Quantum anomalous Hall effect in magnetic topological insulators

The search for topologically non-trivial states of matter has become an important goal for condensed matter physics. Here, we give a theoretical introduction to the quantum anomalous Hall (QAH) effect based on magnetic topological insulators in two-dimensions (2D) and three-dimensions (3D). In 2D topological insulators, magnetic order breaks the symmetry between the counter-propagating helical edge states, and as a result, the quantum spin Hall effect can evolve into the QAH effect. In 3D, magnetic order opens up a gap for the topological surface states, and chiral edge state has been predicted to exist on the magnetic domain walls. We present the phase diagram in thin films of a magnetic topological insulator and review the basic mechanism of ferromagnetic order in magnetically doped topological insulators. We also review the recent experimental observation of the QAH effect. We discuss more recent theoretical work on the coexistence of the helical and chiral edge states, multi-channel chiral edge states, the theory of the plateau transition, and the thickness dependence in the QAH effect.