Anyon condensation and its applications

Bose condensation is central to our understanding of quantum phases of matter. Here we review Bose condensation in topologically ordered phases (also called topological symmetry breaking), where the condensing bosons have non-trivial mutual statistics with other quasiparticles in the system. We give a non-technical overview of the relationship between the phases before and after condensation, drawing parallels with more familiar symmetry-breaking transitions. We then review two important applications of this phenomenon. First, we describe the equivalence between such condensation transitions and pairs of phases with gappable boundaries, as well as examples where multiple types of gapped boundary between the same two phases exist. Second, we discuss how such transitions can lead to global symmetries which exchange or permute anyon types. Finally we discuss the nature of the critical point, which can be mapped to a conventional phase transition in some -- but not all -- cases.