Holographic Rényi entropy for CFT with W symmetry

Self Cite

We evaluate one-interval Rényi entropy and entanglement entropy for the excited states of two-dimensional conformal field theory (CFT) on a cylinder, and examine their differences from the ones for the thermal state. We assume the interval to be short so that we can use operator product expansion (OPE) of twist operators to calculate Rényi entropy in terms of sum of one-point functions of OPE blocks. We find that the entanglement entropy for highly excited state and thermal state behave the same way after appropriate identification of the conformal weight of the state with the temperature. However, there exists no such universal identification for the Rényi entropy in the short-interval expansion. Therefore, the highly excited state does not look thermal when comparing its Rényi entropy to the thermal state one. As the Rényi entropy captures the higher moments of the reduced density matrix but the entanglement entropy only the average, our results imply that the emergence of thermality depends on how refined we look into the entanglement structure of the underlying pure excited state.

Section: Jhep11(2016)116mentioning

confidence: 99%

Thermality and excited state Rényi entropy in two-dimensional CFT

Lin

Wang

Zhang

2016

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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%

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.

“…Another is to the case of higher spin gravity, which is the case that we focus on here, in particular its 3d version and corresponding 2d CFT dual. The study of entanglement entropy in higher spin theories was initiated in [7,8] and further work appears in [9][10][11][12][13][14][15][16][17]. Our two main goals are the following: first to attempt to derive from first principles the prescriptions advanced in [7,8], 1 and second to extend these considerations to the case of Rényi entropy in higher spin theories.…”

Section: Jhep07(2015)168 1 Introductionmentioning

confidence: 99%

Higher spin entanglement and W N $$ {\mathcal{W}}_{\mathrm{N}} $$ conformal blocks

Boer

Castro

Hijano

et al. 2015

158

Two-dimensional conformal field theories with extended W-symmetry algebras have dual descriptions in terms of weakly coupled higher spin gravity in AdS 3 at large central charge. Observables that can be computed and compared in the two descriptions include Rényi and entanglement entropies, and correlation functions of local operators. We develop techniques for computing these, in a manner that sheds light on when and why one can expect agreement between such quantities on each side of the duality. We set up the computation of excited state Rényi entropies in the bulk in terms of Chern-Simons connections, and show how this directly parallels the CFT computation of correlation functions. More generally, we consider the vacuum conformal block for general operators with ∆ ∼ c. When two of the operators obey ∆ c ≪ 1, we show by explicit computation that the vacuum conformal block is computed by a bulk Wilson line probing an asymptotically AdS 3 background with higher spin fields excited, the latter emerging as the effective bulk description of the excited state produced by the heavy operators. Among other things, this puts a previous proposal for computing higher spin entanglement entropy via Wilson lines on firmer footing, and clarifies its relation to CFT. We also study the corresponding computation in Toda theory and find that this provides yet another independent way to arrive at the same result.