Pairing transitions in the finite-temperature relativistic Hartree-Bogoliubov theory

We formulate the finite-temperature relativistic Hartree-Bogoliubov theory for spherical nuclei based on a point-coupling functional, with the Gogny or separable pairing force. Using the functional PC-PK1, the framework is applied to the study of pairing transitions in Ca, Ni, Sn, and Pb isotopic chains. The separable pairing force reproduces the gaps calculated with the Gogny force not only at zero temperature, but also at finite temperatures. By performing a systematic calculation of the even-even Ca, Ni, Sn, and Pb isotopes, it is found that the critical temperature for a pairing transition generally follows the rule $T_c =0.6 \Delta_n(0)$, where $\Delta_n(0)$ is the neutron pairing gap at zero temperature. This rule is further verified by adjusting the pairing gap at zero temperature with a strength parameter.