Berry-Chern monopoles and spectral flows

Abstract

This lecture note adresses the correspondence between spectral flows, often associated to unidirectional modes, and Chern numbers associated to degeneracy points. The notions of topological indices (Chern numbers, analytical indices) are introduced for non specialists with a wave physics or condensed matter background. The correspondence is detailed with several examples, including the Dirac equations in two dimensions, Weyl fermions in three dimensions, the shallow water model and other generalizations.

About 50 pages. This manuscript was partially written from my Habilitation thesis (2019) and from lectures I gave at the Ecole Normale Superieure de Lyon (since 2018)