Abstract
This lecture was devoted to the search for concrete manifestations of the topology of an energy band in a one-dimensional problem. The focus has been on the existence of robust edge states, appearing at the interface between two phases of different topologies. Initially, we focused on the model of an SSH chain, whose characterization in terms of wrapping around the Bloch sphere was further developed. We showed the possible existence of robust edge states, for a finite segment of the chain or in a semi-infinite geometry. We then turned to the description of an experiment conducted with light in 2017 in France, which took advantage of edge states to realize a topological laser. In the final part of the lecture, we compared this SSH chain with another remarkable model system, introduced by Alexei Kitaev to describe a topological one-dimensional superconductor. We have shown that the edge states have very special properties, described by quasiparticle modes for which creation and destruction operators coincide. These modes are called Majorana modes, by analogy with the formalism introduced by Ettore Majorana to analyze the Dirac equation. The study of these two models has given us the opportunity to discuss the importance of symmetries in the stability of topological phases. These are sublattice symmetry for the SSH model, and particle-hole symmetry for the Kitaev model. As long as the system possesses this symmetry, the topological phases found are robust: they can only be transformed into a normal phase by closing a gap between two energy bands, which requires a radical modification of the Hamiltonian parameters.
