Note on non-vacuum conformal family contributions to R\'enyi entropy in two-dimensional CFT

We calculate the contributions of a general non-vacuum conformal family to R\'enyi entropy in two-dimensional conformal field theory (CFT). The primary operator of the conformal family can be either non-chiral or chiral, and we denote its scaling dimension by $\Delta$. For the case of two short intervals on complex plane, we expand the R\'enyi mutual information by the cross ratio $x$ to order $x^{2\Delta+2}$. For the case of one interval on torus with the temperature being low, we expand the R\'enyi entropy by $q=\exp(-2\pi\beta/L)$, with $\beta$ being the inverse temperature and $L$ being the spatial period, to order $q^{\Delta+2}$. To make the result meaningful, we require that the scaling dimension $\Delta$ cannot be too small. For two intervals on complex plane we need $\Delta>1$, and for one interval on torus we need $\Delta>2$. We work in small Newton constant limit in gravity side and so large central charge limit in CFT side, and find matches of gravity and CFT results.


Introduction
The investigation of entanglement entropy has drawn more and more attentions in the past decade, not only because it is interesting in its own right [1], but also because it opens a new angle in the investigation of AdS/CFT correspondence [2,3]. To calculate the entanglement entropy in a quantum field theory, one can use the replica trick [4,5]. One first calculates the general n-th Rényi entropy with n > 1 and being an integer, and then takes the n → 1 limit to get the entanglement entropy. For a conformal field theory (CFT) that has gravity dual in anti-de Sitter (AdS) spacetime [6][7][8][9], one can use the Ryu-Takayanagi formula and just calculate the area of a minimal surface in gravity side [2,3]. The Ryu-Takayanagi area formula of holographic entanglement entropy is the leading classical result in the limit of small Newton constant, and one can also consider the quantum corrections [10][11][12][13].
In AdS 3 /CFT 2 correspondence, small Newton constant limit in gravity side corresponds to large central charge limit in CFT side [14]. There are many investigations of Rényi entropy and holographic Rényi entropy in AdS 3 /CFT 2 correspondence. In gravity side, one uses the partition function of Einstein gravity in handlebody background [15][16][17], and calculates the classical and one-loop parts of the holographic Rényi entropy [12,18,19].
In CFT side one uses different methods to calculate Rényi entropy for the cases of two intervals on complex plane and one interval on torus. For the former case, one uses the operator product expansion (OPE) of twist operators [11,[20][21][22]. For the latter case, one uses the low temperature expansion of density matrix [19,23].
In AdS/CFT correspondence different operators in CFT are dual to different fields in gravity side, and it is interesting to compute the contributions of some specific operators to Rényi entropy in CFT side and compare the contributions of corresponding fields in gravity side to holographic Rényi entropy. The cases of some specific operators have been investigated in literature, for examples stress tensor [12,19,22], W operators [24,25,32], logarithmic partner of stress tensor [26], general scalars [27], supersymmetric partners of stress tensor [34,38], and current operator [38]. There are also some investigations of the contributions of a general primary operator to Rényi entropy [12,20,23,25], and in this paper we generalize the results to higher orders. We consider the contributions of a general non-vacuum conformal family to the Rényi entropy, with the primary operator of the conformal family being non-chiral or chiral. The non-chiral primary operator with conformal weights (h,h) has scaling dimension ∆ = h +h, and the chiral primary operator with conformal weights (h, 0) has scaling dimension ∆ = h. For the case of two short intervals on complex plane, we expand the Rényi mutual information by the cross ratio x to order x 2∆+2 . For the case of one interval on torus with the temperature being low, we expand the Rényi entropy by q = exp(−2πβ/L), with β being the inverse temperature and L being the spatial period, to order q ∆+2 .
The following part of this paper is arranged as follows. In section 2 we consider the Rényi mutual information of two intervals on complex plane. In section 3 we consider the Rényi entropy of one interval on torus. We conclude with discussion in section 4. In appendix A we review some useful properties of the non-vacuum conformal family.

Rényi mutual information of two intervals on complex plane
We calculate the Rényi mutual information of two short intervals on complex plane in expansion of the small cross ratio x. In gravity side we calculate the one-loop holographic Rényi mutual information using the method in [12], and in CFT side we calculate the Rényi mutual information using the OPE of twist operators [11,20,22].
In the CFT calculation we will use some results in [26].

Non-chiral primary operator
The classical part of the holographic Rényi mutual information only depends on the graviton, however the field in gravity dual to a nonidentity primary operator X changes the one-loop result. The non-chiral primary operator X has conformal weights (h,h) with h = 0 andh = 0, and its conformal weight is ∆ = h +h. The gravity Euclidean space is the quotient of global AdS 3 by a Schottky group Γ, and the one-loop partition function is multiplied by with P being a set of representatives of the primitive conjugacy classes of Γ. The form of q γ can be found in [12]. We get the contributions to the one-loop holographic Rényi mutual information with the definition In (2.2) we only incorporated the contributions of the so called consecutively decreasing words (CDW's) of the Schottky generators [12], and the order x 4∆ result that is omitted is from the 2-CDW's. To make the order · · · · · · · · · · · · Table 1: The quasiprimary operators in CFT n from the conformal family of non-chiral primary operator X in the original CFT. The indices j 1 , j 2 , j 3 take values from 0 to n − 1.
x 2∆+2 part meaningful, we need 4∆ > 2∆ + 2 and so ∆ > 1. Using [20] lim we get the contributions to the one-loop holographic mutual information The holographic mutual information is in accord with the result in [27] when the primary operator X is a scaler.
In CFT side, we use the OPE of twist operators in the n-fold CFT that is called CFT n . The Rényi mutual information can be calculated as [22,24,25] with K being all the orthogonalized quasiprimary operators Φ K in CFT n . The coefficients α K and d K are, respectively, the normalization factors and OPE coefficients of Φ K . In CFT n except the quasiprimary operators that are constructed solely by the vacuum conformal family of the original CFT, we have to consider the extra ones that are listed in Table 1. In the table we have the definitions We get contributions to Rényi mutual information from conformal family of X and the ranges of summations can be found in Table 1. The order x 3∆ result that is omitted in the above result is from contributions of the CFT n operators X j1 X j2 X j3 with 0 ≤ j 1 < j 2 < j 3 ≤ n − 1.
We have the normalization factors [26] where the factor i 4s = (−1) 2s aries from the minus sign when X is an fermionic operator. Note that there is always i 8s = 1. We also have the OPE coefficients [26] d j1j2 Here for simplicity we have defined s j1j2 ≡ sin π(j1−j2) n , c j1j2 ≡ cos π(j1−j2) n , · · · . Besides, we also need the normalization factors and OPE coefficients for the operators T j andT j with j = 0, 1, · · · , n − 1 With these coefficients and the formula (2.8), in large c limit we can reproduce the one-loop holomorphic Rényi mutual information (2.2).

Chiral primary operator
The case of chiral primary operator X , with conformal weights (h, 0) and h = 0, is similar to but a little different from the non-chiral operator case, and we discuss this briefly in the subsection. Note that we only consider  Similar to (2.1), the one-loop partition function is multiplied by We get the contributions to the one-loop holographic Rényi mutual information as well as the one-loop holographic mutual information 14) The holographic mutual information is in accord with the result in [27].
In CFT side, we have to consider the extra quasiprimary operators that are listed in We get contributions to Rényi mutual information from conformal family X We have the normalization factors 17) and the OPE coefficients Using the formula (2.16), we can reproduce the one-loop holomorphic Rényi mutual information (2.13).

Rényi entropy of one interval on torus
We calculate the contributions of a non-vacuum conformal family to Rényi entropy of one interval with length ℓ on torus in low temperature limit. The torus has spatial period L and temporal period β, with the temperature being 1/β, and in low temperature we have β/L ≫ 1. In gravity side we use the method in [12,19], and in CFT side we use the method in [19,23].

Non-chiral primary operator
In gravity side the one-loop partition function (2.1) still applies, and we use a different Schottky group that can be found in [12,19]. We get the contributions of a non-chiral conformal family to the one-loop holographic with q = e −2πβ/L ≪ 1. In (3.1) the omitted order q 2∆ result is from the 2-CDW's [12]. To make the order q ∆+2 part meaningful, we need 2∆ > ∆ + 2 and so ∆ > 2. Taking n → 1 limit, we get the one-loop holographic entanglement entropy Then we calculate the Rényi entropy in CFT side. For the vacuum conformal family we have the density matrix Note that we only consider the case without chemical potential. Considering the contributions of the conformal family of a non-chiral primary operator X we have the density matrix In CFT side, we get the Rényi entropy Note that in ∞ j and 0 j the subscript j denotes different replicas, and the ∞ and 0 without any subscript mean the special j = 0 case. The correlation functions on the n-fold complex plane C n can be calculated by mapping it to an ordinary complex plane C by a conformal transformation. The results are In the last two correlation functions we have omitted the O(1/c) terms in large c limit. Also we have to use [19] T (∞) we reproduce the one-loop gravity result (3.1) in large c limit.

Chiral primary operator
Using the one-loop partition function (2.12) and the Schottky group in [12,19], we get the contributions of the fields dual to the conformal family of a chiral primary operator to the one-loop holographic Rényi entropy with q = e −2πβ/L ≪ 1. Taking n → 1 limit, we get the one-loop holographic entanglement entropy Then we calculate the Rényi entropy in the CFT side. For the vacuum conformal family we have the holomorphic part of the density matrix Considering the contributions of the conformal family of a chiral primary operator X we have the density matrix (3.14) Then we get the Rényi entropy We need the correlation functions  Taking the large c limit we reproduce the one-loop gravity result (3.10).

Conclusion and discussion
In this paper we have considered the contributions of a general non-vacuum conformal family to the Rényi mutual information of two intervals on complex plane and the Rényi entropy of one interval on torus in twodimensional CFT. The primary operator of the conformal family can be either non-chiral or chiral. We got the results to the orders higher than those in literature, and found matches of gravity and CFT results.
We have only considered the contributions of one non-vacuum conformal family, and this is not complete for a concrete CFT. The algebra of the operators in the vacuum conformal family and one non-vacuum family is not close. For example at level 2∆ there may be a new conformal family with primary operator O = (X X ) + · · · . (4.1) To make the result in this paper meaningful, we have to require that the scaling dimension ∆ of the primary operator cannot be too small. For two intervals on complex plane we need ∆ > 1, and for one interval on torus we need ∆ > 2. For the contributions of a primary operators with a smaller scaling dimension and the contributions of more than two non-vacuum conformal families, further investigations are needed.

Acknowledgment
The author would like to thank Bin Chen for valuable discussions and thank Peking University for hospitality.

A Review of non-vacuum conformal family
In this appendix we review some properties of the non-vacuum conformal family that are useful for this paper, including the conformal family of a non-chiral primary operator and the conformal family of a chiral primary operator. Details can be found in [39,40], or can be easily derived from the results therein.

A.1 Non-chiral primary operator
The one-loop partition function of the vacuum conformal family is Considering the contributions of the conformal family of a non-chiral primary operator X , one has to multiply the result (A.1) by .