Microscopic derivation of density functional theory for superfluid systems based on effective action formalism

Density-functional theory for superfluid systems is developed in the framework of the functional renormalization group based on the effective action formalism. We introduce the effective action for the particle-number and nonlocal pairing densities and demonstrate that the Hohenberg-Kohn theorem for superfluid systems is established in terms of the effective action. The flow equation for the effective action is then derived, where the flow parameter runs from $0$ to $1$, corresponding to the non-interacting and interacting systems. From the flow equation and the variational equation that the equilibrium density satisfies, we obtain the exact expression for the Kohn-Sham potential generalized to including the pairing potentials. It is shown that our Kohn-Sham potential gives the ground-state energy of the Hartree-Fock-Bogoliubov theory by neglecting the correlations.