Singular continuous Cantor spectrum for magnetic quantum walks

摘要

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 and its type depend sensitively on the value of the magnetic flux : While for /(2p) rational the spectrum is known to consist of bands, we show that for /(2p) irrational, the spectrum is a zero-measure Cantor set and the spectralmeasures have no pure point part.