Thermodynamics of two-dimensional conformal field theory dual to black holes

In this note we investigate the first law of thermodynamics of the two-dimensional conformal field theory (CFT) that is dual to black holes. We start from the Cardy formula and get the CFT thermodynamics with minimal reasonable assumptions. We use both the microcanonical ensemble and canonical ensemble versions of the Cardy formula. In the black hole/CFT correspondence the black hole is dual to a CFT with excitations, and the black hole mass $M$ and charge $N$ correspond to the energy and charge of the excited CFT. The CFT left- and right-moving central charges $c_{L,R}$ should be quantized, and so we assume that they are mass-independent. Also we assume the difference of the left- and right-moving sector levels $N_L-N_R$ is mass-independent dual to level matching condition. The thermodynamics of two-dimensional CFT we get is universal and supports the thermodynamics method of black hole/CFT correspondence.Comment: 10 page

User Cooperation in Wireless Powered Communication Networks

This paper studies user cooperation in the emerging wireless powered communication network (WPCN) for throughput optimization. For the purpose of exposition, we consider a two-user WPCN, in which one hybrid access point (H-AP) broadcasts wireless energy to two distributed users in the downlink (DL) and the users transmit their independent information using their individually harvested energy to the H-AP in the uplink (UL) through time-division-multiple-access (TDMA). We propose user cooperation in the WPCN where the user which is nearer to the H-AP and has a better channel for DL energy harvesting and UL information transmission uses part of its allocated UL time and DL harvested energy to help to relay the far user's information to the H-AP, in order to achieve more balanced throughput optimization. We maximize the weighted sum-rate (WSR) of the two users by jointly optimizing the time and power allocations in the network for both wireless energy transfer in the DL and wireless information transmission and relaying in the UL. Simulation results show that the proposed user cooperation scheme can effectively improve the achievable throughput in the WPCN with desired user fairness.Comment: 7 figure

On short interval expansion of R\'enyi entropy

R\'enyi entanglement entropy provides a new window to study the AdS/CFT correspondence. In this paper we consider the short interval expansion of R\'enyi entanglement entropy in two-dimensional conformal field theory. This amounts to do the operator product expansion of the twist operators. We focus on the vacuum Verma module and consider the quasiprimary operators constructed from the stress tensors. After obtaining the expansion coefficients of the twist operators to level 6 in vacuum Verma module, we compute the leading contributions to the R\'enyi entropy, to order 6 in the short interval expansion. In the case of one short interval on cylinder, we reproduce the first several leading contributions to the R\'enyi entropy. In the case of two short disjoint intervals with a small cross ratio $x$, we obtain not only the classical and 1-loop quantum contributions to the R\'enyi entropy to order $x^6$, both of which are in perfect match with the ones found in gravity, but also the leading $1/c$ contributions, which corresponds to 2-loop corrections in the bulk.Comment: V1, 23 pages, 5 figures; V2, published version, typos corrected, references adde

Thermodynamics in Black-hole/CFT Correspondence

The area law of Bekenstein-Hawking entropy of the black hole suggests that the black hole should have a lower-dimensional holographic description. It has been found recently that a large class of rotating and charged black holes could be holographically described a two-dimensional (2D) conformal field theory (CFT). We show that the universal information of the dual CFT, including the central charges and the temperatures, is fully encoded in the thermodynamics laws of both outer and inner horizons. These laws, characterizing how the black hole responds under the perturbation, allows us to read different dual pictures with respect to different kinds of perturbations. The remarkable effectiveness of this thermodynamics method suggest that the inner horizon could play a key role in the study of holographic description of the black hole.Comment: 8 pages. Essay awarded honourable mention in the Gravity Research Foundation 2013 Awards for Essays on Gravitatio