Charged Rényi entropies and holographic superconductors

Charged Rényi entropies were recently introduced as a measure of entanglement between different charge sectors of a theory. We investigate the phase structure of charged Rényi entropies for CFTs with a light, charged scalar operator. The charged Rényi 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 Rényi entropies can be non-analytic in the Rényi 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.