Analytic critical points of charged Renyi entropies from hyperbolic black holes

We analytically study phase transitions of holographic charged Renyi entropies in two gravitational systems dual to the $\mathcal{N}=4$ super-Yang-Mills theory at finite density and zero temperature. The first system is the Reissner-Nordstrom-AdS$_5$ black hole, which has finite entropy at zero temperature. The second system is a charged dilatonic black hole in AdS$_5$, which has zero entropy at zero temperature. Hyperbolic black holes are employed to calculate the Renyi entropies with the entangling surface being a sphere. We perturb each system by a charged scalar field, and look for a zero mode signaling the instability of the extremal hyperbolic black hole. Zero modes as well as the leading order of the full retarded Green's function are analytically solved for both systems, in contrast to previous studies in which only the IR (near horizon) instability was analytically treated.