论文标题
乐队理论的拓扑和几何方面
Topological and geometrical aspects of band theory
论文作者
论文摘要
本文在发现石墨烯和拓扑绝缘子后，对几何和拓扑结构理论的最新发展提供了教学介绍。有趣的是，这些发展中的许多发展与Dirac高能物理学的贡献有联系。综述首先是对狄拉克磁性单极的呈现，在两级系统中与浆果相位以及晶体中Bloch电子的几何/拓扑结构理论。接下来，介绍了在各种空间维度中产生dirac方程的晶格版本的特定示例：以1D（Su-Schrieffer-Heeger和Rice-Mele型号），2D（石墨烯，硝酸盐，Haldane模型）和3D（Weyl Semi-Metals）。重点是拓扑绝缘子和拓扑半金属。后者的Fermi表面被描述为拓扑缺陷。对于拓扑绝缘子，开发了扭曲的纤维束和拓扑纹理的两个替代观点。在需要时提供拓扑的最小数学背景（基本上是在同型组和纤维束上）。主题很少综述包括：定期与规范的Bloch Hamiltonian（基础I/II问题），Zak与Berry阶段，Su-Schrieffer-Heeger模型的消失电化和Dirac绝缘子的消失。
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.
