Cantor spectra of magnetic chain graphs

TL;DR

This paper shows that a one-dimensional magnetic chain graph can have a Cantor spectrum, especially when exposed to a magnetic field with an irrational slope, linking it to the almost Mathieu operator.

Contribution

It introduces a new example of a magnetic chain graph with a Cantor spectrum, connecting spectral properties to the almost Mathieu operator.

Findings

The spectrum can be Cantor-type under certain magnetic conditions.

The spectral behavior relates to the almost Mathieu operator.

The magnetic field's irrational slope influences the spectral structure.

Abstract

We demonstrate a one-dimensional magnetic system can exhibit a Cantor-type spectrum using an example of a chain graph with $\delta$ coupling at the vertices exposed to a magnetic field perpendicular to the graph plane and varying along the chain. If the field grows linearly with an irrational slope, measured in terms of the flux through the loops of the chain, we demonstrate the character of the spectrum relating it to the almost Mathieu operator.