Phase transitions in two-dimensional disordered systems

The low-energy limits of models with disorder are frequently described by sigma models. In two dimensions, most sigma models admit either a Wess-Zumino-Witten term or a theta term. When such a term is present the model can have a stable critical point with gapless excitations. We describe how such a critical point appears, in particular in two-dimensional superconductors with disorder. The presence of such terms is required by the underlying (anomalous) symmetries of the original electron model. This indicates that the usual symmetry classes of disordered systems in two dimensions can be further refined. Conversely, our results also indicate that models previously thought to be in different universality classes are in fact the same once the appropriate extra terms are included.