Abstract
For the type of lattices considered in this lecture, obtaining a non-zero Chern index requires breaking time-reversal invariance. For a spin-free problem, this requires complex tunnel coefficients. There are two ways of doing this. One is to transpose the geometry of the Hall effect to the lattice, i.e. to apply a uniform magnetic field to the lattice. This field breaks the translational invariance and can give rise to a very rich energy spectrum, with a large number of energy sub-bands. The other approach, proposed by Haldane in 1988, consists in keeping the same unit cell as the initial lattice, i.e. a two-band model, while (slightly) increasing the number of tunnel couplings. This is the approach we have explored in this fifth lecture. After laying down the principles of the Haldane model, we studied its implementation with cold atoms, carried out in various laboratories between 2012 and 2017. We then used this model to explain core-edge correspondence, i.e. the inevitable appearance of edge channels at the interface between two zones of different topology. We ended this lecture by describing the experimental demonstration of these edge channels on photonic systems, published in early 2018.
