higher spin black holes. It is interesting that another choice of the Killing parameter leads to a new entropy for a higher spin black hole. In this paper we will not discuss this case.
A higher spin black hole corresponds to a CFT ensemble at finite temperature and with (higher spin) chemical potentials. For a higher spin black hole with a spin 3 chemical potential turning on, assuming the chemical potential µ is small 8 , one can evaluate the partition function and reproduce the tree level results in the gravity side [34,35]. Here we give the perturbative partition function for spin 3 black hole in hs[λ] theory, Here α = −µτ . For λ = 3, it reproduces the partition function of sl(3) black hole. The order O(c) part can be viewed as the tree level result and can be found either from gravity side [34] 9 or CFT side [34,35]. The quantum correction begins with O(c 0 ), it should be there and there is no explicit result so far. In our perturbative approach, we will find quantum corrections at the order µ 4 . The partition function should be We have used a subscript β to represent the thermal ensemble. One can expand the exponential according to the order of µ and calculate the partition function directly, Z 0 is the thermal partition function without higher spin deformation. A direct and rigorous perturbative analysis of this method is still lack so far 10 . We will develop this perturbation method to the order µ 4 and find the exact answer in the following section.
Here "exact" means that we can calculate the all loop higher spin thermodynamics at a finite order µ k , including the tree level and quantum corrections to the partition function. The tree level results (proportional to central charge c) will reproduce the gravity answer.

Holographic Higher Spin Entanglement Entropy
As discussed in section 2, we can define the entanglement entropy for a QFT. Due to AdS/CF T correspondence, there should be a concept which is dual to entanglement entropy in CFT. In [38], the authors introduced holographic entanglement entropy(HEE) to resolve this problem. In short, for Einstein-Hilbert theory, the static HEE is the minimal area of the surfaces which is homologous to the boundary region A, up to a coefficient 1 4G N . In AdS 3 /CF T 2 , for one-interval [u, v], this is just the length of the geodesics which connects the points u and v. But in this case the story is more interesting. The entanglement entropy is also equal to the logarithmic of a Wilson line [39,28]which connects the two points u and v. Since Wilson line can be defined for arbitrary higher spin theory, it is natural to conjecture this object is the holographic higher spin entanglement entropy. We discuss [39] in detail, since it also provides the holomorphic result. The one-interval holographic higher spin entanglement entropy is conjectured to be where P and Q are two bulk points. When ρ 0 → ∞, they tend to the points u and v. σ 1/2 is a constant and can be determined by the theory. The Wilson line W R (P, Q) is 9 In the gravity side, there are canonical formalism or holomorphic formalism, the result we list above is from holomorphic formalism. But one expects the same answer after one identifies the charges and potentials properly in the canonical formalism [36]. 10 There are some works along this line [37].
defined to be where P means path ordering and R denotes the representation. The representation can be found by matching the HEE to thermal entropy. In the holomorphic formalism, Wilson line (38) is replaced by For spin 3 black hole in sl(3) theory, σ 1/2 = 1 and R is chosen to be the adjoint representation. The higher spin entanglement entropy in the holomorphic formalism is where U = e 2π∆ β and ∆ = |u − v|. We have expanded the higher spin entanglement entropy to order µ 4 . The first term on the right hand side is the usual entanglement entropy (12). There is no contribution from odd power of µ. Actually this is related to the fact that the correlation function of odd number of spin 3 operators is zero in the CFT side. For each even power of µ, the ( µ β ) 2k correction is of the form where g[2k, i; U ] is a rational function of U. In the limit U → ∞, the terms g[2k, dominate. This is because that the higher spin entanglement entropy should be proportional to the thermal entropy in the large interval limit, Before ending this section, we remark that the Wilson line conjecture only reproduce the O(c) part of the higher spin entanglement entropy. It would be an interesting issue to find out the quantum correction in the bulk side.

Partition Function of Higher Spin Black Hole
The first step towards the higher spin entanglement entropy is to find the higher spin partition function as indicated by formula (3). We develop a different method to calculate (34), including its quantum correction. This is also a warmup exercise before we tackle the more tough problem on higher spin Rényi and entanglement entropy. Let us consider a spin J chemical potential in (35). We expand the exponential in (35), using the fact (23)(24)(25) and (29) on the correlator of spin J operator and the conformal map (11), then the partition function log Z can be written out order by order All the integrals are definite integrals ranging from 0 to ∞. However, there are divergences originating from the infinite length of the cylinder. We introduce an IR cutoff by setting the length of the cylinder to be L. Define M = 2πL β , then the answer can be written as a function of M.
For spin 3, the three point function is zero. We need to evaluate I 2 [3], I 4 [3, j](j = 0, · · · , 6) , they are Including the spin 3 coefficients (28), we find the partition function to be The interesting fact is that the higher order divergence cancel, leaving out the linear divergence, which shows the extensive property of the partition function. If we choose the large c limit, c → ∞, the O(c) partition function is exactly the same in (34). We can also read out the quantum correction in O(µ 4 ), Some remarks follows.
1. At the order µ 4 , the quantum correction to the higher spin black hole partition function is independent of λ.
2. In the large c limit, the quantum correction contributes a finite term This is the one-loop contribution. Of course, one can expand (53) in terms of 1/c and find the higher loop contribution. (53) contains all loop corrections to the partition function of spin 3 black hole at O(µ 4 ).

Higher Spin Rényi Entropy and Entanglement Entropy
In this section we compute the higher spin Rényi and entanglement entropy to O(µ 4 ).
From (3), the unknown quantity is log Z n .

O(µ 2 ) Correction
As a first step, let us consider the O(µ 2 ) correction. It is Here we use a spin J chemical potential deformation. β in the subscript means that we are in a thermal ensemble now. On the right hand side of the equality, we use the conformal map (11) from cylinder to complex plane. In this map, a point s i is mapped β . To find < W J (z 1 )W J (z 2 ) > Rn , we can use the conformal map (8) from R n to complex plane. Since W (z) = n−1 j=0 W (z j ), the field in j-th sheet should be mapped to a value ω j (z), hence The summation can be found by the residual theorem, where ξ is defined to be ξ = where the integral F [J, j] is defined to be the j = 0 term is cancelled in n-th Rényi entrpy (3), so finally The constantsb[J, j; n] = b[J,j;n] 1−n whose value can be found in Appendix A. The integral F [J, j] can be evaluated, please find Appendix B for details. There the reader can find the integral from spin 3 to spin 6. For spin 3 , the O(µ 2 ) correction of Rényi entropy is  11 Since the Rényi and entanglement entropy is a function of the distance |u − v|, we can safely set u = 0, v = ∆. In this convention, the two points are mapped to l1 = 1, l2 = U .
The definition of c[J, j, i] can be found in Appendix B. Taking the limit n → 1, the entanglement entropy is where e[2, i] is the same ones in (40). Once we choose normalization convention N 3 = − 5c 6π 2 and take into account of the contribution from anti-holomorphic part, (63) is equal to order µ 2 term in (40).
As the function b[J, j; n] can be determined by the residual theorem and F [J, j] can always be integrated out, the µ 2 J correction from the spin J deformation to the Rényi and entanglement entropy can be obtained. In Appendix C, we list these entropies for other spins, including the spin 4, 5 and 6 cases.

O(µ 3 ) Correction
We just introduce the method briefly as the three point function of spin 3 field is vanishing. However, for other kinds of fields or when there are many chemical potentials, the three point function may not be zero. So they can contribute to the Rényi and entanglement entropy. There is only one single chemical potential µ J in our example, but the reader can extend it to the cases with arbitrary number of chemical potentials.
As in the previous subsection, we first map finite temperature n-sheeted Riemann surface R n,β to R n and then map the R n to complex plane. After carefully collecting the conformal transformation factor and noticing the multi-value of the second transformation, we find the correction is and x ij is The summation function S[a, b, c; x, y, z] is more involved, please find it in Appendix A. The µ 3 correction depends only on the three point function of the operators. The general type of integration is with k ≤ 2 and is just a rational function of t where P [t] and Q[t] are polynomials of t. All such kind of integrals can be reduced to those discussed in Appendix B. For spin 3, the three point function is just zero. At least for this special example, we need not do the summation and integral at all. But such kind of consideration is useful to compute the µ 4 correction, which is indeed relevant even for spin 3.

O(µ 4 ) Correction
After some algebra, we find the partition function of n-sheeted Riemann surface R n is where O(µ 2 J ) is (58). The definition of S can be found in Appendix. The number of S increases linearly with spin J. We choose the smallest number, J = 3 to study. In this case, there are 7 kinds of S. The number of independent terms is of order O(10 2 ) for j = 1, 2, · · · , 6 and O(10) for j = 0. We note that the gravity result is of order c. While in Rényi entropy, there is O(c 2 ) contribution superficially. However, we expect this O(c 2 ) contribution should vanish, especially when we compute the entanglement entropy. Therefore we throw out the O(c 2 ) terms before computation. The large c expansion is determined by N J and a[J, j]. For spin 3, the terms with j = 0, 5, 6 are proportional to c 2 , so we simply ignore them. This reduces the relevant terms to j = 1, 2, 3, 4. Next, we look for O(c) term, which is the combination 12 There are also quantum corrections, which are contributed from j = 2 term.

O(c)
As mentioned above, we need to evaluate S for j = 1, 2, 3, 4. Since there are too many terms, we will not write them down explicitly here. All of the terms can be written as the product of some functions like coth a ij (log √ x ij ) with a ij being positive integers, 1 ≤ i < j ≤ 4. Using an identity then (69) becomes a function P(y ij ), where each term in P(y ij ) is with a, b, c are positive integers. Since i < 3, j < 4, there are actually six type of integrals in (68). Each type of integrals can be done, please find more details in Appendix B.
Finally, the answer is There is a IR divergence term in log Z n | µ 4 ,c . It is canceled by the similar term in n log Z 1 | µ 4 ,c , so the Rényi entropy is finite at this order. We plug (73),(52) into 13 (3) and find the O(c) Rényi entropy to be Taking the limit n → 1, the entanglement entropy is Note that e [4, i] are those in (40). After including the anti-holomorphic part, we reproduce exactly the large c limit of higher spin entanglement entropy (40)!

Quantum Correction
As we have mentioned, there is quantum correction at O(µ 4 ). It is from j = 2 term in (68). So it is important to derive this term seperately. The method is the same as before. We give the result of log Z n | µ 4 ,quan below. .
That means the quantum correction to the Rényi and entanglement entropy at O(µ 4 ) is universal in the sense that it is independent of λ. This is similar to the quantum correction to the thermal partition function, which is also λ independent at this order. There is also a IR divergence term here, it is canceled by n log Z 1 | µ 4 ,quan exactly. So to find the Rényi entropy, we just delete the term related to M in (76) and then divide 1 − n. The functions γ j [i; U ] are The quantum correction to the entanglement entropy is where the functions q[i; U ] are respectively

W ∞ (λ) Theory
We find the difference between W ∞ (λ) and W 3 theory is just the j = 2 coefficient of a [3, j] at O(µ 4 ). So the Rényi and entanglement entropy of W ∞ (λ) theory can be read out directly without any further computation. The difference of the Rényi entropy between W ∞ (λ) and W 3 theory is Here δ is the difference of a [3, j] for arbitrary λ and λ = 3, it is Taking the limit n → 1 and choosing the normalization N 3 , we find the difference of the entanglement entropy is Note the difference is of O(c), indicating the quantum correction is the same for different λ at this order. However, it is not clear whether this property holds to higher order of µ.

Conclusion and Discussion
In this work, we have developed a perturbation formulation to calculate HSRE and HSEE at finite temperature and with finite chemical potential. By using a multi-valued conformal map from R n to complex plane, the correlation functions of primary operators on R n is mapped to a multi-summation of the correlation functions of the same operators in the complex plane. After doing tedious summation and integration, we proved the universality argued in [40] at the order O(µ 2 ) up to a normalization constant. This university holds not only for spin 3 theory but also for arbitrary kinds of higher spin deformation and arbitrary number of higher spin deformations. The results for spin 4 to spin 6 are given in Appendix C, but we can extend the computation to any spin without difficulty. We also check the holographic higher spin entanglement entropy [28,39]  theory, the exist prescription [28,39] faces technical problem due to the infinite dimension of the group. However, our CFT calculation overcome this difficulty without trouble. In fact, in this work, we have calculated the HSRE and HSEE for W ∞ (λ) up to O(µ 4 ). In the holographic prescription, there is no explicit computation so far. We also found that at O(µ 4 ), the difference of HSEE(and HSRE) between W ∞ and W 3 theory is purely classical, without any quantum correction! It would be interesting to check this fact at higher order of µ.
4. There is another point we do not discuss extensively in this paper. When there are many higher spin potentials, the O(µ 3 ) can be non-zero. Except for a theory dependent three point function constant C J 1 J 2 J 3 , our method indicates that the total correction of HSRE and HSEE at this order is also universal. This is similar to µ 2 correction. In conformal field theory, the structure of two and three point function of primary operators are determined by global conformal symmetry, as shown in (23) and (24). This means the universal property found in O(µ 2 ) and O(µ 3 ) actually originates from the global conformal symmetry in the theory.
Besides the previous concrete conclusion, we can give some discussions below.
1. We can calculate the correction to arbitrary order O(µ k ) with finite k > 4 . To O(µ k ), we only need to know the k-point function of higher spin operators which is determined by the Wald identity, this may be complicated but can be evaluated.
2. Our formulation does not use the usual prescription of the twist operator. Instead, we just used a conformal map from n-sheeted Riemann surface to complex plane.
Since the twist operator method should give the same answer as here, one may read out some interesting information about the twist operator from our result. For instance, the most singular term of the OPE between a spin J(J > 2) operator W J with twist operator σ n should be where σ ′ n is another primary operator with dimension c 24 (n − 1 n ) + 1. We expect to find a complete OPE between W J and σ n , with the results found in this work.
3. When the interval number N > 1, there is no similar conformal map. In that case, we can not get the answer for all interval lengths. However, for short intervals,we can use all the technics [41,42,43,44] to calculate higher interval Rényi and entanglement entropy.

Our method to evaluate HSEE(and HSRE) is perturbative, it give new results
which is beyond the holographic method. However, we notice that the holographic description provides the classical correction to all order of µ in the large c limit. In CFT side, to get a result analogously is quite difficult due to the complicated summation and integrals shown in the Appendix. It would be interesting to evaluate the all order µ result from CFT side directly.
In (57), we use the two point function in complex plane and notice the multi-valued solution of the conformal map (8), then where the cross ratio of the four points z 1 , z 2 , l 1 , l 2 is defined to be (83)is universal for different theory up to a spin J normalization. This confirms the conclusion argued in [40]. Let's define a function .
Since the summation depends on the difference between j 1 and j 2 , we get a factor n by eliminating one summation index, Since a is an integer, the summation can be converted to a contour integral, in which the contour can be chosen as follows, The contour C includes the singularities z = πij n , j = 0, · · · , n − 1 and z = z * ,with z * = − log √ x n . The height of the contour is π, since under a shift of z → z + πi the function to be integrated is invariant when a is an integer. The total contour integral is zero so that We use (7) to organize the answer. The two point function on R n should proportional to a four point function with operators W J , W J , σ n ,σ n . The same reason as (29) tells us Note the maximal power of ξ is J − 1, this is just a direct consequence of the residual (88). We evaluate the residual (88)  The terms with j = 0 is always n, because these terms should contribute to the divergent terms (proportional to M) and cancel with the terms coming from n log Z 1 . To simplify notation for Rényi entropy, we also introduce constantsb Appendix A.2: Three Point Function on R n As in the previous subsection, we map R n to complex plane, Here and S is a multi-summation, This summation can be done similar to (85). In (90), which is not the case for general x, y, z in (92). So for the special summation in (90), the summation of j 3 is contributed by the poles at log √ r 13 n + πij 1 n and log √ r 23 n + πij 2 n . This doesn't causing new poles. Then the j 2 summation is contributed from the poles at log √ r 12 n + πij 1 n . The summation j 1 contributes a factor n as there is no pole now.

Appendix A.3: Four Point Function on R n
We only consider the case with four higher spin operators are the same.
a[J, j]S[2J − 2j, j, j, j, j, 2J − 2j; r 12 , r 13 , r 14 , r 23 , r 24 , r 34 ], S is defined as This summation can be done using the residual theorem as before. In this work, we only need the four summations each S are quite lengthy, so we will not give the explicit result here.

Appendix A.4: Higher Point Function on R n
As the m(m ≥ 5) point function in complex plane can be solved by using Wald identities recursively. The m point function on R n can be transformed to a set of summations. The basic k summation function is (97) This function is reduced to two,three and four point case when m = 2, 3, 4. Though residual theorem can be used to compute it, it would become tedious quickly as m increases. As an example, for four spin 3 point function on R n , the number of independent terms is expected to be O(10 2 ) as it is proportional to six point functions(four spin 3 and two twist operator). So for m > 4, to find an explicit answer is horrible. It is interesting to study this summation in other methods, this would be important to read out the information of µ m correction to Rényi and entanglement entropy.
In our work, a is always positive, there seems to be imaginary terms in (102). As the final partition function should be real, the imaginary part of (102) should be canceled by other terms. Since −π 2 − 3 log[− 1 a ] 2 always appears at the same time, we argue that we can set them to zero. Therefore we find a rule before taking the limit of x → ∞, The same reasoning as p = 1 case leads us to a replacement rule when x → ∞, 3. p = 3, Then the replacement rule is The three replacement rules (103),(107)and (109) are all we need when we take the x → ∞ limit.

Appendix B.1: Integral F [J, j]
Here, we list the integral of F [J, j] up to spin 6.
1. J = 3. Since j = 0 term is canceled, there are only two integrals. (a) with with with  The basic integral is In this work, a, b, c are positive integers and 1 ≤ i ≤ 2, 1 ≤ j ≤ 3. It is convenient to define a set of integrals as 14 14 We omit the dependence of J to simplify notation. In the following, our calculation is done for J = 3.
Then after integrating out t 4 , t 3 , t 2 , the terms y a 12 y b i3 y c j4 are replaced by So the building blocks are these defined functions. One can integrate out them term by term, though very tedious. In our computation, we use a slightly different method. Instead of integrating out t 4 , t 3 , t 2 term by term, we just integrate out t 4 , t 3 at first. And then we sum over the results. After that, the integral t 2 , t 1 are done. That means we just do the following replacement, The relevant m a s are the ones with a = 1, 2, · · · , 5, and the relevant g bc are b + c ≤ 6, b ≥ 1, c ≥ 1 So the number of integrals we need to do is (5 + 5 + 4 + 3 + 2 + 1) =20. Then we classify the function according to the power of n and y 12 . A general function is n 2α+1 y a 12 F α,a (t 1 , t 2 )/n 8 where F α,a (t 1 , t 2 ) is determined by m a , g ab . α, a satisfy 1 ≤ α ≤ 4, 1 ≤ a ≤ 2α + 2 So there are (4 + 6 + 8 + 10 =)28 different integrals of t 2 in (126). In the work, we calculate the O(c) part and the quantum correction, there is no divergence after we sum over the integrals of t 4 , t 3 , t 2 . After all these have been done, we need to integrate t 1 . We can integrate it according to the power of n. So there are 4 integrals of t 1 . Since the Rényi entropy at this order should be finite, the integration should cancel the divergent term in the partition function n log Z 1 . So the divergence should come from α = 4 terms. For α = 4, we should be careful to seperate the divergence term. There is no other subtlety except these. In total, we need to do (28 + 20 + 4 =)52 integrals. We just mention that the first 28 and the last 4 integrals is relatively simple, whilst the other 20 integrals of t 2 is a bit complicated. As an illustration, we just give the results of m 1 [t], the other functions m a [t], g ab [t] are similar.
Appendix C: O(µ 2 ) Correction to Rényi Entropy and Entanglement Entropy of Other Spins The formula for Rényi entropy to O(µ 2 J ) is (61). One can plug the value ofb[J, j; n] and F [J, j] to find their explicit expression. Taking the limit n → 1, we can list the O(µ 2 J ) corrections of entanglement entropies from spin 4 to spin 6 below,