Elucidating the finite temperature quasiparticle random phase approximation

In numerous astrophysical scenarios, such as core-collapse supernovae and neutron star mergers, as in well as heavy-ion collision experiments, transitions between thermally populated nuclear excited states have been shown to play an important role. Due to its simplicity and excellent extrapolation ability, the finite-temperature quasiparticle random phase approximation (FT-QRPA) presents itself as an efficient method to study the properties of hot nuclei. The statistical ensembles in the FT-QRPA make the theory much richer than its zero-temperature counterpart, but also obscure the meaning of various physical quantities. In this work, we clarify several aspects of the FT-QRPA, including notations seen in the literature, and demonstrate how to extract physical quantities from the theory. To exemplify the correct treatment of finite-temperature transitions, we place special emphasis on the charge-exchange transitions described within the proton-neutron FT-QRPA (FT-PNQRPA). With the FT-PNQRPA built on the nuclear energy-density functional theory, we obtain solutions using a relativistic matrix approach and also the non-relativistic finite amplitude method. We show that the Ikeda sum rule is fulfilled with the proper treatment of de-excitations from thermally populated excited states. Additionally, we demonstrate the impact of these transitions on stellar electron capture (EC) rates in ${}^{58,78}$Ni. While their inclusion does not influence the EC rates in ${}^{58}$Ni, the rates in ${}^{78}$Ni are dominated by de-excitations for temperatures $T > 0.5$ MeV. In systems with a large negative $Q$-value, the inclusion of de-excitations within the FT-QRPA is necessary for a complete description of reaction rates at finite temperature.