Quantum Hall Conductivity in a Landau Type Model with a Realistic Geometry II

We use a mathematical framework that we introduced in a previous paper to study geometrical and quantum mechanical aspects of a Hall system with finite size and general boundary conditions. Geometrical structures control possibly the integral or fractionnal quantization of the Hall conductivity depending on the value of $NB/2\pi$ ($N$ is the number of charge carriers and $B$ is the magnetic field). When $NB/2\pi$ is irrationnal, we show that monovalued wave functions can be constructed only on the graph of a free group with two generators. When $NB/2\pi$ is rationnal, the relevant space becomes a puncturated Riemann surface. We finally discuss our results from a phenomenological viewpoint.