Abstract
An important class of problems concerns orbital magnetism in the presence of a spatially periodic potential. This type of question arises, for example, when we study the effect of a strong magnetic field on the electron fluid in a crystal. The richness of this problem is linked to the existence of two length scales that can be of the same order: the first is the mesh of the periodic lattice, the second is the size of an elementary cyclotron orbit. The "competition" between these two length scales induces radically new physics, with an energy spectrum of fractal structure (Hofstadter butterfly). In this lecture, we show how this regime can be explored with synthetic materials and/or artificial magnetic fields, in particular cold atoms confined in an optical lattice.
