Symmetry classification of continuous phase transitions in two dimensions

Extending earlier work by Domany et al., this paper provides a group-theoretical classification of continuous phase transitions of systems of arbitrary $d=2$ space-group symmetry with a scalar ordering density. Based on the Lifshitz condition, the classification finds the symmetries for which continuous phase transitions are possible and identifies the corresponding universality classes according to the associated Landau-Ginzburg-Wilson Hamiltonians. Tentatively unidentified universality classes appear, associated with the space groups p4gm, $p3$, $p31m$, and $p6$. Applications to atomic and molecular adsorption and to surface reconstruction are given.