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arXiv:2412.00192 · Physical review. D/Physical review. D.

Rényi entropy of single-character CFTs on the torus

Luis Alberto León Andonayre, Rahul Poddar

Published Jun 17, 2025• 0 citations
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Abstract

We introduce a nonperturbative approach to calculate the Rényi entropy of a single interval on the torus for single-character (meromorphic) conformal field theories. Our prescription uses the Wrońskian method of Mathur, Mukhi, and Sen [], in which we construct differential equations for torus conformal blocks of the twist two-point function. As an illustrative example, we provide a detailed calculation of the second Rényi entropy for the E 8 , 1 Wess-Zumino-Witten (WZW) model. We find that the Z 2 cyclic orbifold of a meromorphic conformal field theory (CFT) results in a four-character CFT, which realizes the toric code modular tensor category. The Z 2 cyclic orbifold of the E 8 , 1 WZW model, however, yields a three-character CFT since two of the characters coincide. We then compute the torus conformal blocks and find that the twist two-point function, and therefore the Rényi entropy, is two-periodic along each cycle of the torus. The second Rényi entropy for a single interval of the E 8 , 1 WZW model has the universal logarithmic divergent behavior in the decompactification limit of the torus, as expected as well as the interval approaches the size of the cycle of the torus. Furthermore, we see that the q -expansion is UV finite, apart from the leading universal logarithmic divergence. We also find that there is a divergence as the size of the entangling interval approaches the cycle of the torus, suggesting that gluing two tori along an interval the size of a cycle is a singular limit.

DOI
10.1103/qt5p-sl5f

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We introduce a nonperturbative approach to calculate the Rényi entropy of a single interval on the torus for single-character (meromorphic) conformal field theories. Our prescription uses the Wrońskian method of Mathur, Mukhi, and Sen [], in which we construct differential equations for torus conformal blocks of the twist two-point function. As an illustrative example, we provide a detailed calculation of the second Rényi entropy for the E 8 , 1 Wess-Zumino-Witten (WZW) model. We find that the Z 2 cyclic orbifold of a meromorphic conformal field theory (CFT) results in a four-character CFT, which realizes the toric code modular tensor category. The Z 2 cyclic orbifold of the E 8 , 1 WZW model, however, yields a three-character CFT since two of the characters coincide. We then compute the torus conformal blocks and find that the twist two-point function, and therefore the Rényi entropy, is two-periodic along each cycle of the torus. The second Rényi entropy for a single interval of the E 8 , 1 WZW model has the universal logarithmic divergent behavior in the decompactification limit of the torus, as expected as well as the interval approaches the size of the cycle of the torus. Furthermore, we see that the q -expansion is UV finite, apart from the leading universal logarithmic divergence. We also find that there is a divergence as the size of the entangling interval approaches the cycle of the torus, suggesting that gluing two tori along an interval the size of a cycle is a singular limit.
We investigate the short interval expansion of the Rényi entropy for two-dimensional conformal field theory (CFT) on a torus. We require the length of the interval ℓ 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 Rényi entropy at any temperature. In particular, we pay special attention to the large c CFTs dual to the AdS3 gravity and its cousins. At both low and high temperature limits, we read the Rényi entropies to order ℓ6, 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.
We analytically evaluate the Rényi entropies for the two dimensional free boson CFT. The CFT is considered to be compactified on a circle and at finite temperature. The Rényi entropies S n are evaluated for a single interval using the two point function of bosonic twist fields on a torus. For the case of the compact boson, the sum over the classical saddle points results in the Riemann-Siegel theta function associated with the A n−1 lattice. We then study the Rényi entropies in the decompactification regime. We show that in the limit when the size of the interval becomes the size of the spatial circle, the entanglement entropy reduces to the thermal entropy of free bosons on a circle. We then set up a systematic high temperature expansion of the Rényi entropies and evaluate the finite size corrections for free bosons. Finally we compare these finite size corrections both for the free boson CFT and the free fermion CFT with the one-loop corrections obtained from bulk three dimensional handlebody spacetimes which have higher genus Riemann surfaces as its boundary. One-loop corrections in these geometries are entirely determined by quantum numbers of the excitations present in the bulk. This implies that the leading finite size corrections contributions from one-loop determinants of the Chern-Simons gauge field and the Dirac field in the dual geometry should reproduce that of the free boson and the free fermion CFT respectively. By evaluating these corrections both in the bulk and in the CFT explicitly we show that this expectation is indeed true.
In this paper, we study the holographic R\'enyi entropy of a large interval on a circle at high temperature for the two-dimensional conformal field theory (CFT) dual to pure AdS3{\mathrm{AdS}}_{3}AdS3​ gravity. In the field theory, the R\'enyi entropy is encoded in the CFT partition function on nnn-sheeted torus connected with each other by a large branch cut. As proposed by Chen and Wu [Large interval limit of R\'enyi entropy at high temperature, arXiv:1412.0763], the effective way to read the entropy in the large interval limit is to insert a complete set of state bases of the twist sector at the branch cut. Then the calculation transforms into an expansion of four-point functions in the twist sector with respect to ${e}^{\ensuremath{-}\frac{2\ensuremath{\pi}TR}{n}}$. By using the operator product expansion of the twist operators at the branch points, we read the first few terms of the R\'enyi entropy, including the leading and next-to-leading contributions in the large central charge limit. Moreover, we show that the leading contribution is actually captured by the twist vacuum module. In this case by the Ward identity the four-point functions can be derived from the correlation function of four twist operators, which is related to double interval entanglement entropy. Holographically, we apply the recipe in [T. Faulkner, The entanglement R\'enyi entropies of disjoint intervals in AdS/CFT, arXiv:1303.7221] and [T. Barrella et al., Holographic entanglement beyond classical gravity, J. High Energy Phys. 09 (2013) 109] to compute the classical R\'enyi entropy and its one-loop quantum correction, after imposing a new set of monodromy conditions. The holographic classical result matches exactly with the leading contribution in the field theory up to ${e}^{\ensuremath{-}4\ensuremath{\pi}TR}$ and l6{l}^{6}l6, while the holographical one-loop contribution is in exact agreement with next-to-leading results in field theory up to ${e}^{\ensuremath{-}\frac{6\ensuremath{\pi}TR}{n}}$ and l4{l}^{4}l4 as well.
In this paper, we propose a novel expansion to compute the large interval limit of the R\'enyi entropy of 2D conformal field theory (CFT) at high temperature. Via the replica trick, the single interval R\'enyi entropy of 2D CFT at finite temperature could be read from the partition function on nnn-sheeted torus connected with each other along a branch cut. We calculate the partition function by inserting a complete basis across the branch cut. Because of the monodromy condition across the branch cut in the large interval limit, the basis of the states should be the ones in the twist sector. We study the twist sector of a general module of CFT and find that there is a one-to-one correspondence between the twist sector states and the normal sector states. As an application, we revisit the noncompact free scalar theory and discuss the large interval limit of the R\'enyi entropy of this theory by using our proposal. We find complete agreement in the leading and next-leading orders with direct expansion of the exact partition function. Moreover, we prove the relation (10) between thermal entropy and the entanglement entropy for a generic CFT with discrete spectrum.
Abstract In this paper we study the properties of two-dimensional CFTs defined by cyclic and symmetric orbifolds of free Dirac fermions, especially by focusing on the partition function and entanglement entropy. Via the bosonization, we construct the twist operators which glue two complex planes to calculate the partition function of ℤ2 orbifold CFT on a torus. We also find an expression of ℤN cyclic orbifold in terms of Hecke operators, which provides an explicit relation between the partition functions of cyclic orbifolds and those of symmetric ones. We compute the entanglement entropy and Renyi entropy in cyclic orbifolds on a circle both for finite temperature states and for time-dependent states under quantum quenches. We find that the replica method calculation is highly non-trivial and new because of the contributions from replicas with different boundary conditions. We find the full expression for the ℤ2 orbifold and show that the periodicity gets doubled. Finally, we discuss extensions of our results on entanglement entropy to symmetric orbifold CFTs and make a heuristic argument towards holographic CFTs.
We compute the N = 2 Rényi entanglement entropy of two intervals at equal time in a circle, for the theory of a 2D compact complex free scalar at finite temperature. This is carried out by performing functional integral on a genus 3 ramified cover of the torus, wherein the quantum part of the integral is captured by the four point function of twist fields on the worldsheet torus, and the classical piece is given by summing over winding modes of the genus 3 surface onto the target space torus. The final result is given in terms of a product of theta functions and certain multi-dimensional theta functions. We demonstrate the T-duality invariance of the result. We also study its low temperature limit. In the case in which the size of the intervals and of their separation are much smaller than the whole system, our result is in exact agreement with the known result for two intervals on an infinite system at zero temperature [5]. In the case in which the separation between the two intervals is much smaller than the interval length, the leading thermal corrections take the same universal form as proposed in [9, 10] for Rényi entanglement entropy of a single interval.