Representations of the Invariance Group for a Bloch Electron in a Magnetic Field

Abstract

The invariance translation-operator group for the Hamiltonian of an electron in a periodic electric field and a uniform magnetic field is examined in the context of the theory of infinite-dimensional representations. It is shown that this group is of Type I if and only if the magnetic field has rational components relative to the lattice. The case of a nonrational field along a lattice vector direction is studied in detail. Here, the group is the direct product of a factor involving the translations along the field and a factor expressible as a semidirect product in a way that depends on the choice of basic lattice vector pair for the translations across the field. For each choice, Mackey's theory of induced representations is applied to obtain an infinite set of physical irreducible representations (based on both transitive and strictly ergodic measures); it is found that for different choices the sets are different unless the vector pairs are related in a simple way. The representation carried by the state space L2(R3) is decomposed into a direct integral of primary representations, using Landau functions for a basis in L2(R3). These primary representations are not of Type I, for it is shown explicitly that each has an infinite number of direct integral decompositions into irreducible representations, such that representations from any two decompositions are inequivalent. Here, different decompositions involve Landau functions for propagation along different lattice vector directions. The invariance translation-operator group for a system with a uniform magnetic field only is also discussed; it is of Type I, and its physical irreducible representations are given.