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Magnetic operator groups of an electron in a crystal Tam, Wing Gay

Abstract

In this thesis the problem of an electron in a crystal in the presence of a uniform magnetic field is investigated using group theory method. A group of operators [ symbol omitted ] commuting with the Hamiltonian of an electron in the presence of a uniform magnetic field and a crystal electric potential is constructed. This group is homomorphic to the group [ symbol omitted ] (a magnetic space group) of space time transformations that leave the magnetic field and the crystal electric potential invariant. The property of the subgroup [ symbol omitted ] of [ symbol omitted ] that under the above homomorphism is mapped onto the lattice [ symbol omitted ] of [ symbol omitted ] is studied in detail. It turns out the structure of [ symbol omitted ] depends on the magnitude and the orientation of the magnetic field, so that, in fact one has to deal with an infinite class of groups. In particular, it is useful to divide this infinite class of groups into two subclasses: one subclass is then referred to as corresponding to "rational" magnetic fields, the other as corresponding to "irrational" magnetic field. The group [ symbol omitted ] is a generalisation of the "magnetic translation group" recently introduced by Zak for the special case of a symmetric gauge. He also constructed "physical" irreducible representations of the "magnetic translation group" for the special case of a "rational" magnetic field. In this case a group [ symbol omitted ] always has a maximal Abelian subgroup with a finite index. (The term "physical" representation simply means a representation which can be generated by functions of spatial coordinates.) In this thesis no such restriction is introduced: the "physical" irreducible representations of [ symbol omitted ] are also constructed for the case of irrational magnetic field, in which case the index of a maximal Abelian subgroup is always infinite; the "physical" irreducible representations are then always infinite dimensional. Using a complete set of Landau functions the basis functions generating "physical" irreducible representations of [ symbol omitted ] are found for the special case when the crystal is simple cubic and the magnetic field is parallel to a lattice vector. It turns out when the field is "irrational" the basis functions are countably infinite sets of Landau functions, and the energy spectrum depends only on one of the parameters labelling the "physical" irreducible representations of [ symbol omitted ]. The problem of perturbation produced by a weak periodic potential on the Landau levels for a free electron in a magnetic field is also considered. In this connection we make plausible the validity of certain quite general selection rules for an arbitrary periodic potential.

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