Fermionic propagators for 2D systems with singular interactions

We analyze the form of the fermionic propagator for 2D fermions interacting with massless overdamped bosons. Examples include a nematic and Ising ferromagnetic quantum-critical points, and fermions at a half-filled Landau level. Fermi liquid behavior in these systems is broken at criticality by a singular self-energy, but the Fermi surface remains well defined. These are strong- coupling problems with no expansion parameter other than the number of fermionic species, $N$. The two known limits, $N >>1$ and $N=0$ show qualitatively different behavior of the fermionic propagator $G(\epsilon_k, \omega)$. In the first limit, $G(\epsilon_k, \omega)$ has a pole at some $\epsilon_k$, in the other it is analytic. We analyze the crossover between the two limits. We show that the pole survives for all $N$, with residue $Z = O(1)$, however at small $N$ it only exists in a range $O(N^2)$. At $N=0$, the range collapses and the behavior of $G (\epsilon_k, \omega)$ becomes analytic.