Coherence, incoherence, and correlations of electrons in magnetic systems

This thesis describes three model calculations involving magnetic fields. In the first, the magnetoresistivity of a magnetic-breakdown linked-orbit network with a stochastic distribution of defects is computed within a semiclassical approach that uses the Boltzmann equation. The magnetoresistivity is shown to have a pathological dependence on the defect distribution for a network with a one-dimensional topology. The second calculation involves a two-dimensional gas of fermions in a uniform magnetic field. In contrast to the first model, the magnetic field is strong enough to realize the extreme quantum limit. The Hamiltonian describing the system may be solved exactly, even when attractive harmonic interactions between each pair of fermions are included. These interactions break the degeneracy of the free-particle Landau levels, giving rise to oscillations of the Fermi energy that are periodic in the reciprocal of the magnetic field strength. The third model is an approach to the problem of electronic correlations in metals. If only a small number of high-symmetry k-points in the Brillouin zone are considered, a Hamiltonian including both band-structure effects and the two-body Coulomb interaction may be solved exactly; this is equivalent to considering a small cluster with periodic boundary conditions