Topological and geometrical aspects of band theory

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