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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, andto determine how perturbations that couple to electron spin impact them. Wethen study two specific realizations of such systems: quantum wires ofrectangular cross-section and a rectangular slab in a magnetic field. In theformer 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 bottomsurfaces, which lead to gapless chiral states appearing on the lateralsurfaces. In the presence of a magnetic field, we examine how Landau levelstates on surfaces perpendicular to the field join onto confined states of thelateral surfaces. We show that an imbalance in the number of states propagatingin each direction on the lateral surface is sufficient to stabilize a quantizedHall effect if there are processes that equilibrate the distribution of currentamong these channels.

Electronic States of Wires and Slabs of Topological Insulators: Quantum Hall Effects and Edge Transport