Singular Fermi Surfaces II. The Two--Dimensional Case

Details Singular Fermi Surfaces II. The Two--Dimensional Case Singular Fermi Surfaces II. The Two--Dimensional Case

2007-06-12

arxiv.org/abs/0706.1788v1

Physics

Mathematical Physics

68 pages LaTeX with figures

Scientific paper

10.1142/S0129055X08003304

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 $d \ge 3$. 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.

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