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arXiv:1610.09685 · Physics Letters B

Free compact boson on branched covering of the torus

Feihu Liu

Published May 29, 2017• 1 citation
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Abstract

We have studied the partition function of a free compact boson on a n-sheeted covering of torus gluing along m branch cuts. It is interesting because when the branched cuts are chosen to be real, the partition function is related to the n-th Rényi entanglement entropy of m disjoint intervals in a finite system at finite temperature. After proposing a canonical homology basis and its dual basis of the covering surface, we find that the partition function can be written in terms of theta functions.

DOI
10.1016/j.physletb.2017.05.058

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We have studied the partition function of a free compact boson on a n-sheeted covering of torus gluing along m branch cuts. It is interesting because when the branched cuts are chosen to be real, the partition function is related to the n-th Rényi entanglement entropy of m disjoint intervals in a finite system at finite temperature. After proposing a canonical homology basis and its dual basis of the covering surface, we find that the partition function can be written in terms of theta functions.
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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.
We compute the single-interval Rényi entropy (replica partition function) for free fermions in 1+1d at finite temperature and finite spatial size by two methods: (i) using the higher-genus partition function on the replica Riemann surface, and (ii) using twist operators on the torus. We compare the two answers for a restricted set of spin structures, leading to a non-trivial proposed equivalence between higher-genus Siegel Θ-functions and Jacobi θ-functions. We exhibit this proposal and provide substantial evidence for it. The resulting expressions can be elegantly written in terms of Jacobi forms. Thereafter we argue that the correct Rényi entropy for modular-invariant free-fermion theories, such as the Ising model and the Dirac CFT, is given by the higher-genus computation summed over all spin structures. The result satisfies the physical checks of modular covariance, the thermal entropy relation, and Bose-Fermi equivalence.
We study the Rényi entropies of N disjoint intervals in the conformal field theories given by the free compactified boson and the Ising model. They are computed as the 2N point function of twist fields, by employing the partition function of the model on a particular class of Riemann surfaces. The results are written in terms of Riemann theta functions. The prediction for the free boson in the decompactification regime is checked against exact results for the harmonic chain. For the Ising model, matrix product states computations agree with the conformal field theory result once the finite size corrections have been taken into account.
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.
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.