Holographic Phases of Renyi Entropies

We consider Renyi entropies of conformal field theories in flat space, with the entangling surface being a sphere. The AdS/CFT correspondence relates this Renyi entropy to that of a black hole with hyperbolic horizon; as the Renyi parameter $n$ increases the temperature of the black hole decreases. If the CFT possesses a sufficiently low dimension scalar operator the black hole will be unstable to the development of hair. Thus, as $n$ is varied, the Renyi entropies will exhibit a phase transition at a critical value of $n$. The location of the phase transition, along with the spectrum of the reduced density matrix $\rho$, depends on the dimension of the lowest dimension non-trivial scalar operator in the theory.
Comment: 21 pages, 7 figures, v3 - References added. Neumann boundary conditions added in section 5