We study a model in 2+1 dimensions composed of a Fermi surface of N-f flavors of fermions coupled to scalar fluctuations near quantum critical points (QCPs). The N-f -> 0 limit allows us to non-perturbatively calculate the long-range behavior of fermion correlation functions. We use this to calculate charge, spin and pair susceptibilities near different QCPs at zero and finite temperatures, with zero and finite order parameter gaps. While fluctuations smear out the fermionic quasiparticles, we find QCPs where the overall effect of fluctuations leads to enhanced pairing. We also find QCPs where the fluctuations induce spin and charge density wave instabilities for a finite interval of order parameter fluctuation gaps at T = 0. We restore a subset of the diagrams suppressed in the N-f -> 0 limit, all diagrams with internal fermion loops with at most 2 vertices, and find that this does not change the long-range behavior of correlators except right at the QCPs.

We study a model in 1 + 2 dimensions composed of a spherical Fermi surface of N-f flavors of fermions coupled to a massless scalar. We present a framework to non-perturbatively calculate general fermion n-point functions of this theory in the limit N-f -> 0 followed by k(F) -> infinity where k(F) sets both the size and curvature of the Fermi surface. Using this frame-work we calculate the zero-temperature fermion density-density correlation function in real space and find an exponential decay of Friedel oscillations.