Spectral analysis near a Dirac type crossing in a weak non-constant magnetic field

TL;DR

This paper investigates how a weak, nearly constant magnetic field affects the spectral properties near a Dirac crossing in 2D Bloch eigenvalues, revealing gap formations similar to Landau levels of a magnetic Dirac operator.

Contribution

It provides a detailed spectral analysis of 2D Bloch eigenvalues near a Dirac crossing under a weak magnetic field, extending previous work on the Peierls substitution.

Findings

Spectral gaps form near the Dirac crossing in a weak magnetic field.

The gaps resemble Landau levels of a massless magnetic Dirac operator.

The study enhances understanding of magnetic effects on band crossings.

Abstract

This is the last paper in a series of three in which we have studied the Peierls substitution in the case of a weak magnetic field. Here we deal with two $2d$ Bloch eigenvalues which have a conical crossing. It turns out that in the presence of an almost constant weak magnetic field, the spectrum near the crossing develops gaps which remind of the Landau levels of an effective mass-less magnetic Dirac operator.