Journal
JOURNAL OF PHYSICS-MATERIALS
Volume 4, Issue 3, Pages -Publisher
IOP Publishing Ltd
DOI: 10.1088/2515-7639/abf0b5
Keywords
topological band theory; graphene; topological insulators; Weyl semimetals; Dirac monopole; Berry phase
Categories
Funding
- Quantum Matter Network at Bordeaux University under project TaQuaMaUC
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This paper provides a pedagogical introduction to recent developments in geometrical and topological band theory following the discovery of graphene and topological insulators. It explores the connection of these developments to contributions in high-energy physics by Dirac. The focus is on topological insulators and topological semi-metals, with specific examples of lattice versions of the Dirac equation in different space dimensions.
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.
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