Dirac Fermions and Topological Phases in Magnetic Topological Insulator Films

We develop a Dirac fermion theory for topological phases in magnetic topological insulator films. The theory is based on exact solutions of the energies and the wave functions for an effective model of the three-dimensional topological insulator (TI) film. It is found that the TI film consists of a pair of massless or massive Dirac fermions for the surface states, and a series of massive Dirac fermions for the bulk states. The massive Dirac fermion always carries zero or integer quantum Hall conductance when the valence band is fully occupied while the massless Dirac fermion carries a one-half quantum Hall conductance when the chemical potential is located around the Dirac point for a finite range. The magnetic exchange interaction in the magnetic layers in the film can be used to manipulate either the masses or chirality of the Dirac fermions and gives rise to distinct topological phases, which cover the known topological insulating phases, such as quantum anomalous Hall effect, quantum spin Hall effect and axion effect, and also the novel topological metallic phases, such as half quantized Hall effect, half quantum mirror Hall effect, and metallic quantum anomalous Hall effect.