Spin-orbit coupled Fermi superfluid on a spherical shell with a phase transition in the BCS-BEC crossover

We consider the ground-state properties of a two-component Fermi superfluid with attractive pairing interactions and isotropic spin-orbit coupling (SOC) on a thin spherical shell. The presence of SOC changes the underlying dispersions and suppresses the chemical potential on the BCS side when the pairing is weak. Further increases of the SOC result in negative chemical potential for the whole BCS-BEC crossover. The compressibility of the Fermi superfluid with SOC shows a kink within the BCS-BEC crossover, which indicates a phase transition. An exotic feature of the transition is that the gap function remains finite throughout the crossover, thereby the Fermi superfluid retains its order parameter while exhibiting signatures of a thermodynamic phase transition.