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.
Highlights
We shall refer to that concept as the monopole - spectral flow correspondence. This correspondence relates the topological properties of point defects in three dimensional (3D) space in which eigenstates of the system are parametrized, to the existence of peculiar modes that transit from an energy branch to another one; a phenomenon which is referred to as a spectral flow. Those point defects correspond to degeneracy points, that act as the source of a Berry curvature, and whose flux through a close surface surrounding them is quantized and expressed by a topological invariant, the first Chern number
The energy spectrum of (16) is already shown in figure 1, and displays the same spectral flow as the 2D massive Dirac fermions. Such a spectral flow of Landau levels is currently associated with the condensed matter signature of the chiral anomaly in 3D Weyl semimetals: Assuming the Fermi energy lies around E = 0, applying an electric field along the z direction generates a bulk current of electrons
Chern numbers C±K and C±K of the Berry-Chern monopoles can be computed for each valley, like previously, and are opposite to each other. It follows that the interface problem, that is when ∆ is varied in space and changes sign, manifests a double spectral flow which is opposite in each valley, as illustrated by the numerical spectrum of the Haldane model for a sharp interface in figure 4
Summary
In his successful popular book La valeur de la science [1], the mathematician physicist Henri Poincaré pointed out in 1905 that : “we can see mathematical analogies between phenomena which have no physical relation neither apparent nor real, so that the laws of one of these phenomena help us to guess those of the other. [...] The goal of mathematical physics is to facilitate the physicist in numerical computation [...]. This correspondence relates the topological properties of point defects in three dimensional (3D) space in which eigenstates of the system are parametrized, to the existence of peculiar modes that transit from an energy (or frequency) branch to another one; a phenomenon which is referred to as a spectral flow Those point defects correspond to degeneracy points (or band crossing points), that act as the source of a Berry curvature, and whose flux through a close surface surrounding them is quantized and expressed by a topological invariant, the first Chern number. We use a minimal twoband model to illustrate this ubiquitous correspondence and describe simultaneously interface states in 2D systems such as Chern insulators, quantum valley Hall insulators and other classical waves systems, together with the dispersive Landau levels that trigger the chiral anomaly in 3D Weyl semimetals [17, 18] This first part is mainly dedicated to a broad community of quantum and classical physicists that are not familiar with the topological concepts. For let us introduce the second example where the same spectral flow appears
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