The discovery of topological effects in quantum solid states has revolutionized condensed matter studies. From the fractional Quantum Hall effect to topological superconductors and insulators, whether in one, two or three dimensions, they offer a wide variety of models to study. One of the main characteristics of such systems is the presence of non-trivial edge states, with fractional spin and charge. In this talk, I will first present a general introduction to these exotic materials, with a slight focus on the works that were awarded this year’s Nobel prize.
In a second time, I will specialize on Z2 superconductors that presents Majorana modes, either at the edges or in vortices. These zero-modes are of particular interest in conjunction with the rise of quantum information. The topological nature of these modes, preventing any effect from small local perturbations, protects them from decoherence and make them perfect candidates for the realization of quantum bits. Several proposals have indeed been made to realize mesoscopic heterostructures for one-dimensional topological wires, relying on controlled interactions between several of such wires to implement the different required logical gates. I will present some results on the interplay between topological phases and interactions in a specific model, leading to the opening of new complex phases.