  Over a long period of exploration, the successful observation of quantized
version of anomalous Hall effect (AHE) in thin film of magnetically-doped
topological insulator completed a quantum Hall trio---quantum Hall effect
(QHE), quantum spin Hall effect (QSHE), and quantum anomalous Hall effect
(QAHE). On the theoretical front, it was understood that intrinsic AHE is
related to Berry curvature and U(1) gauge field in momentum space. This
understanding established connection between the QAHE and the topological
properties of electronic structures characterized by the Chern number. With the
time reversal symmetry broken by magnetization, a QAHE system carries
dissipationless charge current at edges, similar to the QHE where an external
magnetic field is necessary. The QAHE and corresponding Chern insulators are
also closely related to other topological electronic states, such as
topological insulators and topological semimetals, which have been extensively
studied recently and have been known to exist in various compounds.
First-principles electronic structure calculations play important roles not
only for the understanding of fundamental physics in this field, but also
towards the prediction and realization of realistic compounds. In this article,
a theoretical review on the Berry phase mechanism and related topological
electronic states in terms of various topological invariants will be given with
focus on the QAHE and Chern insulators. We will introduce the Wilson loop
method and the band inversion mechanism for the selection and design of
topological materials, and discuss the predictive power of first-principles
calculations. Finally, remaining issues, challenges and possible applications
for future investigations in the field will be addressed.
