This paper provides a pedagogical introduction to recent developments in geometrical and topological band theory following the discovery of graphene and topological insulators. Amusingly, many of these developments have a connection to contributions in high-energy physics by Dirac. The review starts by a presentation of the Dirac magnetic monopole, goes on with the Berry phase in a two-level system and the geometrical/topological band theory for Bloch electrons in crystals. Next, specific examples of tight-binding models giving rise to lattice versions of the Dirac equation in various space dimension are presented: in 1D (Su–Schrieffer–Heeger (SSH) and Rice–Mele models), 2D (graphene, boron nitride, Haldane model) and 3D (Weyl semi-metals). The focus is on topological insulators and topological semi-metals. The latter have a Fermi surface that is characterized as a topological defect. For topological insulators, the two alternative view points of twisted fiber bundles and of topological textures are developed. The minimal mathematical background in topology (essentially on homotopy groups and fiber bundles) is provided when needed. Topics rarely reviewed include: periodic versus canonical Bloch Hamiltonian (basis I/II issue), Zak versus Berry phase, the vanishing electric polarization of the SSH model and Dirac insulators.

本文对石墨烯和拓扑绝缘体的发现后几何和拓扑能带理论的最新发展进行了教学性介绍。有趣的是，这些发展中有许多都与狄拉克在高能物理学中的贡献有关。该评论首先介绍狄拉克磁单极子，接着介绍两能级系统中的 Berry 相和晶体中布洛赫电子的几何/拓扑能带理论。接下来，给出了在各种空间维度上产生狄拉克方程晶格版本的紧束缚模型的具体例子：一维（Su-Schrieffer-Heeger (SSH) 和 Rice-Mele 模型）、二维（石墨烯、氮化硼、 Haldane 模型）和 3D（外尔半金属）。重点是拓扑绝缘体和拓扑半金属。后者具有费米面，其特征是拓扑缺陷。对于拓扑绝缘体，开发了扭曲纤维束和拓扑纹理的两个替代观点。在需要时提供拓扑学中的最小数学背景（主要是同伦群和纤维丛）。很少回顾的主题包括：周期性与经典 Bloch Hamiltonian（基础 I/II 问题）、Zak 与 Berry 相、SSH 模型的消失电极化和 Dirac 绝缘体。