Landau quantum systems: an approach based on symmetry

Abstract

We show that the Landau quantum systems (or integer quantum Hall effect systems) in a plane, sphere or a hyperboloid, can be explained in a complete meaningful way from group-theoretical considerations concerning the symmetry group of the corresponding configuration space. The crucial point in our development is the role played by locality and its appropriate mathematical framework, the fiber bundles. In this way the Landau levels can be understood as the local equivalence classes of the symmetry group. We develop a unified treatment that supplies the correct geometric way to recover the planar case as a limit of the spherical or the hyperbolic quantum systems when the curvature goes to zero. This is an interesting case where a contraction procedure gives rise to nontrivial cohomology starting from a trivial one. We show how to reduce the quantum hyperbolic Landau problem to a Morse system using horocyclic coordinates. An algebraic analysis of the eigenvalue equation allow us to build ladder operators which can help in solving the spectrum under different boundary conditions.