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arXiv:1510.05993

Rényi Entropy for the $\sun1$ WZW model on the torus

Howard J. Schnitzer

Published Oct 20, 2015• 4 citations
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Abstract

The $\sun1$ WZW model is constructed on a n-sheeted branched torus, which allows the investigation of the Rényi entropy for a single interval at finite temperature. The small and large interval limits, as well as the low temperature expansion are presented for this theory.

DOI
10.48550/arxiv.1510.05993

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The \sun1\sun1\sun1 WZW model is constructed on a n-sheeted branched torus, which allows the investigation of the Rényi entropy for a single interval at finite temperature. The small and large interval limits, as well as the low temperature expansion are presented for this theory.
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.
Rényi Entropy for a 2d\bf 2d2d CFT with a gauge field: SU^(N)1\bf \widehat{\rm SU}(N)_1SU(N)1​ WZW theory on a branched torus
The Rényi entropy for the SU^(N)1\widehat{\rm SU}(N)_1SU(N)1​ WZW model as described by NNN free fermions coupled to a U(1)U(1)U(1) constraint field is computed on an nnn-sheeted branched torus. The boundary condition of the harmonic component of the gauge field on the homology cycles of the genus ggg Riemann surface is central to the final result. This calculation is complementary to that of arXiv:1510.059931510.059931510.05993, which presents the bose side of the bose-fermi equivalence.
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.
The recent study of Rényi entropy opens a new window to study the AdS3/CFT2 correspondence. Here we review the latest developments in this area. In particular, we focus on the computations of the Rényi entropy in the cases of two short disjoint intervals on a complex plane and one single interval on a torus. We also discuss the large interval limit of the latter case and propose a new expansion framework in the CFT computation and a new set of monodromy conditions in the holographic computation.
In this paper, we reconsider the single-interval R\'enyi entropy of a free compact scalar on a torus. In this case, the contribution to the entropy could be decomposed into a classical part and a quantum part. The classical part includes the contribution from all the saddle points, while the quantum part is universal. After considering a different monodromy condition from the one in the literature, we reevaluate the classical part of the R\'enyi entropy. Moreover, we expand the entropy in the low-temperature limit and find the leading thermal correction term, which is consistent with the universal behavior suggested in [J. Cardy and C. P. Herzog, Phys. Rev. Lett. 112, 171603 (2014)]. Furthermore, we investigate the large-interval behavior of the entanglement entropy and show that the universal relation between the entanglement entropy and thermal entropy holds in this case.
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