Abstract
In this last lecture, we tackled the problem that actually gave rise to the notion of topological bands: the quantum Hall effect. It was in fact the analysis of the two-dimensional quantum motion of a set of charges placed in a magnetic field that showed the quantization of transport-related quantities, such as the Hall conductivity. This analysis also highlighted the importance of edge states, and led to the notion of topological robustness in Quantum Condensed Matter Physics. A priori, the study of the quantum Hall effect does not require an underlying lattice. However, to make the connection with the preceding lectures, we have focused on the case where a periodic potential is also present. Treating this potential in the limit of strong bonds simplifies the analysis considerably: this is the Harper-Hofstadter model. Having established this model, we studied its recent implementations (2015-2018) in atomic physics and photonics. We have shown how they have enabled explicit measurement of the Chern number of energy bands. We have also described how the edge states associated with this non-trivial topology enable the realization of new devices, topological lasers, which exploit the robustness of the bands thus formed.
