Phase Structure of Higher Spin Black Hole
Bin Chen, Jiang Long, Yi-Nan Wang
TL;DR
The paper develops a universal thermodynamic framework for AdS3 higher-spin black holes by deriving an entropy formula from dimensional analysis and the first law, then applying it to spin-3 and spin-tilde-4 cases. Solving the holonomy equations yields multiple branches of solutions, with detailed entropy functions and phase diagrams that reproduce known spin-3 results and reveal charge-conjugation asymmetries in the spin-tilde-4 case. The work demonstrates that BTZ-like behavior is confined to low temperatures, while high-temperature phases explore a rich landscape of branches and transitions, providing a foundation for studying larger higher-spin algebras and RG-flow between CFT fixed points. These insights advance understanding of HS/CFT thermodynamics and offer a tractable path to exploring phase structures in more complex higher-spin gravities.
Abstract
In this paper, we investigate the phase structures of the black holes with one single higher spin hair, focusing specifically on the spin 3 and spin tilde 4 black holes. Based on dimensional analysis and the requirement of having consistent thermodynamics, we derive an universal formula relating the entropy and the conserved charges for arbitrary AdS3 higher spin black holes. Then we use it to study the phase structure of the higher spin black holes. We find that there are six branches of solutions in the spin 3 gravity, eight branches of solutions in the spin tilde 4 gravity and twelve branches of solutions in the G2 gravity. In each case, all branches are related by a simple angle shift in the entropy functions. In the spin 3 case, we reproduce all the results found before. In the spin tilde 4 case, we find that in the low temperature it is at the BTZ branch while in the high temperature it transits to one of two other branches, depending on the signature of the chemical potential, a reflection of charge conjugate asymmetry found before.