There is a useful page on the BKT (or KT) transition here which describes the role of vortices in 2d equilibrium systems.
I am interested in what happens in 2d non-equilibrium driven-dissipative systems like microcavity polaritons. This is an active area of research, but I find the papers difficult to understand, e.g. PRX 5, 041028, PRX 7, 041006. It would be great to have a concise, straightforward explanation of the mechanisms. It doesn't have to be mathematically vigorous.