Heisenberg Groups in the Theory of the Lattice Peierls Electron: the Irrational Flux Case

Abstract

It is shown that the quantum mechanics of a charged particle moving in a uniform magnetic field in the plane (Landau) or on a planar lattice (Peierls) is described in all detail by the projective representation theory of the "euclidean" group of the appropriate configuration space. In the Landau case, a detailed description of the state space as well as the determination of the correct Hamiltonian follows from the properties of the real Heisenberg group, especially the fact that it has an essentially unique irreducible representation. In the Peierls case, the corresponding groups are infinite discrete translation groups centrally extended by the circle group. For irrational flux/plaquette (in units of the flux quantum) these groups are "almost Heisenberg" in the sense that they have a distinguished irreducible representation which plays, in the quantum theory, the role of the unique representation of the real Heisenberg group. The physics is fully determined by, and is periodic in, the value of the flux/plaquette. The Hamiltonian for nearest neighbour hopping is the Harper Hamiltonian. Vector potentials are not introduced.