Abstract
Starting with this lecture, we turned to the study of two-dimensional topological systems. This two-dimensional geometry played a major role in the emergence of topological concepts in physics, with the discovery of the quantum Hall effect for electron gases confined in quantum wells. It allows topological phases to be characterized by their transport properties, which was not possible in one dimension; moreover, it gives rise to topological numbers (Chern indices) that are more "robust" than the Zak phase, which remained dependent on the parameterization chosen for a given physical problem. We considered an isolated energy band and sought to characterize its topology. We came up with two types of answer. The first is mathematical, and is directly inspired by our understanding of geometric pumps, concerning the coverage of the Bloch sphere. The second is physical, and concerns the transport properties that can be expected for these systems. Of course, both types of response ultimately lead to the same characterization. To simplify our analysis, we have mainly studied discrete systems with tunnel couplings allowing jumps only between neighboring sites. This enabled us to use the strong-bond approximation and carry out almost all calculations analytically. We also looked for the simplest geometries allowing non-trivial topology to appear. As in 1D, this led us to consider a network with two possible sites per unit cell. This type of network naturally leads to the emergence of Dirac points in a two-dimensional geometry, and so it was with this notion that we began our study, before moving on to characterizing the topology.
