Replica Phase Transition with Quantum Gravity Corrections

Motivated by bulk replica wormholes, we study the boundary effective theory that describes the near-horizon fluctuations of a near-extremal Reissner-Nordstr\"om black hole. This theory consists of a Schwarzian mode and a $U(1)$ phase mode. We compute the partition function of this boundary theory on connected geometries, from which the entropy is derived. Our analysis reveals a rich phase structure, in which the dominance of connected or disconnected configurations leads to a phase transition controlled by the temperature and the coupling constants $C$, $K$, and $\mathcal{E}$ of the 1d effective theory.