能带理论的拓扑和几何方面
Topological and geometrical aspects of band theory
摘要：
本文介绍了石墨烯和拓扑绝缘体发现后几何和拓扑能带理论的最新发展。有趣的是，这些发展中的许多都与狄拉克在高能物理学中的贡献有关。本文首先介绍了狄拉克磁共振，接着介绍了二能级系统中的Berry相和晶体中Bloch电子的几何/拓扑能带理论。接下来，紧束缚模型的具体例子在不同的空间维度上产生了狄拉克方程的晶格版本：在1D（Su-Schrieffer-Heeger和Rice-Mele模型），2D（石墨烯，氮化硼，Halflion模型）和3D（Weyl半金属）。重点是拓扑绝缘体和拓扑半金属。后者有一个费米表面，其特征在于作为一个拓扑缺陷。对于拓扑绝缘体，提出了扭曲纤维束和拓扑织构两种观点。最低限度的数学背景拓扑（基本上对同伦群和纤维束）提供时，需要的。很少审查的主题包括：周期与正则布洛赫哈密顿量（基础I/II问题），Zak与Berry相位，Su-Schrieffer-Heeger模型和Dirac绝缘体的消失电极化的强度。
英文摘要：
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 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 Su-Schrieffer-Heeger model and Dirac insulators.
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