Abstract
This first lecture has been devoted to the introduction of notions which play a central role in the whole of this lecture. The first concerns the geometric phase, also known as the Berry phase , which tells us how a physical system evolves when a control parameter is slowly varied; this notion is particularly important when the parameter returns to its initial value at the end of the evolution. We use this notion here in the context of quantum physics, but it is also relevant to classical physics, the Foucault pendulum being a fine illustration. The second important notion concerns the formalism describing "two-level" quantum systems, i.e. systems whose state space is of dimension 2. We also recalled some important elements of spin 1/2 physics, in particular the representation of a spin state using the Bloch sphere. Finally, we summarized the specific properties of periodic space systems, in particular Bloch's theorem and the notion of energy band. We illustrated these properties on a first example of a topological system, the model proposed by Su, Schrieffer and Heeger (SSH).
