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 bar right arrow e( k) vanishes. In a previous paper, we have treated the case of spatial dimension d >= 3. In this paper, we focus on the more singular case 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.