  We investigate a model many-body system of spinless Fermi gas in two
dimensions, where the bare two-body interaction is repulsive and takes the form
of a soft-core disk potential. We obtain the zero temperature phase diagram of
this model by numerical functional renormalization group (FRG), which retains
the effective interaction vertices in all channels to provide a detailed
picture of how Cooper pairing emerges under the renormalization flow. The
repulsion drives the system to a series of superfluid states with higher
angular momentum paring, for example in the $f$- and $h$-wave channels instead
of the $p$-wave channel. This is in sharp contrast to the original
Kohn-Luttinger mechanism where pairing of very large angular momenta and
exponentially small transition temperature was predicted. We trace the
stabilization and enhancement of $f$- and $h$-wave pairing back to the momentum
dependence of the bare interaction. A perturbative calculation is carried out
to show that while the second order Kohn-Luttinger diagrams provide a
qualitative understanding of the onsets of the various superfluid phases, they
are unable to accurately capture the phase boundaries predicted by FRG. Our
findings suggest that tuning the shape of the interaction potential offers a
promising route to achieve stronger ``pairing glue" and to realize nontrivial
superfluid phases in repulsive Fermi gases beyond the scope of the original
Kohn-Luttinger analysis.
