Abstract
In this lecture, we have gone beyond the SSH model to examine what happens when subnet symmetry is lost. There are, of course, multiple ways to make this generalization; initially, we added a minimum number of ingredients, staying in the strong-binding regime and relaxing only the equivalence constraint between sites, which led us to the Rice-Mele model. We then described how to implement this general class of bipartite 1D networks with cold atoms trapped in standing light waves. In particular, we presented the principle of the Zak phase measurement, made in 2013 by Atala et al. We then addressed the concept of the adiabatic pump, which emerges when we allow ourselves to slowly vary in time the lattice parameters, i.e. the couplings between sites or the on-site energies. We have seen that, for a periodic variation of these parameters and for an adequate preparation of the particles, the displacement of the center of mass of these particles is quantified. We have described an experiment carried out in Japan in 2016 that highlighted this quantization in a cold atom experiment. The adiabatic pump is in fact a two-dimensional problem, with one dimension of space and one dimension of time, and we have shown that it is possible to express the quantization found in terms of the integral of a Berry curvature. We have thus taken a first step into topological problems of dimension greater than 1: the results obtained in this lecture will resurface in a slightly modified form for the truly 2D problems tackled in the following.
