Honeycomb structures in magnetic fields

TL;DR

This paper analyzes the spectral and transport properties of honeycomb lattice models in magnetic fields, revealing fractal spectra, Dirac cones, and disorder effects relevant to quantum Hall phenomena.

Contribution

It provides a comprehensive spectral analysis of honeycomb lattices in magnetic fields, including fractal spectra, Dirac cones, and disorder-induced localization, advancing understanding of quantum Hall effects.

Findings

Discovery of a fractal Cantor spectrum for irrational flux

Proof of zero energy Dirac cones at rational flux

Identification of mobility edges under disorder

Abstract

We consider reduced-dimensionality models of honeycomb lattices in magnetic fields and report results about the spectrum, the density of states, self-similarity, and metal/insulator transitions under disorder. We perform a spectral analysis by which we discover a fractal Cantor spectrum for irrational magnetic flux through a honeycomb, prove the existence of zero energy Dirac cones for each rational flux, obtain an explicit expansion of the density of states near the conical points, and show the existence of mobility edges under Anderson-type disorder. Our results give a precise description of de Haas-van Alphen and Quantum Hall effects, and provide quantitative estimates on transport properties. In particular, our findings explain experimentally observed asymmetry phenomena by going beyond the perfect cone approximation.