  Over the past two decades, many studies in the Density Functional Theory
context revealed new aspects and properties of strongly correlated superfluid
quantum systems in numerous configurations that can be simulated in
experiments. This was made possible by the generalization of the Local Density
Approximation to superfluid systems by Bulgac in [Phys. Rev. C 65, 051305,
(2002), Phys. Rev. A 76, 040502, (2007)]. In the presented work, we propose an
extension of the Superfluid Local Density Approximation systematically
improvable and applicable to a large range of many-body quantum problems
getting rid of the fitting procedures of the functional parameters. It turns
out that only the knowledge of the density dependence of the quasi-particle
properties, namely, the chemical potential, the effective mass, and the pairing
gap function, are enough to obtain an explicit and accurate local functional of
the densities without any adjustment a posteriori. This opens the way toward an
Effective Field Theory formulation of the Density Functional Theory in the
sense that we obtain a universal expansion of the functional parameters
entering in the theory as a series in pairing gap function. Finally, we discuss
possible applications of the developed approach allowing precise analysis of
experimental observations. In that context, we focus our applications on the
static structure properties of superfluid vortices.
