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We analyze the form of the fermionic propagator for 2D fermions interactingwith massless overdamped bosons. Examples include a nematic and Isingferromagnetic quantum-critical points, and fermions at a half-filled Landaulevel. Fermi liquid behavior in these systems is broken at criticality by asingular self-energy, but the Fermi surface remains well defined. These arestrong-coupling problems with no expansion parameter other than the number offermionic species, N. The two known limits, N >>1 and N=0 show qualitativelydifferent behavior of the fermionic propagator G(\epsilon_k, \omega). In thefirst limit, G(\epsilon_k, \omega) has a pole at some \epsilon_k, in the otherit is analytic. We analyze the crossover between the two limits. We show thatthe 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 systemevolves and G(\epsilon_k, \omega) becomes regular. At N=0, the range where thepole exists collapses and G(\epsilon_k, \omega) becomes regular everywhere.

Fermionic propagators for 2D systems with singular interactions