Singular Fermi Surfaces II. The Two--Dimensional Case

We consider many–fermion systems with singular Fermi surfaces, which contain Van Hove points where the gradient of the band function $k\mapsto e(k)$ vanishes. In a previous paper, we have treated the case of spatial dimension $d \ge 3$\mathrm{.}Inthispaper,wefocusonthemoresingularcase$d=2$ and establish properties of the fermionic self–energy to all orders in perturbation theory. We show that there is an asymmetry between the spatial and frequency derivatives of the self–energy. The derivative with respect to the Matsubara frequency diverges at the Van Hove points, but, surprisingly, the self–energy is $C^1$ in the spatial momentum to all orders in perturbation theory, provided the Fermi surface is curved away from the Van Hove points. In a prototypical example, the second spatial derivative behaves similarly to the first frequency derivative. We discuss the physical significance of these findings.