  Interacting Fermi gas provides an ideal model system to understand
unconventional pairing and intertwined orders relevant to a large class of
quantum materials. Rydberg-dressed Fermi gas is a recent experimental system
where the sign, strength, and range of the interaction can be controlled. The
interaction in momentum space has a negative minimum at $q_c$ inversely
proportional to the characteristic length-scale in real space, the soft-core
radius $r_c$. We show theoretically that single-component (spinless)
Rydberg-dressed Fermi gas in two dimensions has a rich phase diagram with novel
superfluid and density wave orders due to the interplay of the Fermi momentum
$p_F$, interaction range $r_c$, and interaction strength $u_0$. For repulsive
bare interactions $u_0>0$, the dominant instability is $f$-wave superfluid for
$p_Fr_c\lesssim 2$, and density wave for $p_Fr_c\gtrsim 4$. The $f$-wave
pairing in this repulsive Fermi gas is reminiscent of the conventional
Kohn-Luttinger mechanism, but has a much higher $T_c$. For attractive bare
interactions $u_0<0$, the leading instability is $p$-wave pairing. The phase
diagram is obtained from functional renormalization group that treats all
competing many-body instabilities in the particle-particle and particle-hole
channels on equal footing.
