Berry-Chern monopoles and spectral flows

TL;DR

This paper explores the relationship between spectral flows and Chern numbers in topological physics, providing examples across wave physics and condensed matter to clarify their correspondence and topological significance.

Contribution

It introduces topological indices like Chern numbers and analytical indices for non-specialists and details their correspondence through various physical examples.

Findings

Spectral flows relate to Chern numbers at degeneracy points.

Examples include Dirac equations, Weyl fermions, and shallow water models.

The paper clarifies topological indices for wave physics and condensed matter.

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.