Periodic quantum graphs: not always what a common wisdom would suggest
Spectral Methods in Mathematical Physics
08 February 15:00 - 16:00
Pavel Exner - Nuclear Physics Institute ASCR
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. While it is often the case, my aim here is to show that if the systems in question are quantum graphs, many different situations 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, and discuss some consequences of this property.
University of Zurich, UZH