Spectra of periodic quantum graphs: some unusual features
Seminars
Joint Seminar “Modern Mathematical Physics” and “Mathematical Problems of Theoretical Physics”
Date and Time: Thursday, 27 September 2018, at 2:30 PM
Venue: Blokhintsev Hall (4th floor), Bogoliubov Laboratory of Theoretical Physics
Seminar topic: “Spectra of periodic quantum graphs: some unusual features”
Speaker: Pavel Exner (Doppler Institute for Mathematical Physics and Applied Mathematics, Prague)
Abstract:
Spectra of periodic quantum systems are usually expected to be absolutely continuous, consisting of bands and gaps, the number of the latter being determined by the dimensionality. Our aim is to show that if the systems in question are quantum graphs, some unusual spectral features may arise. Using simple examples, we show that the spectrum may then have a pure point or a fractal character, and also that it may have only a finite but nonzero number of open gaps. Furthermore, motivated by recent attempts to model the anomalous Hall effect, we investigate a class of vertex couplings that violate the time reversal invariance. We will find spectra of lattice graphs with the simplest coupling of this type, the one with `maximum’ non invariance, and demonstrate that it depends substantially on the lattice topology. Finally, we will discuss an interpolation between this “maximal” coupling and the usual Δ-type one.