Hannover 2020 – wissenschaftliches Programm

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Q: Fachverband Quantenoptik und Photonik

Q 22: Posters: Quantum Optics and Photonics II

Q 22.46: Poster

Dienstag, 10. März 2020, 16:30–18:30, Empore Lichthof

Regrouping invariance of lattice systems in a rational magnetic field — Tobias Geib, •Pablo Tieben, and Reinhard F. Werner — Institut für theroetische Physik, Leibniz Universität Hannover, Hannover, Deutschland

The traditional way of analyzing lattice systems in a magnetic field is to choose the fluxes for one cell to be rational. Then a suitable periodic grouping makes the system translation invariant, and therefore susceptible to the usual methods of band structure analysis. This method works for Hamiltonian (continuous time) systems as well as discrete time systems, so called quantum walks. Typical results obtained in this way are Hofstadter*s butterfly, representing the spectrum as a function of the field parameter, and the characterization of bands by Chern numbers.

A problem with this approach is that the results depend prima facie on the chosen regrouping. E.g. in two dimensions the regrouping can always be chosen along the x- or y- axis, but also other partitions into skew parallelograms can be used.

We show that under a technical assumption, namely the existence of a periodic gauge for the chosen regrouping, not only the spectrum is independent of the regrouping, but also the isomorphism class of the vector bundles over the Brillouin zone associated with the bands. In particular, the Chern numbers mentioned earlier are invariants. The technical assumption can be satisfied for some regroupings in any lattice dimension, and for all regroupings of two dimensional lattices. We conjecture the latter is true in every dimension.