Singular continuous Cantor spectrum for magnetic quantum walks

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 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.

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Notes

  1. 1.

    One can also overcome this difficulty in the unitary setting by using the holomorphic functional calculus for completely non-unitary contractions instead of the concrete construction of a decoupling used here [47].

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Acknowledgements

C. Cedzich acknowledges support by the project PIA-GDN/QuantEx P163746-484124 and by DGE – Ministère de l’Industrie. J. Fillman thanks Svetlana Jitomirskaya for helpful conversations. T. Geib acknowledge support from the DFG SFB 1227 DQmat. A. H. Werner thanks the VILLUM FONDEN for its support with a Villum Young Investigator Grant (Grant No. 25452) and its support via the QMATH Centre of Excellence (Grant No. 10059).

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Cedzich, C., Fillman, J., Geib, T. et al. Singular continuous Cantor spectrum for magnetic quantum walks. Lett Math Phys 110, 1141–1158 (2020). https://doi.org/10.1007/s11005-020-01257-1

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Keywords

  • Quantum walks
  • Discrete electromagnetism
  • Spectral theory
  • Singular continuous spectrum
  • Cantor spectrum

Mathematics Subject Classification

  • 37N20
  • 81Q10
  • 81S99