Berezinskii-Kosterlitz-Thouless phase transitions in two-dimensional systems with internal symmetries

The Berezinskii-Kosterlitz-Thouless (BKT) phase transitions in two-dimensional systems with internal continuous Abelian symmetries are investigated. In order for phase transitions to occur, the kinetic part of the action of the system must have conformal invariance, and the vacuum manifold must be degenerate and have a discrete Abelian homotopy group π1. In this case topological excitations have a logarithmically divergent energy and can be described by effective theories that generalize the two-dimensional Euclidean sine-Gordon theory, which is an effective theory of the original XY model. In particular, the effective actions are found for chiral models on the maximal Abelian tori TG of the simple compact Lie groups G. The critical properties of the possible effective theories are found, and it is shown that they are characterized by the Coxeter numbers hG of lattices of the A, D, E, and Z series and can be interpreted as properties of conformal theories with an integer central charge C=n, where n is the rank of the groups π1 and G. The possibility of reconstructing the complete symmetry of G in the massive phase is also discussed