Singular Fermi Surfaces II. The Two--Dimensional Case

TL;DR

This paper analyzes two-dimensional fermionic systems with Van Hove singularities, revealing that the self-energy's frequency derivative diverges at these points while the spatial derivative remains smooth, highlighting unique anisotropic properties.

Contribution

It establishes that in 2D systems with Van Hove points, the self-energy's frequency derivative diverges, whereas the spatial derivative remains bounded, extending previous 3D results to the more singular 2D case.

Findings

Frequency derivative of self-energy diverges at Van Hove points.

Self-energy is $C^1$ in spatial momentum away from Van Hove points.

Spatial derivatives behave similarly to frequency derivatives in certain cases.

Abstract

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.