Crossover from Fermi to Non-Fermi Liquid in Two-Dimensional Interacting Fermions

Self-energy at zero temperature is investigated up to the third-order of interaction using one-patch model in two dimensions, whose interaction process corresponds to $g_4$-process of $g$-ology model in one dimension. The self-energy ${\mathrm{\Sigma }}^R(\text{mib}k,ϵ)$ diverges at $ϵ={\xi }_k$, and the contribution from the particle-hole process in third-order self-energy diagrams has a stronger divergence compared to the one from the particle-particle process. This implies that the loop-cancellation in the forward scattering is insufficient due to the effect of the warping of the Fermi surface. The strong energy dependence of the self-energy in the vicinity of $ϵ={\xi }_k$ implies the existence of the crossover from Fermi to non-Fermi liquid behavior as the momentum becomes away from the Fermi momentum, and this crossover is enhanced as interaction becomes stronger.