We develop a simple model of surface states for topological insulators,
developing matching relations for states on surfaces of different orientations.
The model allows one to write simple Dirac Hamiltonians for each surface, and
to determine how perturbations that couple to electron spin impact them. We
then study two specific realizations of such systems: quantum wires of
rectangular cross-section and a rectangular slab in a magnetic field. In the
former case we find a gap at zero energy due to the finite size of the system.
This can be removed by application of exchange fields on the top and bottom
surfaces, which lead to gapless chiral states appearing on the lateral
surfaces. In the presence of a magnetic field, we examine how Landau level
states on surfaces perpendicular to the field join onto confined states of the
lateral surfaces. We show that an imbalance in the number of states propagating
in each direction on the lateral surface is sufficient to stabilize a quantized
Hall effect if there are processes that equilibrate the distribution of current
among these channels.Comment: 14 pages, 9 figures include
