Interacting fermions in two dimensions: Beyond the perturbation theory

We consider a system of 2D fermions with a short-range interaction. A straightforward perturbation theory is shown to be ill defined even for an infinitesimally weak interaction, as the perturbative series for the self-energy diverges near the mass shell. We show that the divergences result from the interaction of fermions with the zero-sound collective mode. By resumming the most divergent diagrams, we obtain a closed form of the self-energy near the mass shell. The spectral function exhibits a threshold feature at the onset of the emission of the zero-sound waves. We also show that the interaction with the zero sound does not affect a nonanalytic, T² part of the specific heat.