  Charged Renyi entropies were recently introduced as a measure of entanglement
between different charge sectors of a theory. We investigate the phase
structure of charged Renyi entropies for CFTs with a light, charged scalar
operator. The charged Renyi entropies are calculated holographically via areas
of charged hyperbolic black holes. These black holes can become unstable to the
formation of scalar hair at sufficiently low temperature; this is the
holographic superconducting instability in hyperbolic space. This implies that
the Renyi entropies can be non-analytic in the Renyi parameter n. We find the
onset of this instability as a function of the charge and dimension of the
scalar operator. We also comment on the relation between the phase structure of
these entropies and the phase structure of a holographic superconductor in flat
space.
