Short interval expansion of Rényi entropy on torus

Abstract

Abstract:We investigate the short interval expansion of the Rényi entropy for twodimensional conformal field theory (CFT) on a torus. We require the length of the interval 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 AdS 3 gravity and its cousins. At both low and high temperature limits, we read the Rényi entropies to order 6 , 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.

Thermality and excited state Rényi entropy in two-dimensional CFT

“…After that, one can obtain the short-interval expansion of the Rényi entropy by evaluating the corresponding one-point or multi-point functions of the OPE blocks. This trick has been adopted to calculate the Rényi entropy for some special cases [11][12][13][14][15][16][17][18][19][20], such as the Rényi entropy of two intervals on complex plane and of the one-interval on torus. In this paper, we adopt the same trick to investigate the short-interval expansion of excited state Rényi entropy on a cylinder, and then compare our results with the thermal state Rényi entropy.…”

“…This is the case considered in this paper and the setup is shown in figure 1. The results will then depend on the details of the CFT, e.g., on the spectrum and structure constants [20,[27][28][29][30]. In this case, we need to find some ways to obtain the approximate results.…”

“…Our description will be brief as the method is in parallel with the details in [20] for Rényi entropy on torus. We will then review the results in [20] for latter comparisons.…”

“…with · · · φ := φ| · · · |φ cyl , c n is the normalization of the twist operators, and the coefficients b's are given as follows [20] …”

Holographic description of 2D conformal block in semi-classical limit

“…On the CFT side one can use the operator product expansion to do short interval expansion and get the subleading terms [42,43,46]. Naively the next leading order contribution should correspond to the quantum correction around the cosmic brane.…”