Note on non-vacuum conformal family contributions to Rényi entropy in two-dimensional CFT

We calculate the contributions of a general non-vacuum conformal family to Rényi 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 Δ. For the case of two short intervals on a complex plane, we expand the Rényi mutual information by the cross ratio x to order x2Δ+2. For the case of one interval on a torus with low temperature, we expand the Rényi entropy by q=exp(−2πβ/L), with β being the inverse temperature and L being the spatial period, to order qΔ+2. To make the result meaningful, we require that the scaling dimension Δ cannot be too small. For two intervals on a complex plane we need Δ > 1, and for one interval on a torus we need Δ > 2. We work in the small Newton constant limit on the gravity side and so a large central charge limit on the CFT side, and find matches of gravity and CFT results.


Introduction
The investigation of entanglement entropy has drawn more and more attention 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 take 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 on the 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 quantum corrections [10][11][12][13].
In AdS/CFT correspondence, different operators in CFT are dual to different fields on the gravity side, and it is interesting to compute the contributions of some specific operators to Rényi entropy on the CFT side and compare the contributions of corresponding fields on the gravity side to holographic Rényi entropy. The cases of some specific operators have been investigated in the literature, for example the stress tensor [12,19,22], W operators [24,25,32], logarithmic partner of stress tensor [26], general scalars [27], supersymmetric partners of the stress tensor [34,38], and current operators [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 a 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 a torus with low temperature, 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 rest of this paper is arranged as follows. In Section 2 we consider the Rényi mutual information of two intervals on a complex plane. In Section 3 we consider the Rényi entropy of one interval on a torus. We conclude with discussion in Section 4. In the Appendix we review some useful properties of the non-vacuum conformal family.

Rényi mutual information of two intervals on a complex plane
We calculate the Rényi mutual information of two short intervals on a complex plane in expansion of the small cross ratio x. On the gravity side we calculate the one-loop holographic Rényi mutual information using the method in Ref. [12], and on the 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 from Ref. [26].

Non-chiral primary operator
The classical part of the holographic Rényi mutual information only depends on the graviton, but 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 and h = 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 Ref. [12]. We get the contributions to the one-loop holographic Rényi mutual information (2) with the definition In (2) we only incorporated the contributions of the so called consecutively decreasing words (CDWs) of the Schottky generators [12], and the order x 4∆ result that is omitted is from the 2-CDWs. To make the order 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 Ref. [27] when the primary operator X is a scalar.
On the 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 , as well as 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 063103-2 Table 1. The quasiprimary operators in CFT n from the conformal family of non-chiral primary operator X in the original CFT. The indices j1, j2, j3 take values from 0 to n − 1.
We get contributions to Rényi mutual information from the 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 opera- We have the normalization factors [26] α X X = i 4s α 2 where the factor i 4s = (−1) 2s arises 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] Here for simplicity we have defined 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 Eq. (8), in the large c limit we can reproduce the one-loop holomorphic Rényi mutual information (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. Note that we only consider the contributions of the conformal family X , and we do not count the contributions of the possible conformal family of the anti-holomorphic operatorX with conformal weights (0, h).
Similar to (1), the one-loop partition function is multiplied by We get the contributions to the one-loop holographic Rényi mutual information (13) as well as the one-loop holographic mutual information The holographic mutual information is in accord with the result in Ref. [27].
On the CFT side, we have to consider the extra quasiprimary operators that are listed in Table 2. In the table we have the definitions We get contributions to Rényi mutual information from the conformal family X

063103-4
We have the normalization factors and the OPE coefficients Using Eq. (16), we can reproduce the one-loop holomorphic Rényi mutual information (13).

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

Discussion and conclusions
In this paper we have considered the contributions of a general non-vacuum conformal family to the Rényi mutual information of two intervals on a complex plane and the Rényi entropy of one interval on a torus in twodimensional CFT. The primary operator of the conformal family can be either non-chiral or chiral. We got the results to orders higher than those in the previous literature, and found matches of gravity and CFT results.
We have only considered the contributions of one nonvacuum 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 con-formal family with primary operator 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 a complex plane we need ∆ > 1, and for one interval on a torus we need ∆ > 2. For the contributions of primary operators with a smaller scaling dimension and the contributions of more than two non-vacuum conformal families, further investigations are needed.
The author would like to thank Bin Chen for valuable discussions, and Peking University for hospitality.
The author thanks Matthew Headrick for his Mathematica code Virasoro.nb, which can be downloaded at http://people.brandeis.edu/%7E headrick/Mathematica/index.html.

Review of non-vacuum conformal family
In this Appendix we review some properties of the nonvacuum 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 Refs. [39,40], or can be easily derived from the results therein. 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 (A1) by .
Note that αX is the normalization factor of X , and that we consider a CFT with equaling holomorphic and antiholomorphic central charges c =c. Under a general conformal transformation z → f (z),z →f (z), the primary operator X transforms as with the Schwarzian derivatives