Instabilities of 2d quantum critical metals in the Nf->0 limit

We study a Fermi surface coupled to fluctuations of a critical order parameter in 1+2 dimensions. The limit of vanishing fermion flavors gives a controlled way of studying this strongly coupled theory. We show how to calculate low-energy fermion n-point functions. We find that certain four-fermion correlators grow exponentially with spatial separation indicating that these theories have an instability in the limit of 0 fermion flavors. We show that these instabilities survive when some of the Nf corrections are added.