Singular continuous Cantor spectrum for magnetic quantum walks

TL;DR

This paper investigates the spectral properties of a 2D quantum walk under a magnetic field, revealing a transition from band spectrum to a zero-measure Cantor set depending on the rationality of magnetic flux.

Contribution

It demonstrates that the spectrum becomes a Cantor set with no pure point part when the magnetic flux ratio is irrational, extending understanding of quantum walk spectra.

Findings

For rational flux, the spectrum consists of bands.

For irrational flux, the spectrum is a zero-measure Cantor set.

Spectral measures lack pure point components in the irrational case.

Abstract

In this note, we consider a physical system given by a two-dimensional quantum walk in an external magnetic field. In this setup, we show that both the topological structure as well as its type depend sensitively on the value of the magnetic flux $\Phi$: while for $\Phi/(2{\pi})$ rational the spectrum is known to consist of bands, we show that for $\Phi/(2{\pi})$ irrational the spectrum is a zero-measure Cantor set and the spectral measures have no pure point part.