Optimal superconductivity near a Lifshitz transition in strained (La,Pr)$_3$Ni$_2$O$_7$

High-temperature superconductivity in pressurized and strained bilayer nickelates (La,Pr)$_3$Ni$_2$O$_7$ has emerged as a new frontier. One of the key unresolved issues concerns the fermiology that underlies superconductivity. On both theoretical and experimental sides, no general consensus has been reached, and conflicting results exist regarding whether the relevant Fermi surface involves a hole $γ$ pocket, or not. Here, we address this issue by unveiling a Janus-faced role of the $γ$ pocket in spin-fluctuation-mediated superconductivity. We show that this pocket simultaneously induces dominant pair-breaking and pair-forming fluctuations for the leading $s_\pm$-wave pairing. Consequently, an optimal superconducting transition temperature $T_\mathrm{c}$ is achieved when the $γ$ pocket surfaces at the Fermi level, placing the system near a Lifshitz transition. This suggests that superconductivity can emerge provided the maximum energy level of the $γ$ pocket lies sufficiently close to the Fermi level, either from below or above. Our finding not only reconciles two opposing viewpoints on the fermiology, but also naturally explains recent experiments on (La,Pr)$_3$Ni$_2$O$_7$ thin films, including the superconductivity under compressive strain, two conflicting measurements on the Fermi surface, and the dome shape of $T_\mathrm{c}$ as a function of hole doping.