Stochastic Behaviour of the Magnetisation in a Classically Integrable Quantum System

We present a two-dimensional model of noninteracting electrons in an external magnetic field. This model is of common use in the study of the Hall effect. In the classical limit the system is integrable. Nevertheless at the quantum level, the magnetisation as a function of the number of electrons exhibits a chaotic behaviour. The origin of this phenomenon can be traced back to the "quasi-degeneracy" of the energy levels. The susceptibility is, on the contrary, much less sensitive to this quasi-degeneracy and exhibits a more regular behaviour. Interestingly these phenomena show up for electronic densities and system sizes characteristic of mesoscopic systems. It may be therefore possible to observe them in real experiments.