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Singular continuous Cantor spectrum for magnetic quantum walks. (English) Zbl 1445.81025

Summary: 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 \(\Phi : \) 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.

MSC:

81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81V10 Electromagnetic interaction; quantum electrodynamics
78A30 Electro- and magnetostatics
35P05 General topics in linear spectral theory for PDEs
82D25 Statistical mechanics of crystals
82D37 Statistical mechanics of semiconductors
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
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