Cantor spectrum of graphene in magnetic fields

TL;DR

This paper models graphene in magnetic fields using quantum graphs and demonstrates that with irrational magnetic flux, the spectrum forms a Cantor set of measure zero, revealing complex spectral properties.

Contribution

It provides a complete spectral analysis of graphene in magnetic fields for all flux values, highlighting the Cantor spectrum in the irrational flux case.

Findings

Continuous spectrum is a Cantor set of Lebesgue measure zero for irrational flux.

Spectrum analysis applies to all constant magnetic fluxes.

The model captures complex spectral phenomena in graphene under magnetic influence.

Abstract

We consider a quantum graph as a model of graphene in magnetic fields and give a complete analysis of the spectrum, for all constant fluxes. In particular, we show that if the reduced magnetic flux $\Phi/2\pi$ through a honeycomb is irrational, the continuous spectrum is an unbounded Cantor set of Lebesgue measure zero.