Singular Fermi Surfaces II. The Two--Dimensional Case

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.

68 pages LaTeX with figures

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• Quantum criticality of reconstructing Fermi surfaces in antiferromagnetic metals
• High-order Van Hove singularities and their connection to flat bands
• Lifshitz transitions and crystallization of fully-polarised dipolar Fermions in an anisotropic 2D lattice
• Renormalization in Condensed Matter: Fermionic Systems - from Mathematics to Materials
• Low-Energy Effective Theory at a Quantum Critical Point of the Two-Dimensional Hubbard Model: Mean-Field Analysis
• Singular Fermi Surfaces I. General Power Counting and Higher Dimensional Cases
• Fermions in $(1+2)$-dimensions modified by nonminimal coupling and its applications to condensed matter physics
• Honeycomb Hubbard Model at van Hove Filling