Fermionic propagators for two-dimensional systems with singular interactions

We analyze the form of the fermionic propagator for two-dimensional 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, but at small N it only exists in a range O(N(2)) near the mass shell. At larger distances from the mass shell, the system evolves and G(epsilon(k),omega) becomes regular. At N=0, the range where the pole exists collapses and G(epsilon(k),omega) becomes regular everywhere.