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Singular perturbation theory for interacting fermions in two dimensions

We consider a system of interacting fermions in two dimensions beyond the second-order perturbation theory in the interaction. It is shown that the mass-shell singularities in the self-energy, arising already at the second order of the perturbation theory, manifest a non-perturbative effect: an interaction with the zero-sound mode. Resumming the perturbation theory for a weak, short-range interaction and accounting for a finite curvature of the fermion spectrum, we eliminate the singularities and obtain the results for the quasi-particle self-energy and the spectral function to all orders in the interaction with the zero-sound mode. A threshold for emission of zero-sound waves leads a non-monotonic variation of the self-energy with energy (or momentum) near the mass shell. Consequently, the spectral function has a kink-like feature. We also study in detail a non-analytic temperature dependence of the specific heat, $C(T)\propto T^2$. It turns out that although the interaction with the collective mode results in an enhancement of the fermion self-energy, this interaction does not affect the non-analytic term in $C(T)$ due to a subtle cancellation between the contributions from the real and imaginary parts of the self-energy. For a short-range and weak interaction, this implies that the second-order perturbation theory suffices to determine the non-analytic part of $C(T)$. We also obtain a general form of the non-analytic term in $C(T)$, valid for the case of a generic Fermi liquid, \emph{i.e.}, beyond the perturbation theory.