Holographic calculation for large interval Rényi entropy at high temperature

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Abstract:In this work, we study the universal behaviors in the mutual information of two disjoint spheres in a conformal field theory (CFT). By using the operator product expansion of the spherical twist operator in terms of the conformal family, we show that the large distance expansion of the mutual information can be cast in terms of the conformal blocks. We develop the 1/n prescription to compute the coefficients before the conformal blocks. For a single conformal family, the leading nonvanishing contribution to the mutual information comes from the bilinear operators. We show that the coefficients of these operators take universal forms and such universal behavior persists in the bilinear operators with derivatives as well. Consequently the first few leading order contributions to the mutual information in CFT take universal forms. To illustrate our framework, we discuss the free scalars and free fermions in various dimensions. For the free scalars, we compute the mutual information to the next-to-leading order and find good agreement with the improved numerical lattice result. For the free fermion, we compute the leading order result, which is of universal form, and find the good match with the numerical study. Our formalism could be applied to any CFT potentially.

Section: Jhep06(2017)096mentioning

confidence: 99%

On the mutual information in conformal field theory

Chen

Hao

et al. 2017

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“…This leads to the proof of the RyuTakayanagi formula for the holographic entanglement entropy [23,24]. More interestingly, from the study on the Rényi entropy of double intervals with a small cross-ratio and the single interval on a torus [25][26][27][28][29][30][31][32][33][34], it turns out that the holographic computation is even correct at 1-loop level. It is found that the c 0 order contribution to the Rényi entropy from…”

Section: Jhep12(2015)109mentioning

confidence: 99%

“…For a general higher genus Riemann surface, this is a very difficult problem. However, for the Riemann surface resulted from the replica trick in computing the Rényi entropy, the problem has been solved explicitly in a perturbative way in the double-interval case [22,26] and single interval on a torus case [26,33].…”

Section: Schottky Uniformizationmentioning

confidence: 99%

1-loop partition function in AdS 3/CFT 2

Chen

2015

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The 1-loop partition function of the handlebody solutions in the AdS 3 gravity have been derived some years ago using the heat kernel techniques and the method of images. In the semiclassical limit, such partition function should correspond to the order O(c 0 ) part in the partition function of dual conformal field theory(CFT) on the boundary Riemann surface. The higher genus partition function could be computed by the multipoint functions in the Riemann sphere via sewing prescription. In the large central charge limit, the CFT is effectively free in the sense that to the leading order of c the multi-point function is further simplified to be a summation over the products of two-point functions of single-particle states. Correspondingly in the bulk, the graviton is freely propagating without interaction. Furthermore the product of the two-point functions may define the links, each of which is in one-to-one correspondence with the conjugacy class of the Schottky group of the Riemann surface. Moreover, the value of a link is determined by the multiplier of the element in the conjugacy class. This allows us to reproduce exactly the gravitational 1-loop partition function. The proof can be generalized to the higher spin gravity and its dual CFT.

“…Note that this formula holds as long as L − β, and was obtained by using the method of twist operators [28,30]. The long interval Rényi entropy has also been investigated in [36,37] and [29], and we just adopt the result in [29] that was obtained using conformal transformations. In the thermodynamic limit, the result is…”

Section: Canonical Ensemble Statementioning

confidence: 99%

“…For a long interval, we can still use the OPE of twist operators [36,37], and we give the calculation details in Appendix B. The long interval Rényi entropy in microcanonical ensemble state is…”

Section: Microcanonical Ensemble Statementioning

confidence: 99%

Rényi entropy at large energy density in 2D CFT

Guo

Lin

Zhang

2019

We investigate the Rényi entropy and entanglement entropy of an interval with an arbitrary length in the canonical ensemble, microcanonical ensemble and primary excited states at large energy density in the thermodynamic limit of a two-dimensional large central charge c conformal field theory. As a generalization of the recent work [Phys. Rev. Lett. 122 (2019) 041602], the main purpose of the paper is to see whether one can distinguish these various large energy density states by the Rényi entropies of an interval at different size scales, namely, short, medium and long. Collecting earlier results and performing new calculations in order to compare with and fill gaps in the literature, we give a more complete and detailed analysis of the problem. Especially, we find some corrections to the recent results for the holographic Rényi entropy of a medium size interval, which enlarge the validity region of the results. Based on the Rényi entropies of the three interval scales, we find that Rényi entropy cannot distinguish the canonical and microcanonical ensemble states for a short interval, but can do the job for both medium and long intervals. At the leading order of large c the entanglement entropy cannot distinguish the canonical and microcanonical ensemble states for all interval lengths, but the difference of entanglement entropy for a long interval between the two states would appear with 1/c corrections. We also discuss Rényi entropy and entanglement entropy differences between the thermal states and primary excited state. Overall, our work provides an up-to-date picture of distinguishing different thermal or primary states at various length scales of the subsystem.