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arXiv:1501.00373 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Rényi entropy of a free compact boson on a torus

Bin Chen, Jie-qiang Wu

Published May 18, 2015• 17 citations
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Abstract

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.

DOI
10.1103/physrevd.91.105013

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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.
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.
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 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 calculate the Rényi entropy of one single interval on a circle at finite temperature in 2D CFT. In the low temperature limit, we expand the thermal density matrix level by level in the vacuum Verma module, and calculate the first few leading terms in e −π/T L explicitly. On the other hand, we compute the same Rényi entropy holographically. After considering the dependence of the Rényi entropy on the temperature, we manage to fix the interval-independent constant terms in the classical part of holographic Rényi entropy. We furthermore extend the analysis in [9] to higher orders and find exact agreement between the results from field theory and bulk computations in the large central charge limit. Our work provides another piece of evidence to support holographic computation of Rényi entropy in AdS3/CFT2 correspondence, even with thermal effect.
Using a Wigner-function-based approach, we study the Renyi entropy of a subsystem AAA of a system of bosons interacting with a local repulsive potential. The full system is assumed to be in thermal equilibrium at a temperature TTT and density $\ensuremath{\rho}$. For a U(N)\mathcal{U}(N)U(N)-symmetric model, we show that the Renyi entropy of the system in the large-NNN limit can be understood in terms of an effective noninteracting system with a spatially varying mean field potential, which has to be determined self-consistently. The Renyi entropy is the sum of two terms: (a) the Renyi entropy of this effective system and (b) the difference in thermal free energy between the effective system and the original translation-invariant system, scaled by TTT. We determine the self-consistent equation for this effective potential within a saddle-point approximation. We use this formalism to look at one- and two-dimensional Bose gases on a lattice. In both cases, the potential profile is that of a square well, taking one value in subsystem AAA and a different value outside it. The potential varies in space near the boundary of subsystem AAA on the scale of density-density correlation length. The effect of interaction on the entanglement entropy density is determined by the ratio of the potential barrier to the temperature and peaks at an intermediate temperature, while the high- and low-temperature regimes are dominated by the noninteracting answer.
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.