Singular Fermi Surfaces I. General Power Counting and Higher Dimensional Cases

TL;DR

This paper proves that in three or more dimensions, systems with singular Fermi surfaces containing Van Hove points still exhibit regular self-energy behavior under perturbation, maintaining quasiparticle properties.

Contribution

It extends overlapping loop bounds to singular Fermi surfaces with Van Hove points in dimensions d ≥ 3 under a no-nesting condition, ensuring regularity of the self-energy.

Findings

Self-energy is continuously differentiable in frequency and momentum.

Quasiparticle weight and Fermi velocity remain close to noninteracting values.

Regularity results hold despite Van Hove singularities in higher dimensions.

Abstract

We prove regularity properties of the self-energy, to all orders in perturbation theory, for systems with singular Fermi surfaces which contain Van Hove points where the gradient of the dispersion relation vanishes. In this paper, we show for spatial dimensions $d \ge 3$ that despite the Van Hove singularity, the overlapping loop bounds we proved together with E. Trubowitz for regular non--nested Fermi surfaces [J. Stat. Phys. 84 (1996) 1209] still hold, provided that the Fermi surface satisfies a no-nesting condition. This implies that for a fixed interacting Fermi surface, the self-energy is a continuously differentiable function of frequency and momentum, so that the quasiparticle weight and the Fermi velocity remain close to their values in the noninteracting system to all orders in perturbation theory. In a companion paper, we treat the more singular two-dimensional case.