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 (TI) 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 the 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 (TRS) 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 TIs 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.

中文翻译：

---

量子反常霍尔效应及相关拓扑电子态

经过长期探索，成功观察到磁掺杂拓扑绝缘体（TI）薄膜中异常霍尔效应（AHE）的量子化版本，完成了量子霍尔三重奏——量子霍尔效应（QHE）、量子自旋霍尔效应（QSHE） ) 和量子反常霍尔效应 (QAHE)。在理论方面，据了解，内在 AHE 与动量空间中的 Berry 曲率和 U(1) 规范场有关。这种理解在 QAHE 与以陈数为特征的电子结构的拓扑性质之间建立了联系。由于时间反转对称性 (TRS) 被磁化破坏，QAHE 系统在边缘承载无耗散的充电电流，类似于需要外部磁场的 QHE。QAHE 和相应的陈绝缘体也与其他拓扑电子态密切相关，例如 TI 和拓扑半金属，它们最近已被广泛研究并且已知存在于各种化合物中。第一性原理电子结构计算不仅对于理解该领域的基础物理学，而且对于现实化合物的预测和实现都具有重要作用。在本文中，将重点介绍 QAHE 和陈绝缘体，从各种拓扑不变量的角度对 Berry 相机制和相关的拓扑电子态进行理论回顾。我们将介绍用于选择和设计拓扑材料的威尔逊环法和能带反转机制，并讨论第一性原理计算的预测能力。最后，将解决该领域未来调查的剩余问题、挑战和可能的应用。