Short interval expansion of R\'enyi entropy on torus

We investigate the short interval expansion of the R\'enyi entropy for two-dimensional conformal field theory (CFT) on a torus. We require the length of the interval $\ell$ to be small with respect to the spatial and temporal sizes of the torus. The operator product expansion of the twist operators allows us to compute the short interval expansion of the R\'enyi entropy at any temperature. In particular, we pay special attention to the large $c$ CFTs dual to the AdS$_3$ gravity and its cousins. At both low and high temperature limits, we read the R\'enyi entropies to order $\ell^6$, and find good agreements with holographic results. Moreover, the expansion allows us to read $1/c$ contribution, which is hard to get by expanding the thermal density matrix. We generalize the study to the case with the chemical potential as well.


Introduction
The entanglement entropy characterizes the entanglement between a system and its environment. It could be defined by the von Neumann entropy of the reduced density matrix (1.1) In practice, it is often more convenient to compute the entanglement entropy by taking the n → 1 limit of the n-th order Rényi entropy where the Rényi entropy is defined by The study of the Rényi entropy is interesting on its own, as it encodes the spectral information of the reduced density matrix.
The entanglement entropy in quantum field theory is difficult to compute, due to the fact that there are infinite degrees of freedom in a field theory [1], as nicely reviewed in [2]. If the quantum field theory is conformal invariant, there are more tools to use, in particular for a two-dimensional (2D) conformal field theory (CFT) [3]. The standard way of computing the entanglement entropy in a quantum field theory is the replica trick [4]. For a 2D CFT, one could insert the twist operators at the branch points to impose the nontrivial boundary conditions on the fields in applying the replica trick [1,3]. Then the Rényi entropy is determined either by the partition function on the resulting Riemann surface, which is generically of higher genus, or equivalently by the correlation function of the twist operators on the original spacetime manifold but in an orbifold CFT.
The Rényi entropy on complex plane has been well studied. It is well known that when there is a single interval of length ℓ, the Rényi entropy has a universal form, which is independent of details of the CFT and depends only on the central charge c [1] S n = c(n + 1) 6n log ℓ ǫ , (1.4) with ǫ being the UV cutoff. But for the case with two and more intervals, the Rényi entropy would depend on the details of the CFT [5][6][7]. For the case of two disjoint intervals there have been some studies. The analytical results of Rényi entropy S n , n ≥ 2 for a free compactified boson and Ising model have been presented in [5] and [7] respectively. For a general CFT, it was proposed in [6] that one can use the operator product expansion (OPE) of the twist operators to compute the Rényi entropy S n , n ≥ 2. This proposal was generalized to find the leading term of Rényi entropy with small cross ratio in [7]. This method was also generalized to higher dimensions in [8]. More interestingly, the proposal has been applied to the large c CFT with a gravity dual. In [9][10][11][12][13], the Rényi mutual information has been computed and reorganized by the powers of 1/c. 1 It was found that the leading order terms agree exactly with the holographic Rényi entropy (HRE) [16][17][18][19], the next-to-leading terms are in exact agreement with the one-loop correction to HRE [20,21], and moreover there are nonvanishing next-to-next-to-leading terms which should be identified with the two-loop corrections to HRE [22,23]. The study suggests that the holographic computation should be correct even beyond classical gravity, as shown in recent study [24] of the one-loop partition function of any handle-body configuration from dual CFT.
Another interesting case is the Rényi entropy on a torus. Now the CFT is in a thermal state and within a finite size. There are both thermal and finite-size effects in the entanglement entropy. In [25], the universal thermal correction to the single interval entanglement entropy has been found. In [26][27][28][29], the Rényi entropy of free boson on a torus has been discussed. In [30][31][32][33], the Rényi entropy of free fermion on a torus has been studied. In [34], the Rényi entropy for the SU(N ) 1 Wess-Zumino-Witten model on the torus has been discussed. In [35], the thermal effect to the Rényi entropy of locally excited states has been discussed. In particular, the single interval Rényi entropy on a torus in the large c CFT dual to the AdS 3 gravity has been investigated in [36][37][38] and the exact agreement up to the next-to-leading order with the holographic computation has been found.
The studies in [25,[36][37][38] rely on the expansion of the thermal density matrix at the low temperature or at the high temperature. In the low temperature case, the thermal density matrix could be expanded where the summation is over all the normalized and orthogonalized excitations in the theory. For the large c CFT, the contribution from the vacuum Verma module may dominate [39] such that one can focus just on the states in the vacuum module. Then the computation is reduced to the multi-point functions on a cylinder. This treatment could be applied to the interval of almost any length, even though in the large interval limit one needs the expansion with respect to the twist sector [28,38]. The disadvantage in the treatment is that one has to work in the low temperature or high temperature limit, and the computation becomes complicated beyond the first few excited states, and it is hard to read the 1/c terms.
In this work, we would like to apply another strategy to compute the single interval Rényi entropy on torus.
We focus on the case that the interval is short so that we can apply the OPE of the twist operators. In this way, the computation boils down to the one-point functions of various operators on a torus. Such a strategy could be applied to any CFT one is interested in. Here we pay more attention to the large c CFT. One advantage of this approach is that it works for any temperature, or more generally for any modulus of the torus. By expanding further in the low temperature and high temperature limits, we find consistent picture with the results in [36,37].
Furthermore, this new approach allows us to read the 1/c terms easily.
The remaining parts of the paper are organized as follows. In Section 2, we introduce the general framework of our treatment. In Section 3, we focus on the vacuum conformal family of a large c CFT and discuss the low and high temperature limits, as well as the effect of chemical potential. In Section 4 we consider a large c CFT with W(2, 3) symmetry. In Section 5 we consider a large c N = (1, 1) superconformal field theory (SCFT). We conclude with some discussions in Section 6. In Appendix A there are some useful CFT basics. In Appendix B there is an alternative calculation of low temperature expansion of the one-point functions (3.5).

Rényi entropy on torus
In this section, we introduce the general strategy to compute the single interval Rényi entropy on a torus. We consider the case that the interval is so short that we may first use the OPE of the twist operators and then sum over the expectation values of one-point functions on the torus. We first briefly review the OPE of the twist operators, and then calculate the single interval Rényi entropy on torus.

OPE of twist operators
To calculate the n-th order Rényi entropy using the replica trick one has to calculate the partition function of the CFT on a Riemann surface, or equivalently one can calculate the correlation functions of twist operators σ,σ in an orbifold CFT [1], which is denoted by CFT n . The twist operators have conformal weights In CFT n one has OPE of the twist operators [6,7,9] σ(z,z)σ(0, 0) = c n z 2hσz2hσ Here c n is the normalization of the twist operators, and the summation over K is on all the independent quasiprimary operators Φ K in CFT n , and there are definitions of coefficients where the binomial coefficient is , (2.4) and the OPE coefficient d K can be determined by the one-point function on the Riemann surface R n,1 [7] with α K being the normalization coefficient of Φ K . The quasiprimary operators in CFT n can be constructed by the quasiprimary operators in each copy of CFT.
For the vacuum conformal family, to level 8 the CFT n quasiprimary operators have been constructed in [9,10].
In this paper we only need the ones to level 6 that have nonvanishing expectation values on the torus. Because of the translation symmetry, one-point functions on torus are coordinate-independent, and so the derivatives of an operator have vanishing expectation values on the torus. This simplifies the discussion. To level 6, the holomorphic CFT n quasiprimary operators that have nonvanishing expectation values on torus are listed as follows.
• At level 2 we have T j .
• At level 4 we have A j , and T j1 T j2 with j 1 < j 2 .
• At level 6 we have B j , D j , T j1 A j2 with j 1 = j 2 , and T j1 T j2 T j3 with j 1 < j 2 < j 3 .
Here the subscript j labels the replica and all the j's take values from 0 to n − 1. The detailed discussion on the quasiprimary operators can be found in Appendix A. Note that operators at level 5 and some of operators at level 6 have vanishing expectation values on the torus so that they can be ignored. The needed OPE coefficients d K have been calculated in [9,10] (2.7) We may introduce It is similar for the quasiprimary operators in the antiholomorphic sector.
For other modules, the construction of quasiprimary operators is straightforward. In the following discussion, we mainly consider the primary operator φ with conformal weight (h φ , 0), in which case h φ has to be an integer or a half-integer. Such operators appear in the CFT with W symmetry or the superconformal symmetry. In the OPE of the twist operators, there are the quasiprimary operators from the module φ, among which the operators φ j1 φ j2 with j 1 < j 2 have the lowest scaling dimension. The normalization of φ is chosen to be α φ . We have the OPE coefficient . (2.10) For later convenience, we introduce Similarly, we have d j1j2 φφ and bφφ for the antiholomorphic sector.

Rényi entropy
We consider the Rényi entropy of one short interval A = [0, ℓ] on a torus T with the complex coordinate z. The modulus of the torus is τ = τ 1 + iτ 2 , with τ 1 and τ 2 being real. The modulus τ is related to the inverse temperature β and chemical potential µ E by Note that here we are working in Euclidean space, and the Euclidean chemical potential µ E is real. If we want the Minkowski result, in the final Rényi entropy we have to make the analytical continuation µ E = −iµ with the Minkowski chemical potential µ being real. The spatial period of the torus is L so we have Without loss of generality, we may set L = 1. As usual we define q ≡ e 2πiτ . (2.14) The partition function is given by Due to OPE of the twist operators, we have summation over the one-point functions of the quasiprimary operators in CFT n . In principle, this could be applied to any CFT. Here we are more interested in the large c CFT dual to the AdS 3 gravity. In this case, the contribution is dominated by the vacuum module. To the level we are interested in, all quasiprimary operators in the OPE are the products of the quasiprimary operators in a single CFT on different replica. Therefore, their expectation values are just the products of the ones on the torus. The resulting partition function is with the two summations being over all the nonidentity holomorphic and antiholomorphic primary operators respectively. Consequently we can get the short interval expansion of the single interval Rényi entropy Note that φ andφ appear in pairs, and the summation above is over such pairs. Therefore, the computation of the single interval Rényi entropy is reduced to the one-point functions on the torus. Note that because of translation symmetry the one-point functions on torus are independent of the coordinate.
On a torus, the one-point function of the stress tensor is given by [40] T T = 2πi∂ τ log Z, (2.18) with Z being the partition function on the torus. From the Ward identity on torus [40], we get with ℘(x) being the Weierstrass P function, ζ(x) being the Weierstrass ζ function, and (2.20)

From (2.19) and the definitions in (A.1) and (A.2), we get
Here we have introduced where σ a (k) is the divisor function.
Note that the short interval expansion (2.17) holds for any value of torus modulus τ , but the expansions in (2.24) For consistency of the CFT, the partition function Z is invariant under the SL(2, Z) transformation At a low temperature, the holomorphic sector of the density matrix of the vacuum conformal family can be expanded as More generally, there could be other modules in the theory. Thus one can add other holomorphic conformal families and have the density matrix with the definition Here h φ is the conformal weight, k denotes the levels in the φ conformal family, and G ij φ,k is the inverse of the matrix φ, k, i|φ, k, j .
with φ 0 being the 0-th mode of φ. In the low temperature expansion, the leading order contribution of the one-point function is with ψ being the primary operator that has the lowest conformal weight h ψ among the ones with nonvanishing structure constant C ψφψ and α ψ being its normalization. In the Rényi entropy (2.17) we have the following term in the expansion of ℓ and q The behavior of the term in the expansion of 1/c has to be discussed case by case.

Vacuum conformal family
In the discussion above, the prescription can be applied to any CFT at any temperature. In this section we focus on the contributions of the vacuum conformal family to the Rényi entropy, and we temporarily ignore the contributions from other conformal families. We would like to compute the entropy at both the low and high temperature limits, in order to compare with the results got in [36,37].

Low temperature limit
At a low temperature we first consider the case without chemical potential, and we have 3 q =q = e −2πβ ≪ 1. (3.1) We will briefly discuss the case with chemical potential in the end of this subsection. On the gravity side the Rényi entropy can be calculated to the order q 4 [20,36,37], and the gravity result has been confirmed from the CFT side [25,36,37]. We may expand the holographic Rényi entropy by the length of the interval ℓ. The classical part and the one-loop part is The calculation of the Rényi entropy to higher orders of q is very cumbersome. However, since it is exact in ℓ, one can get the Rényi entropy to the higher order of ℓ very easily.
On the CFT side, if we only consider the vacuum conformal family, we have the partition function .
We did not have subscript (2) in the leading part, since it is universal and does not depend on the operator content of the CFT. The leading contribution S L n is proportional to c, and it is in perfect agreement with the classical part of the holographic Rényi entropy S cl n in (3.2). The next-to-leading part S NL n,(2) is of order one, and it agrees exactly with the one-loop gravitational part S 1-loop n, (2) in (3.3). The next-to-next-to-leading part S NNL n,(2) is proportional to 1/c S NNL n,(2) = − 8π 4 (n + 1)(n 2 + 11) 45n 3 c q 4 + O(q 5 ) ℓ 4 + 8π 6 (n + 1)(26n 4 + 271n 2 − 345) which is expected to match the two-loop correction to the holographic Rényi entropy. Note that expansion of Rényi entropy to the higher orders of q is straightforward and can be done easily. However, the expansion of Rényi entropy to higher orders of ℓ is cumbersome. The parameter region of the method in this paper is different from that of the gravity and CFT methods in [20,25,36,37].
When the chemical potential is turned on, one need to make the replacement in the Rényi entropy. In the Minkowski spacetime, we have to make the analytical continuation µ E = −iµ, and the replacement becomes The CFT at a low temperature with a chemical potential is dual to the thermal AdS with angular momentum. It would be nice to see if the Rényi entropy can be reproduced on the gravity side.

High temperature limit without chemical potential
At the high temperature limit without chemical potential, the classical and one-loop gravity results are just making the following replacement [20] in (3.2) and (3.3) except the ℓ in the logarithmic function. The classical part becomes S cl n = c(n + 1) 6n log ℓ ǫ + π 2 c(n + 1) 36n and the one-loop part becomes − 4π 6 (n + 1)(17n 4 − 46n 2 + 31) 945n 5 q ′2 + 2π 6 (n + 1)(492n 4 − 1013n 2 + 527) 945n 5 q ′3 + 4π 6 (n + 1)(1654n 4 − 2903n 2 + 1255) On the CFT side we make the modular transformation Note that if we restore L in (3.5), then Eq.(3.14) equal Eq. (3.5) after the simple substitutions There are similar results for the antiholomorphic operators. Using (2.17) and organizing the Rényi entropy by the powers of 1/c, we find Now S L n and S NL n, (2) are in exact match with the classical part S cl n (3.11) and the one-loop part S 1-loop n,(2) (3.12) respectively. The next-to-next-to-leading part is The CFT at a high temperature with vanishing chemical potential is dual to nonrotating BTZ (Bañados-Teitelboim-Zanelli) black hole [41].

High temperature limit with chemical potential
When we turn the chemical potential on, the calculation is similar. Firstly, we have Note that in the Euclidean space the left-and right-moving inverse temperatures β L,R are complex, but they are real in the Minkowski spacetime. One can see this point, for example, in [42]. At a high temperature β is small, and we work in the parameter region that µ is of order one. We make the modular transformation (3.20) Here the superscripts L, R denote the Rényi entropy from the holomorphic and antiholomorphic sectors respectively, or, in terms of the Minkowski CFT, the left-and right-moving sectors. Formally, S L,L n , S NL,L n, (2) and S NNL,L n, (2) are just half of the previous subsection results S L n , S NL n, (2) , and S NNL n, (2) in (3.16) after setting β → β L . Also S L,R n , S NL,R n, (2) and S NNL,R n, (2) are just S L,L n , S NL,L n, (2) and S NNL,L n, (2) after setting β L → β R , q ′ →q ′ .
The CFT at the high temperature with a chemical potential is dual to the gravitational configuration of rotating BTZ black hole. The holographic computation of the Rényi entropy in this case has not been worked out. However, the classical part of the entanglement entropy can be read from the length of a geodesic in the background of the BTZ black hole [43] S cl = c 6 log Expanding the holographic entanglement entropy by small ℓ, we get
Compared to (3.5), there are corrections starting from the order q 3 . As shown in Appendix A, on torus we have The next primary operator is E (A.7) at level 6, and its leading contribution to the Rényi entropy is (A.10). This is beyond our consideration. From (2.17), we get the Rényi entropy in expansion of 1/c with the leading part S L n equaling S cl n (3.2), the next-to-leading part S NL n,(2) equaling S 1-loop n,(2) (3.3), and the next-toleading part S NL n,(3) equaling S 1-loop n,(3) (4.1).

Conclusion
In this paper we investigated the short interval expansion of the single interval Rényi entropy for a two-dimensional CFT on torus. The length of the interval ℓ is small such that we could apply the OPE of the twist operators to reduce the computation to the one-point functions on the torus. Even though the prescription could be applied to any CFT, we focused on the large c CFT dual to the AdS 3 gravity and we got the Rényi entropy to order ℓ 6 .
We discussed the cases of vacuum conformal family, the CFT with W(2, 3) symmetry, and N = (1, 1) SCFT, and found consistent agreement with the existing results in all the cases. We discussed the case with chemical potential as well.
In the previous studies of the Rényi entropy in a large c CFT on the torus [36][37][38], there is no restriction on the length of the interval except that it should not be comparable with the size the torus , but one has to take the low or high temperature limit in order to expand the thermal density matrix appropriately. It is unwieldy to get the higher orders thermal corrections to the Rényi entropy. In the new prescription based on the short interval expansion, we have to take the small interval limit and only get the first few orders of interval length. However, the higher order thermal corrections, or finite size corrections, of the Rényi entropy can be got easily. In particular, we may read the 1/c correction quite easily, which is hard to find in the old treatment.

A Some basics of CFT
In this appendix we review some useful basics of the two-dimensional CFT. We mainly give some quasiprimary operators in the vacuum conformal family, and some primary operators in CFT W(2, 3) symmetry and N = (1, 1) SCFT.
The operators in the holomorphic sector of the vacuum conformal family can be written as quasiprimary operators and their derivatives. In level 2, one has the quasiprimary operator T , with the usual normalization α T = c 2 . In level 4, we have In level 6, we have Under a general coordinate transformation z → f (z), we have 744s 3 + (70c + 29)(4ss ′′ − 5s ′2 ) , For holomorphic sector of the CFT with W(2, 3) symmetry, we have the primary operator W at level 3 with the usual normalization α W = c 3 . All the operators constructed by T and W are of integer spins, and so for any such quasiprimary operator φ we have Similarly, we can calculate other three-point correlation functions and then get the low temperature expansion of T T . In the way method we can get A T , B T , and D T to the order q 4 . The results are the same with (3.5).